{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_63372/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_63372","id":"gutenberg_63372","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/63372","summary":null,"tags":["Category: Philosophy & Ethics","Category: Science - Physics","Relativity (Physics)"],"text":"TRANSCRIBERS NOTE:\n\n    The plain text version of this ebook includes mathematical\n    formulas in TeX notation. These formulas are enclosed\n    between dollar signs. The HTML version includes the same\n    formulas rendered in their original form.\n\n\n\n\n\n           EINSTEIN'S THEORIES OF RELATIVITY AND GRAVITATION\n\n                A SELECTION OF MATERIAL FROM THE ESSAYS\n                      SUBMITTED IN THE COMPETITION\n                 FOR THE EUGENE HIGGINS PRIZE OF $5,000\n\n\n                          COMPILED AND EDITED,\n                    AND INTRODUCTORY MATTER SUPPLIED\n                                   BY\n                            J. MALCOLM BIRD,\n                 Associate Editor, Scientific American\n\n\n                                NEW YORK\n                  SCIENTIFIC AMERICAN PUBLISHING CO.,\n                               MUNN & CO.\n                                  1921\n\n\n\n\n\n\n\n\n\nPREFACE\n\n\nThe obstacles which the layman finds to understanding Einstein's\nrelativity theories lie not so much in the inherent difficulty of\nthese theories themselves as in the difficulty of preparing the\nmind for their reception. The theory is no more difficult than any\nscientific development of comparable depth; it is not so difficult as\nsome of these. But it is a fact that for a decent understanding of it,\na large background of scientific knowledge and scientific habit of\nthought is essential. The bulk of the writers who have attempted to\nexplain Einstein to the general reader have not realized the great gulf\nwhich lies between the mental processes of the trained mathematician\nand those of the man in the street. They have not perceived that\nthe lay reader must be personally conducted for a long distance from\nthe vestibule of the temple of science before he comes to Einstein,\nand that he cannot by any possibility make this journey unaided. The\nresult has been to pitchfork the reader into the intricacies of the\nsubject without adequate preparation.\n\nThe present volume avoids this mistake with the utmost care. It avoids\nit, in fact, with such deliberation as to make it in order to say a\nword in explanation of what will at first glance seem an extraordinary\narrangement of material. It was to be expected, doubtless, that this\nbook would open with a brief statement of the genesis and the outcome\nof the Einstein Prize Essay Contest for the $5,000 prize offered by\nMr. Eugene Higgins. It was doubtless to be expected that, after this\nhad been dismissed, the winning essay would be given the post of honor\nin advance of all other material bearing actually on the Einstein\ntheories. When the reader observes that this has not been done, he\nwill by all means expect a word of explanation; and it is mainly for\nthe purpose of giving this that we make these introductory remarks.\n\nThe essays submitted in the contest, and in particular the comments of\na few disappointed readers upon Mr. Bolton's prize essay, make quite\nplain what might have been anticipated--that in the small compass\nof 3,000 words it is not possible both to prepare the reader's mind\nfor a discussion of Relativity and to give a discussion that shall be\nadequate. Mr. Bolton himself, in replying to a protest that he had not\ndone all this, has used the word \"miracle\"--we think it a well-advised\none. No miracle was expected as a result of the contest, and none has\nbeen achieved. But in awarding the prize, the Judges had to decide\nwhether it was the best preliminary exposition or the best discussion\nthat was wanted. They decided, and rightly we believe, that the award\nshould go to an actual statement of what the Einstein theories are\nand what they do, rather than to a mere introduction, however well\nconceived and well executed the latter might be. Nevertheless,\nwe should be closing our eyes to a very obvious fact if we did\nnot recognize that, without something in the way of preparation,\nthe general reader is not going to pursue Mr. Bolton's essay, or\nany other essay on this subject, with profit. It is in order the\nmore forcefully to hold out inducements to him to subject himself to\nthis preparation that we place at the head of the book the chapters\ndesigned to give it to him.\n\nChapter II. is intended so to bring the mind of the reader into contact\nwith certain philosophical problems presented to us by our experiences\nwith the external world and our efforts to learn the facts about it,\nthat he may approach the subject of relativity with an appreciation\nof the place it occupies as a phase of human thought and a pillar of\nthe scientific structure. Until the reader is aware of the existence\nof these problems and the directions taken by the efforts, successful\nand unsuccessful, to unravel them, he is not equipped to comprehend\nthe doctrine of relativity at all; he is in much the same case as a\nchild whose education had reached only the primer stage, if asked to\nread the masterpieces of literature. He lacks not alone the vocabulary,\nbut equally the mental background on which the vocabulary is based.\n\nIt will be noted that in this and the chapters immediately following\nit, the Editor has supplied material freely. The obvious interpretation\nis that satisfactory material covering the desired ground was not\nfound in any of the essays; for we are sure the scope and number\nof the credited excerpts will make it clear that all contributions\nwere adequately scrutinized in search of available passages. This\n\"inadequacy\" of the competing essays has been severely commented\nupon by several correspondents, and the inference drawn that as\na whole the offerings were not up to the mark. Such a viewpoint\nis wholly unjust to the contestants. The essays which paid serious\nattention to the business of paving the way to relativity necessarily\ndid so at the expense of completeness in the later paragraphs where\nspecific explanation of the Einstein theories was in order. Mr. Law,\nwhose essay was by all means the best of those that gave much space\nto introductory remarks, found himself left with only 600 words\nin which to tell what it was that he had been introducing. The\nmajority of the contestants appear to have faced the same question\nas to subject matter which the Judges faced, and to have reached\nthe same decision. They accordingly devoted their attention toward\nthe prize, rather than toward the production of an essay that would\nbest supplement that of the winner. It is for this very reason that,\nin these preliminary chapters, so large a proportion of the material\nhas had to be supplied by the Editor; and this very circumstance is\na tribute to the good judgment of the competitors, rather than ground\nfor criticism of their work.\n\nThe general introduction of Chapter II. out of the way, Chapters\nIII. and IV. take up the business of leading the reader into the\nactual subject of relativity. The subject is here developed in what\nmay be called the historical order--the order in which it took form in\nEinstein's own mind. Both in and outside the contest of which this book\nis the outcome, a majority of those who have written on relativity\nhave followed this order, which is indeed a very natural one and\none well calculated to give to the rather surprising assumptions of\nrelativity a reasonableness which they might well appear to the lay\nmind to lack if laid down more arbitrarily. In these two chapters\nno effort is made to carry the argument beyond the formulation of\nthe Special Principle of the relativity of uniform motion, but this\nprinciple is developed in considerably more detail than would be the\ncase if it were left entirely to the competing essayists. The reason\nfor this is again that we are dealing with a phase of the subject\nwhich is of subordinate importance so far as a complete statement of\nthe General Theory of Relativity is concerned, but which is of the\ngreatest significance in connection with the effort of the layman to\nacquire the proper preliminary orientation toward the larger subject.\n\nChapter V. goes back again to general ground. Among the ideas which\nthe competing essayists were forced to introduce into their text on a\nliberal scale is that of non-Euclidean geometry. The entire formulation\nof the General Theory of Relativity is in fact an exercise in this. The\nessayists--good, bad and indifferent alike--were quite unanimous in\ntheir decision that this was one thing which the reader would have\nto assume the responsibility of acquiring for himself. Certainly they\nwere justified in this; for the Editor has been able to explain what\nnon-Euclidean geometry is only by using up considerably more space\nthan the contestants had for an entire essay. No effort has been made\nto set forth any of the details of any of the various non-Euclidean\ngeometries; it has simply been the aim to draw the dividing line\nbetween Euclidean and non-Euclidean, and to make the existence of\nthe latter appear reasonable, so that when the essayists come to talk\nabout it the reader will not feel hopelessly at sea. In other words,\nthis is another case of providing the mental background, but on such\na scale that it has seemed necessary to give a separate chapter to it.\n\nChapter VI. completes the preliminary course in the fundamentals of\nrelativity by tying up together the findings of Chapter V. and those\nof Chapters III. and IV. It represents more or less of a last-minute\nchange of plan; for while it had been the Editor's intent from the\nbeginning to place the material of Chapters II.-V. in its present\nposition, his preliminary impression would have been that the work\nof the present Chapter VI. would be adequately done by the essayists\nthemselves. His reading of the essays, however, convinced him that it\nhad not so been done--that with the possible exception of Mr. Francis,\nthe essayists did not make either a serious or a successful effort to\nshow the organic connection between the Special Theory of Relativity\nand the Minkowski space-time structure, or the utter futility of\ntrying to reconcile ourselves to the results of the former without\nemploying the ideas of the latter. So Chapter VI. was supplied to make\ngood this deficiency, and to complete the mental equipment which the\nreader requires for his battle with the General Theory.\n\nIn laying down a set of general principles to govern the award of\nthe prize, one of the first things considered by the Judges was the\nrelative importance of the Special and the General Theories. It was\ntheir opinion that no essay could possibly qualify for the prize which\ndid not very distinctly give to the General Theory the center of the\nstage; and that in fact discussion of the Special Theory was pertinent\nonly so long as it contributed, in proportion to the space assigned\nit, to the attack upon the main subject. The same principle has been\nemployed in selecting essays for complete or substantially complete\nreproduction in this volume. Writers who dealt with the Special Theory\nin any other sense than as a preliminary step toward the General Theory\nhave been relegated to the introductory chapters, where such excerpts\nfrom their work have been used as were found usable. The distinction\nof publication under name and title is reserved for those who wrote\nconsistently and specifically upon the larger subject--with the one\nexception of Dr. Russell, whose exposition of the Special Theory is so\nfar the best of those submitted and at the same time so distinctive\nthat we have concluded it will appear to better advantage by itself\nthan as a part of Chapters III. and IV.\n\nFollowing after Mr. Bolton's essay we have tried to arrange the various\ncontributions, not at all in any order of merit, but in the order\nthat will make connected reading of the book most nearly possible\nand profitable. Each essay should be made easier of reading by the\nexamination of those preceding it; at the same time each, by the choice\nof ground covered and by the emphasis on points not brought out sharply\nby its predecessors, should throw new light upon these predecessors.\n\nThe reader will find that no two of the essays given thus in full\nduplicate or even come close to duplicating one another. They have\nof course been selected with this in view; each represents the best\nof several essays of substantially the same character. Not all of\nthem require comment here, but concerning some of them a word may\nwell be said.\n\nMr. Francis, we believe, has succeeded in packing more substance\ninto his 3,000 words than any other competitor. Mr. Elliot has come\ncloser than anybody else to really explaining relativity in terms\nfamiliar to everybody, without asking the reader to enlarge his\nvocabulary and with a minimum demand so far as enlarging his mental\noutlook is concerned. Were it not for certain conspicuous defects,\nhis essay would probably have taken the prize. In justice to the\nJudges, we should state that we have taken the liberty of eliminating\nMr. Elliot's concluding paragraph, which was the most objectionable\nfeature of his essay.\n\nDr. Dushman chose for his title the one which we adopted for this\nbook. It became necessary, therefore, for us to find a new title for\nhis essay; aside from this instance, the main titles appearing at the\nheads of the various complete essays are those of the authors. The\nsubtitles have in practically every instance been supplied editorially.\n\nDr. Pickering submitted two essays, one written from the viewpoint\nof the physicist, the other from that of the astronomer. To make\neach complete, he naturally found it necessary to duplicate between\nthem certain introductory and general material. We have run the two\nessays together into a single narrative, with the elimination of this\nduplicated material; aside from this blue-penciling no alteration\nhas been made in Dr. Pickering's text. This text however served as\nthe basis of blue-penciling that of several other contestants, as\nindicated in the foot notes.\n\nFor the reader who is qualified or who can qualify to understand it,\nDr. Murnaghan's essay is perhaps the most illuminating of all. Even\nthe reader who does not understand it all will realize that its author\nbrings to the subject a freshness of viewpoint and an originality of\ntreatment which are rather lacking in some of the published essays,\nand which it will readily be understood were conspicuously lacking\nin a good many of the unpublished ones. Dr. Murnaghan of all the\ncompetitors has come closest to making a contribution to science as\nwell as to the semi-popular literature of science.\n\nIn the composite chapters, the brackets followed by reference numbers\nhave been used as the most practicable means of identifying the\nvarious individual contributions. We believe that this part of the\ntext can be read without allowing the frequent occurrence of these\nsymbols to distract the eye. As to the references themselves, the\nasterisk marks the contributions of the Editor. The numbers are those\nattached to the essays in order of and at the time of their receipt;\nit has been more convenient to use these than to assign consecutive\nnumbers to the quoted essays. The several numbers identify passages\nfrom the essays of the following contestants:\n\n\n 10:   Frederick W. Shurlock, Derby, England.\n 18:   L. L. Whyte, Cambridge, England.\n 24:   Prof. Moritz Schlick, University of Rostock, Germany.\n 30:   C. E. Rose, M.E., Little Rock, Ark.\n 33:   H. Gartelmann, Bremen, Germany.\n 35:   Prof. Joseph S. Ames, Johns Hopkins University, Baltimore.\n 47:   James O. G. Gibbons, East Orange. N. J.\n 82:   Charles H. Burr, Philadelphia.\n101:   L. F. H. de Miffonis. B.A., C.E., Ottawa, Canada.\n102:   Charles A. Brunn, Kansas City.\n106:   J. Elias Fries, Fellow A.I.E.E., Birmingham, Ala.\n114:   Dean W. P. Graham, Syracuse University, Syracuse, N. Y.\n115:   Rev. George Thomas Manley, London.\n116:   Prof. J. A. Schouten, Delft, Netherlands.\n121:   Elwyn F. Burrill, Berkeley, Cal.\n125:   Dorothy Burr, Bryn Mawr, Pa.\n130:   C. W. Kanolt, Bureau of Standards, Washington.\n135:   Robert Stevenson, New York.\n139:   Leopold Schorsch, New York.\n141:   Dr. M. C. Mott-Smith, Los Angeles, Calif.\n147:   Edward A. Clarke, Columbus, O.\n149:   Edward A. Partridge, Philadelphia.\n150:   Col. John Millis, U. S. A., Chicago.\n152:   George F. Marsteller, Detroit.\n156:   D. B. Hall, Cincinnati.\n165:   Francis Farquhar, York, Pa.\n178:   Dr. George de Bothezat, Dayton, O.\n179:   Professor A. E. Caswell, University of Oregon, Eugene, Ore.\n182:   C. E. Dimick, New London, Conn.\n186:   Earl R. Evans, Washington, D. C.\n188:   Norman E. Gilbert, Dartmouth College, Hanover, N. H.\n192:   A. d'Abro. New York.\n194:   L. M. Alexander, Cincinnati.\n197:   Kenneth W. Reed, East Cleveland, O.\n198:   Prof. E. N. da C. Andrade, Ordnance College, Woolwich, England.\n216:   Professor Andrew H. Patterson, University of North Carolina,\n       Chapel Hill, N. C.\n220:   Prof. Arthur Gordon Webster, Clark College, Worcester, Mass.\n221:   Walter van B. Roberts, Princeton University, N. J.\n223:   Paul M. Batchelder, Austin, Tex.\n227:   Prof. R. W. Wood, Johns Hopkins University, Baltimore.\n229:   E. P. Fairbairn, M.C., B.Sc., Glasgow.\n231:   R. F. Deimel, Hoboken, N. J.\n232:   Lieut. W. Mark Angus, U. S. N., Philadelphia.\n235:   Edward Adams Richardson, Kansas City.\n263:   Prof. William Benjamin Smith, Tulane University, New Orleans.\n264:   James Rice, University of London, London.\n267:   William Hemmenway Pratt, Lynn, Mass.\n272:   R. Bruce Lindsay, New Bedford, Mass.\n283:   Frank E. Law, Montclair, N. J.\n\n\nIn addition to the specific credit given by these references for\nspecifically quoted passages, the Editor feels that he ought to\nacknowledge his general indebtedness to the competing essayists,\ncollectively, for the many ideas which he has taken away from their\ntext to clothe in his own words. This does not mean that the Editor has\nundertaken generally to improve upon the language of the competitors,\nbut merely that the reading of all their essays has given him many\nideas of such complex origin that he could not assign credit if\nhe would.\n\n\n\n\n\n\n\n\n\nTABLE OF CONTENTS\n\n\nI.--The Einstein $5,000 Prize: How the Contest Came to be Held,\nand Some of the Details of Its Conduct. By the Editor      1\n\nII.--The World--And Us: An Introductory Discussion of the Philosophy\nof Relativity, and of the Mechanism of our Contact with Time and\nSpace. By various contributors and the Editor      19\n\nIII.--The Relativity of Uniform Motion: Classical Ideas on the\nSubject; the Ether and the Apparent Possibility of Absolute Motion;\nthe Michelson-Morley Experiment and the Final Negation of this\npossibility. By various contributors and the Editor      47\n\nIV.--The Special Theory of Relativity: What Einstein's Study of\nUniform Motion Tells Us About Time and Space and the Nature of the\nExternal Reality. By various contributors and the Editor      76\n\nV.--That Parallel Postulate: Modern Geometric Methods; the Dividing\nLine Between Euclidean and Non-Euclidean; and the Significance of\nthe Latter. By the Editor      111\n\nVI.--The Space-Time Continuum: Minkowski's World of Events, and the\nWay in Which It Fits Into Einstein's Structure. By the Editor and a\nfew contributors      141\n\nVII.--Relativity: The Winning Essay in the Contest for the Eugene\nHiggins $5,000 Prize. By Lyndon Bolton, British Patent Office,\nLondon      169\n\nVIII.--The New Concepts of Time and Space: The Essay in Behalf of\nWhich the Greatest Number of Dissenting Opinions Have Been Recorded. By\nMontgomery Francis, New York      181\n\nIX.--The Principle of Relativity: A Statement of What it is All\nAbout, in Ideas of One Syllable. By Hugh Elliot, Chislehurst, Kent,\nEngland      195\n\nX.--Space, Time and Gravitation: An Outline of Einstein's Theory of\nGeneral Relativity. By W. de Sitter, University of Leyden      206\n\nXI.--The Principle of General Relativity: How Einstein, to a Degree\nNever Before Equalled, Isolates the External Reality from the\nObserver's Contribution. By E. T. Bell, University of Seattle      218\n\nXII.--Force Vs. Geometry: How Einstein Has Substituted the Second\nfor the First in Connection with the Cause of Gravitation. By Saul\nDushman, Schenectady      230\n\nXIII.--An Introduction to Relativity: A Treatment in which the\nMathematical Connections of Einstein's Work are Brought Out More\nStrongly and More Successfully than Usual in a Popular Explanation. By\nHarold T. Davis, University of Wisconsin      240\n\nXIV.--New Concepts for Old: What the World Looks Like After Einstein\nHas Had His Way with It. By John G. McHardy, Commander R. N.,\nLondon      251\n\nXV.--The New World: A Universe in Which Geometry Takes the Place of\nPhysics, and Curvature that of Force. By George Frederick Hemens,\nM.C., B.Sc., London      265\n\nXVI.--The Quest of the Absolute: Modern Developments in Theoretical\nPhysics, and the Climax Supplied by Einstein. By Dr. Francis\nD. Murnaghan, Johns Hopkins University, Baltimore      276\n\nXVII.--The Physical Side of Relativity: The Immediate Contacts between\nEinstein's Theories and Current Physics and Astronomy. By Professor\nWilliam H. Pickering, Harvard College Observatory, Mandeville,\nJamaica      287\n\nXVIII.--The Practical Significance of Relativity: The Best Discussion\nof the Special Theory Among All the Competing Essays. By Prof. Henry\nNorris Russell, Princeton University      306\n\nXIX.--Einstein's Theory of Relativity: A Simple Explanation of His\nPostulates and Their Consequences. By T. Royds, Kodaikanal Observatory,\nIndia      318\n\nXX.--Einstein's Theory of Gravitation: The Discussion of the\nGeneral Theory and Its Most Important Application, from the Essay by\nProf. W. F. G. Swann, University of Minnesota, Minneapolis      327\n\nXXI.--The Equivalence Hypothesis: The Discussion of This, With Its\nDifficulties and the Manner in Which Einstein Has Resolved Them, from\nthe Essay by Prof. E. N. da C. Andrade, Ordnance College, Woolwich,\nEngland      334\n\nXXII.--The General Theory: Fragments of Particular Merit on This\nPhase of the Subject. By Various Contributors      338\n\n\n\n\n\n\n\n\n\nI.\n\nTHE EINSTEIN $5,000 PRIZE\n\nHOW THE CONTEST CAME TO BE HELD, AND SOME OF THE DETAILS OF ITS CONDUCT\n\nBY THE EDITOR\n\n\nIn January, 1909, an anonymous donor who was interested in the spread\nof correct scientific ideas offered through the Scientific American a\nprize of $500 for the best essay explaining, in simple non-technical\nlanguage, that paradise of mathematicians and bugaboo of plain\nordinary folk--the fourth dimension. Many essays were submitted in\nthis competition, and in addition to that of the winner some twenty\nwere adjudged worthy of ultimate publication. It was felt that the\ncompetition had added distinctly to the popular understanding of\nthis significant subject; that it had done much to clear up popular\nmisconception of just what the mathematician means when he talks\nof four or even more dimensions; and that it had therefore been as\nsuccessful as it was unusual in character.\n\nIn November, 1919, the world was startled by the announcement from\nLondon that examination of the photographs taken during the total\nsolar eclipse of May 29th had been concluded, and that predictions\nbased upon the Einstein theories of relativity had been verified. In\nthe reaction from the long surfeit of war news an item of this sort\nwas a thoroughly journalistic one. Long cable dispatches were carried\nin the news columns all over the world; Einstein and his theories\nwere given a prominent place on the front pages day after day; leading\nscientists in great number were called upon to tell the public through\nthe reportorial medium just what the excitement was all about, just\nin what way the classical scientific structure had been overthrown.\n\nInstead of being a mere nine days' wonder, the Einstein theories held\ntheir place in the public mind. The more serious periodicals devoted\nspace to them. First and last, a very notable group of scientific men\nattempted to explain to the general reader the scope and content of\nEinstein's system. These efforts, well considered as they were, could\nbe no more than partially successful on account of the very radical\ncharacter of the revisions which the relativity doctrine demands in\nour fundamental concepts. Such revisions cannot be made in a day;\nthe average person has not the viewpoint of the mathematician which\npermits a sudden and complete exchange of one set of fundamentals\nfor another. But the whole subject had caught the popular attention\nso strongly, that even complete initial failure to discover what it\nwas all about did not discourage the general reader from pursuing\nthe matter with determination to come to some understanding of what\nhad happened to Newton and Newtonian mechanics.\n\n\n\n\nTHE DONOR AND THE PRIZE\n\nIn May, 1920, Mr. Eugene Higgins, an American citizen long resident\nin Paris, a liberal patron of the arts and sciences, and a lifelong\nfriend of the Scientific American and its proprietors, suggested that\nthe success of the Fourth Dimension Prize Contest of 1910 had been\nso great that it might be desirable to offer another prize in similar\nfashion for the best popular essay on the Einstein theories. He stated\nthat if in the opinion of the Scientific American these theories\nwere of sufficient importance, and the probability of getting a\ngood number of meritorious essays were sufficiently great, and the\npublic need and desire for enlightenment were sufficiently present,\nhe would feel inclined to offer such a prize, leaving the conduct\nof the contest to the Scientific American as in the former event. It\nwas the judgment of the editors of the Scientific American that all\nthese provisos should be met with an affirmative, and that Mr. Higgins\naccordingly could with propriety be encouraged to offer the prize.\n\nIn his preliminary letter Mr. Higgins had suggested that in view\nof the apparent greater importance of the subject to be proposed\nfor discussion by the contestants of 1920, the prize offered should\nprobably be more liberal than in the former instance. This view met\nwith the approval of the editors as well; but they were totally\nunprepared for the receipt, late in June, of a cablegram from\nMr. Higgins stating that he had decided to go ahead with the matter,\nand that he was forwarding a draft for $5,000 to represent the amount\nof the prize. Such a sum, exceeding any award open to a professional\nman with the single exception of the Nobel Prize, for which he cannot\nspecifically compete, fairly took the breath of the Editors, and\nmade it immediately clear that the contest would attract the widest\nattention, and that it should score the most conspicuous success. It\nalso made it clear that the handling of the contest would be a more\nserious matter than had been anticipated.\n\nIn spite of the fact that it would not for some time be possible to\nannounce the identity of the Judges, it was felt that the prospective\ncontestants should have every opportunity for extensive preparation;\nso the contest was announced, and the rules governing it printed\nas far as they could be determined on such short shrift, in the\nScientific American for July 10, 1920. Several points of ambiguity\nhad to be cleared up after this initial publication. In particular,\nit had been Mr. Higgins' suggestion that in the very probable event\nof the Judges' inability to agree upon the winning essay, the prize\nmight, at their discretion, be divided between the contributors\nof the best two essays. This condition was actually printed in the\nfirst announcement, but the Post Office Department insisted upon its\nwithdrawal, on the ground that with it in force the contestant would\nnot know whether he were competing for $5,000 or for $2,500, and that\nthis would introduce the \"element of chance\" which alone was necessary,\nunder the Federal statutes, to make the contest a lottery. So this\nprovision was replaced by one to the effect that in the event the\nJudges were not able to agree, the Einstein Editor should cast the\ndeciding vote between the essays respectively favored by them.\n\nThe announcement attracted the widest attention, and was copied in\nnewspapers and magazines all over the world. Inquiries poured in from\nall quarters, and the Einstein Editor found it almost impossible to\nkeep himself supplied with proofs of the conditions and rules to mail\nin response to these inquiries. It was immediately clear that there\nwas going to be a large number of essays submitted, and that many\ndistinguished names would be listed among the competitors.\n\n\n\n\nTHE JUDGES\n\nIn the Scientific American for September 18, announcement was carried\nin the following words:\n\n\"We are assured with complete certainty that the competition for the\nfive-thousand-dollar prize will be very keen, and that many essays\nwill be submitted which, if they bore the names of their authors, would\npass anywhere as authoritative statements. The judges will confront a\ntask of extraordinary difficulty in the effort to determine which of\nthese efforts is the best; and we believe the difficulties are such\nthat multiplication of judges would merely multiply the obstacles to\nan agreement. It is altogether likely that the initial impressions\nof two or three or five judges would incline toward two or three\nor five essays, and that any final decision would be attainable\nonly after much consultation and discussion. It seems to us that by\nmaking the committee as small as possible while still preserving the\nnecessary feature that its decision represent a consensus, we shall\nsimplify both the mental and the physical problem of coming to an\nagreement. We believe that the award should if possible represent\na unanimous decision, without any minority report, and that such a\nrequirement is far more likely to be met among two men than among\nthree or five. At the same time, the bringing together of two men and\nthe details of general administration of their work together are far\nsimpler than if there were three or five. So we have finally decided\nto have but two judges, and in this we have the endorsement of all\nthe competent opinion that we have consulted.\n\n\"The gentlemen who have consented to act as Judges are Professors\nLeigh Page and Edwin Plimpton Adams, of the departments of physics\nof Yale and Princeton Universities, respectively. Both are of the\nyounger generation of physicists that has paid special attention to\nthose phases of mathematics and physics involved in the Einstein\ntheories, and both have paid special attention to these theories\nthemselves. We are gratified to be able to put forward as Judges\ntwo men so eminently qualified to act. We feel that we may here\nappropriately quote Professor Page, who says in his acceptance:\n'As the large prize offers a great inducement, I had thought of\nentering the contest. However I realize that not many people in\nthis country have made a considerable study of Einstein's theory,\nand if all who have should enter the contest, it would be difficult\nto secure suitable Judges.' Without any desire to put the gentleman\nin the position of pleading for himself, we think this suggests very\nwell the extent to which the Scientific American, the contestants,\nand the public at large, are indebted to Professors Page and Adams\nfor their willingness to serve in the difficult capacity of Judges.\"\n\nIt might appropriately have been added to this announcement that it\nwas altogether to the credit of science and the scientific spirit\nthat the first two gentlemen approached with the invitation to act\nas Judges were willing to forego their prospects as contestants in\norder thus to contribute to the success of the contest.\n\n\n\n\nTHREE THOUSAND WORDS\n\nOf the conditions, the one which evoked most comment was that stating\nthe word limit. This limit was decided upon after the most careful\ndiscussion of the possibilities of the situation. It was not imagined\nfor a moment that any contestant would succeed in getting within\n3,000 words a complete discussion of all aspects of the Special\nand the General Theories of Relativity. It was however felt that\nfor popular reading a single essay should not be much if any longer\nthan this. Moreover, I will say quite frankly that we should never\nhave encouraged Mr. Higgins to offer such a prize if we had supposed\nthat the winning essay was the only thing of value that would come\nfrom the contest, or if we had not expected to find in many of the\nother essays material which would be altogether deserving of the\nlight. From the beginning we had in view the present volume, and the\nsevere restriction in length was deliberately imposed for the purpose\nof forcing every contestant to stick to what he considered the most\nsignificant viewpoints, and to give his best skill to displaying the\ntheories of Einstein to the utmost advantage from these viewpoints. We\nfelt that divergent viewpoints would be more advantageously treated in\nthis manner than if we gave each contestant enough space to discuss\nthe subject from all sides; and that the award of the prize to the\nessay which, among other requirements, seemed to the Judges to embody\nthe best choice of material, would greatly simplify the working of the\ncontest without effecting any injustice against those contestants who\ndisplayed with equal skill less happily chosen material. Perhaps on\nthis point I may again quote with profit the editorial page of the\nScientific American:\n\n\"An essay of three thousand words is not long enough to lose a\nreader more than once; if it does lose him it is a failure, and if\nit doesn't it is a competitor that will go into the final elimination\ntrials for the prize. If we can present, as a result of the contest,\nsix or a dozen essays of this length that will not lose the lay reader\nat all, we shall have produced something amply worth the expenditure\nof Mr. Higgins' money and our time. For such a number of essays of\nsuch character will of necessity present many different aspects of\nthe Einstein theories, and in many different ways, and in doing so\nwill contribute greatly to the popular enlightenment.\n\n\"Really the significant part of what has already appeared is\nnot the part that is intelligible, but rather the part that,\nbeing unintelligible, casts the shadow of doubt and suspicion\non the whole. The successful competitor for the prize and\nhis close contestants will have written essays that, without\nany claim to completeness, will emphasize what seems to each\nauthor the big outstanding feature; and every one of them will be\nintelligible. Together they will in all probability be reasonably\ncomplete, and will retain the individual characteristic of\nintelligibility. They will approach the various parts of the field\nfrom various directions--we could fill this page with suggestions as\nto how the one item of the four-dimensional character of Einstein's\ntime-space might be set forth for the general reader. And when a man\nmust say in three thousand words as much as he can of what eminent\nscientists have said in whole volumes--well, the result in some cases\nwill be sheer failure, and in others a product of the first water. The\nbest of the essays will shine through intelligent selection of what\nis to be said, and brilliant success in saying it. It is to get a\ngroup of essays of this character, not to get the single essay which\nwill earn the palm, that the prize is offered.\"\n\n\n\n\nTHE COMPETING ESSAYS\n\nAt all times after the first announcement the Einstein Editor had a\nheavy correspondence; but the first real evidence that the contest was\nunder way came with the arrival of the first essay, which wandered\ninto our office in the middle of September. About a week later they\nbegan to filter in at the rate of one or two per day--mostly from\nforeign contestants who were taking no chances on the mails. Heavy\nreturns did not commence until about ten days before the closing\ndate. The great avalanche, however, was reserved for the morning of\nMonday, November 1st. Here we had the benefit of three days' mail;\nthere were about 120 essays. Among those which were thrown out on\nthe ground of lateness the honors should no doubt go to the man who\nmailed his offering in The Hague on October 31st.\n\nEssays were received in greater quantity from Germany than from any\nother foreign country, doubtless because of the staggering value of\n$5,000 when converted into marks at late 1920 rates. England stood\nnext on the list; and one or more essays were received from Austria,\nCzechoslovakia, Jugoslavia, France, Switzerland, the Netherlands,\nDenmark, Italy, Chile, Cuba, Mexico, India, Jamaica, South Africa\nand the Fiji Islands. Canada, of course, contributed her fair share;\nand few of our own states were missing on the roll-call.\n\nThe general level of English composition among the essays from\nnon-English-speaking sources was about what might have been expected. A\nman may have a thorough utilitarian knowledge of a foreign tongue,\nbut when he attempts intensive literary competition with a man who was\nbrought up in that tongue he is at a disadvantage. We read French and\nGerman with ease and Spanish and Italian without too much difficulty,\nourselves; we should never undertake serious writing in any of these\nlanguages. Not many of the foreign contributions, of course, were\nas ludicrous as the one we quote to some extent in our concluding\nchapter, but most of them were distinctly below par as literary\ncompositions. Drs. De Sitter and Schlick were the notable exceptions\nto this; both showed the ability to compete on a footing of absolute\nequality with the best of the native product.\n\nWe dare say it was a foregone conclusion that many essays should have\nbeen over the limit, and that a few should have been over it to the\npoint of absurdity. The winning essay contains 2,919 words, plus or\nminus a reasonable allowance for error in counting; that it should\ncome so far from being on the ragged edge should be sufficient answer\nto those who protested against the severity of the limitation. One\ninquirer, by the way, wanted to know if 3,000 words was not a misprint\nfor 30,000. Another contestant suggested that instead of disqualifying\nany essay that was over the line, we amputate the superfluous words\nat the end. This was a plausible enough suggestion, since any essay\nable to compete after such amputation must necessarily have been one\nof extreme worth; but fortunately we did not have to decide whether we\nshould follow the scheme. Perhaps twenty of the essays submitted were\nso seriously in excess of the limit that it was not even necessary\nto count their words in detail; most of these offenders ran to 3,500\nwords or thereabouts, and one--a good one, too, from which we use\na good deal of material in this volume--actually had 4,700. On the\nother extreme were a few competitors who seemed to think that the\nshortest essay was necessarily the best, and who tried to dismiss\nthe subject with 500 or 1,000 words.\n\nBy a curious trick of chance there were submitted in competition\nfor the prize exactly 300 essays. Of course a few of these did not\nrequire serious consideration--this is inevitable in a contest of such\nmagnitude. But after excluding all the essays that were admittedly\nnot about the Einstein theories at all, and all those whose English\nwas so execrable as to make them quite out of the question, and all\nthose which took the subject so lightly as not to write reasonably\nclose to the limit of 3,000 words, and all those which were given\nover to explanation of the manner in which Einstein's theories verify\nthose of the writer, and all those in which the writer attempted\nto substitute his own cosmic scheme for Einstein's--after all this,\nthere remained some 275 essays which were serious efforts to explain\nin simple terms the nature and content and consequences of Special\nand General Relativity.\n\n\n\n\nLOOKING FOR THE WINNER\n\nThe Einstein Editor was in sufficiently close touch with the details\nof the adjudication of the essays to have every realization of\nthe difficulty of this work. The caliber of the essays submitted\nwas on the whole high. There were many which would have been well\nworthy of the prize in the absence of others that were distinctly\nbetter--many which it was not possible to eliminate on the ground of\nspecific faults, and which could only be adjudged \"not the best\" by\ndetailed comparison with specific other essays. It was this detailed\ncomparison which took time, and which so delayed the award that we\nwere not able to publish the winning essay any sooner than February\n5th. Especially difficult was this process of elimination after the\nnumber of surviving essays had been reduced to twenty or less. The\nadvantages of plan possessed by one essay had to be weighed against\nthose of execution exhibited in another. A certain essay had to be\ncritically compared with another so like it in plan that the two might\nhave been written from a common outline, and at the same time with\na third as unlike it in scope and content as day and night. And all\nthe time there was present in the background the consciousness that\na prize of $5,000 hung upon the decision to be reached. For anyone\nwho regards this as an easy task we have no worse wish than that he\nmay some day have to attack a similar one.\n\nWe had anticipated that the bulk of the superior essays would be\namong those received during the last day or two of the contest;\nfor we felt that the men best equipped to attack the subject would\nbe the most impressed with its seriousness. Here we were quite off\nthe track. The seventeen essays which withstood most stubbornly the\nJudges' efforts at elimination were, in order of receipt, numbers 8,\n18, 28, 40, 41, 43, 92, 95, 97, 130, 181, 194, 198, 223, 267, 270, 275:\na fairly even distribution. The winner was the 92nd essay received.\n\nThe Judges held their final meeting in the editorial office on\nJanuary 18, 1921. The four essays which were before the committee\nat the start of the session were speedily cut to three, and then\nto two; and after an all-day session the Judges found themselves\nconscientiously able to agree on one of these as the best. This\nunanimity was especially gratifying, the more so since it by no\nmeans was to be confidently expected, on a priori grounds, that it\nwould be possible of attainment. Even the Einstein Editor, who might\nhave been called upon for a final decision but wasn't, can hardly be\nclassed as a dissenter; for with some slight mental reservations in\nfavor of the essay by Mr. Francis which did not enter the Judges'\nfinal discussion at all, and which he rather suspects appeals more\nto his personal taste than to his soundest judgment, he is entirely\nin accord with the verdict rendered.\n\nThe fact that the prize went to England was no surprise to those\nacquainted with the history of Einstein's theories. The Special Theory,\npromulgated fifteen years ago, received its fair share of attention\nfrom mathematicians all over the world, and is doubtless as well\nknown and as fully appreciated here as elsewhere. But it has never\nbeen elevated to a position of any great importance in mathematical\ntheory, simply because of itself, in the absence of its extension\nto the general case, it deserves little importance. It is merely an\ninteresting bit of abstract speculation.\n\nThe General Theory was put out by Einstein in finished form during\nthe war. Owing to the scientific moratorium, his paper, and hence a\nclear understanding of the new methods and results and of the sweeping\nconsequences if the General Theory should prevail, did not attain\ngeneral circulation outside Germany until some time in 1918 or even\nlater. Had it not been for Eddington it is doubtful that the British\nastronomers would have realized that the eclipse expeditions were of\nparticular consequence. Therefore at the time of these expeditions,\nand even as late as the November announcement of the findings, the\ngeneral body of scientific men in America had not adequately realized\nthe immense distinction between the Special and the General Theories,\nhad not adequately appreciated that the latter led to distinctive\nconsequences of any import, and we fear in many cases had not even\nrealized explicitly that the deflection of light and the behavior of\nMercury were matters strictly of the General and in no sense of the\nSpecial Theory. Certainly when the American newspapers were searching\nfrantically for somebody to interpret to their public the great stir\nmade by the British announcement that Einstein's predictions had\nbeen verified, they found no one to do this decently; nor were our\nmagazines much more successful in spite of the greater time they had\nto devote to the search. In a word, there is not the slightest room\nfor doubt that American science was in large measure caught asleep\nat the switch--perhaps for no reason within its control; and that\nAmerican writers were in no such favorable case to write convincingly\non the subject as were their British and continental contemporaries.\n\nSo it was quite in accord with what might have been expected to find,\non opening the identifying envelopes, that not alone the winning essay,\nbut its two most immediate rivals, come from members of that school of\nBritish thought which had been in contact with the Einstein theories\nin their entirety for two years longer than the average American of\nequal competence. This riper familiarity with the subject was bound\nto yield riper fruit. Indeed, had it not been for the handicap of\nwriting in a strange language, it is reasonable to assume that the\nscientists of Germany would have made a showing superior to that of\neither Americans or British--and for the same reason that Britain\nshowed to better advantage than America.\n\n\n\n\nTHE WINNER OF THE PRIZE\n\nMr. Bolton, the winner of the big prize, we suppose may fairly be\nreferred to as unknown in a strict scientific sense. Indeed, at\nthe time of the publication of his essay in the Scientific American\nnothing could be learned about him on the American side of the water\nbeyond the bare facts that he was not a young man, and that he had\nfor a good many years occupied a position of rank in the British\nPatent Office. (It will be recalled that Einstein himself was in the\nSwiss Patent Office for some time.) In response to the request of the\nScientific American for a brief biographical sketch that would serve\nto introduce him better to our readers, Mr. Bolton supplied such a\nconcise and apparently such a characteristic statement that we can\ndo no better than quote it verbatim.\n\n\"I was born in Dublin in 1860, but I have lived in England since\n1869. My family belonged to the landed gentry class, but I owe nothing\nto wealth or position. I was in fact put through school and college\non an income which a workman would despise nowadays. After attending\nsundry small schools, I entered Clifton College in 1873. My career\nthere was checkered, but it ended well. I was always fairly good at\nnatural science and very fond of all sorts of mechanical things. I\nwas an honest worker but no use at classics, and as I did practically\nnothing else for the first four years at Clifton, I came to consider\nmyself something of a dunce. But a big public school is a little\nworld. Everyone gets an opportunity, often seemingly by accident,\nand it is up to him to take it. Mine did not come till I was nearly\n17. As I was intended for the engineering profession, I was sent to\nthe military side of the school in order to learn some mathematics,\nat which subject I was then considered very weak. This was certainly\ntrue, as at that time I barely knew how to solve a quadratic, I was\nonly about halfway through the third book of Euclid, and I knew no\ntrigonometry. But the teaching was inspiring, and I took readily\nto mathematics. One day it came out that I had been making quite a\ngood start with the differential calculus on my own without telling\nanybody. After that all was well. I left Clifton in 1880 with a School\nExhibition and a mathematical scholarship at Clare College, Cambridge.\n\n\"After taking my degree in 1883 as a Wrangler, I taught science and\nmathematics at Wellington College, but I was attracted by what I had\nheard of the Patent Office and I entered it through open competition in\n1885. During my official career I have been one of the Comptroller's\nprivate secretaries and I am now a Senior Examiner. During the\nwar I was attached to the Inventions Department of the Ministry of\nMunitions, where my work related mainly to anti-aircraft gunnery. I\nhave contributed, and am still contributing to official publications\non the subject.\n\n\"I have written a fair number of essays on various subjects, even\non literature, but my only extra-official publications relate to\nstereoscopic photography. I read a paper on this subject before the\nRoyal Photographic Society in 1903 which was favorably noticed by\nDr. von Rohr of Messrs. Zeiss of Jena. I have also written in the\nAmateur Photographer.\n\n\"I have been fairly successful at athletics, and I am a member of\nthe Leander Club.\"\n\nThat Mr. Bolton did not take the prize through default of serious\ncompetition should be plain to any reader who examines the text from\ncompeting essays which is to be found in this volume. The reference\nlist of these competitors, too, supplemented by the names that appear\nat the heads of complete essays, shows a notable array of distinguished\npersonalities, and I could mention perhaps a dozen more very well\nknown men of science whose excellent essays have seemed a trifle too\nadvanced for our immediate use, but to whom I am under a good deal\nof obligation for some of the ideas which I have attempted to clothe\nin my own language.\n\nBefore leaving the subject, we wish to say here a word of appreciation\nfor the manner in which the Judges have discharged their duties. The\nreader will have difficulty in realizing what it means to read such\na number of essays on such a subject. We were fortunate beyond all\nexpectation in finding Judges who combined a thorough scientific\ngrasp of the mathematical and physical and philosophical aspects\nof the matter with an extremely human viewpoint which precluded any\npossibility of an award to an essay that was not properly a popular\ndiscussion, and with a willingness to go to meet each other's opinions\nthat is rare, even among those with less ground for confidence in\ntheir own views than is possessed by Drs. Page and Adams.\n\n\n\n\n\n\n\n\n\nII.\n\nTHE WORLD--AND US\n\nAn Introductory Discussion of the Philosophy of Relativity, and of\nthe Mechanism of Our Contact with Time and Space\n\nBY VARIOUS CONTRIBUTORS AND THE EDITOR\n\n\nFrom a time beyond the dawn of history, mankind has been seeking\nto explain the universe. At first the effort did not concern itself\nfurther probably than to make a supposition as to what were the causes\nof the various phenomena presented to the senses. As knowledge\nincreased, first by observation and later by experiment also,\nthe ideas as to these causes passed progressively through three\nstages--the theological (the causes were thought to be spirits or\ngods); the metaphysical (the causes were thought in this secondary\nor intermediate stage to be some inherent, animating, energizing\nprinciples); and the scientific (the causes were finally thought of\nas simply mechanical, chemical, and magneto-electrical attractions\nand repulsions, qualities or characteristics of matter itself, or of\nthe thing of which matter is itself composed.)\n\nWith increase of knowledge, and along with the inquiry as to the nature\nof causes, there arose an inquiry also as to what reality was. What was\nthe essential nature of the stuff of which the universe was made, what\nwas matter, what were things in themselves, what were the noumena (the\nrealities), lying back of the phenomena (the appearances)? Gradually\nideas explaining motion, force, and energy were developed. At the\nsame time inquiry was made as to the nature of man, the working\nof his mind, the nature of thought, the relation of his concepts\n(ideas) to his perceptions (knowledge gained through the sense)\nand the relations of both to the noumena (realities).]283\n\n[The general direction taken by this inquiry has been that of a\nconflict between two schools of thought which we may characterize\nas those of absolutism and of relativism.]* [The ancient Greek\nphilosophers believed that they could tap a source of knowledge\npure and absolute by sitting down in a chair and reasoning about the\nnature of time and space, and the mechanism of the physical world.]221\n[They maintained that the mind holds in its own right certain concepts\nthan which nothing is more fundamental. They considered it proper to\nconceive of time and space and matter and the other things presented\nto their senses by the world as having a real existence in the mind,\nregardless of whether any external reality could be identified\nwith the concept as ultimately put forth. They could even dispute\nwith significance the qualities which were to be ascribed to this\nabstract conceptual time and space and matter. All this was done\nwithout reference to the external reality, often in defiance of that\nreality. The mind could picture the world as it ought to be; if the\nrecalcitrant facts refused to fit into the picture, so much the worse\nfor them. We all have heard the tale of how generation after generation\nof Greek philosophers disputed learnedly why and how it was that a live\nfish could be added to a brimming pail of water without raising the\nlevel of the fluid or increasing the weight; until one day some common\nperson conceived the troublesome idea of trying it out experimentally\nto learn whether it were so--and found that it was not. True or false,\nthe anecdote admirably illustrates the subordinate place which the\nexternals held in the absolutist system of Greek thought.]*\n\n[Under this system a single observer is competent to examine a\nsingle phenomenon, and to write down the absolute law of nature\nby referring the results to his innate ideas of absolute qualities\nand states. The root of the word absolute signifies \"taking away,\"\nand in its philosophical sense the word implies the ability of the\nmind to subtract away the properties or qualities from things, and\nto consider these abstract qualities detached from the things; for\nexample, to take away the coldness from ice, and to consider pure or\nabstract coldness apart from anything that is cold; or to take away\nmotion from a moving body, and to consider pure motion apart from\nanything that moves. This assumed power is based upon the Socratic\ntheory of innate ideas. According to this theory the mind is endowed\nby nature with the absolute ideas of hardness, coldness, roundness,\nequality, motion, and all other absolute qualities and states, and\nso does not have to learn them. Thus a Socratic philosopher could\ndiscuss pure or absolute being, absolute space and absolute time.]121\n\n\n\n\nGETTING AWAY FROM THE GREEK IDEAS\n\n[This Greek mode of thought persisted into the late Middle Ages,\nat which time it was still altogether in order to dispose of a\ntroublesome fact of the external world by quoting Aristotle against\nit. During the Renaissance, which intellectually at least marks the\ntransition from ancient to modern, there came into being another\ntype of absolutism, equally extreme, equally arbitrary, equally\nunjustified. The revolt against the mental slavery to Greek ideas\ncarried the pendulum too far to the other side, and early modern\nscience as a consequence is disfigured by what we must now recognize\nas gross materialism. The human mind was relegated to the position\nof a mere innocent bystander. The external reality was everything,\nand aside from his function as a recorder the observer did not in the\nleast matter. The whole aim of science was to isolate and classify the\nelusive external fact. The r\u00f4le of the observer was in every possible\nway minimized. It was of course his duty to get the facts right,\nbut so far as any contribution to these was concerned he did not\ncount--he was definitely disqualified. He really played the part of\nan intruder; from his position outside the phenomena he was searching\nfor the absolute truth about these phenomena. The only difference\nbetween his viewpoint and that of Aristotle was that the latter looked\nentirely inside himself for the elusive \"truth,\" while the \"classical\"\nscientist, as we call him now, looked for it entirely outside himself.\n\nLet me illustrate the difference between the two viewpoints which\nI have discussed, and the third one which I am about to outline,\nby another concrete instance. The Greeks, and the medievals as well,\nwere fond of discussing a question which embodies the whole of what\nI have been saying. This question involved, on the part of one who\nattempted to answer it, a choice between the observer and the external\nworld as the seat of reality. It was put in many forms; a familiar\none is the following: \"If the wind blew down a great tree at a time\nand place where there was no conscious being to hear, would there\nbe any noise?\" The Greek had to answer this question in the negative\nbecause to him the noise was entirely a phenomenon of the listener. The\nclassical scientist had to answer it in the affirmative because to him\nthe noise was entirely a phenomenon of the tree and the air and the\nground. Today we answer it in the negative, but for a very different\nreason from that which swayed the Greek. We believe that the noise is\na joint phenomenon of the observer and the externals, so that in the\nabsence of either it must fail to take existence. We believe there are\nsound waves produced, and all that; but what of it? There is no noise\nin the presence of the falling tree and the absence of the observer,\nany more than there would be in the presence of the observer and the\nabsence of the tree and the wind; the noise, a joint phenomenon of\nthe observer and the externals, exists only in their joint presence.\n\n\n\n\nRELATIVISM AND REALITY\n\nThis is the viewpoint of relativism. The statue is golden for\none observer and silver to the other. The sun is rising here and\nsetting in another part of the world. It is raining here and clear in\nChicago. The observer in Delft hears the bombardment of Antwerp and\nthe observer in London does not. If they were to be consistent, both\nthe Greek and the medieval-modern absolutist would have to dispute\nwhether the statue were \"really\" golden or silver, whether the sun\nwere \"really\" rising or setting, whether the weather were \"really\"\nfair or foul, whether the bombardment were \"really\" accompanied by\nloud noises or not; and on each of these questions they would have to\ncome to an agreement or confess their methods inadequate. But to the\nrelativist the answer is simple--whether this or that be true depends\nupon the observer. In simple cases we understand this full well, as\nwe have always realized it. In less simple cases we recognize it less\neasily or not at all, so that some of our thought is absolutist in\nits tendencies while the rest is relativistic. Einstein is the first\never to realize this fully--or if not this, then the first ever to\nrealize it so fully as to be moved toward a studied effort to free\nhuman thought from the mixture of relativism and absolutism and make\nit consistently the one or the other.\n\nThis brings it about that the observed fact occupies a position of\nunexpected significance. For when we discuss matters of physical\nscience under a strictly relativistic philosophy, we must put away as\nmetaphysical everything that smacks of a \"reality\" partly concealed\nbehind our observations. We must focus attention upon the reports\nof our senses and of the instruments that supplement them. These\nobservations, which join our perceptions to their external objects,\nafford us our only objective manifestations; them we must accept as\nfinal--subject always to such correction as more refined observations\nmay suggest. The question whether a \"true\" length or area or mass\nor velocity or duration or temperature exists back of the numerical\ndetermination, or in the presence of a determination that is subject\nto correction, or in the absence of any determination at all, is a\nmetaphysical one and one that the physicist must not ask. Length,\narea, mass, velocity, duration, temperature--none of these has any\nmeaning other than the number obtained by measurement.]* [If several\ndifferent determinations are checked over and no error can be found\nin any of them, the fault must lie not with the observers but with\nthe object, which we must conclude presents different values to\ndifferent observers.]33\n\n[We are after all accustomed to this viewpoint; we do not demand that\nPittsburgh shall present the same distance from New York and from\nPhiladelphia, or that the New Yorker and the Philadelphian come to\nany agreement as to the \"real\" distance of Pittsburgh. The distance\nof Pittsburgh depends upon the position of the observer. Nor do we\ndemand that the man who locates the magnetic pole in one spot in 1900\nand in another in 1921 come to a decision as to where it \"really\"\nis; we accept his statement that its position depends upon the time\nof the observation.\n\nWhat this really means is that the distance to Pittsburgh and the\nposition of the magnetic pole are joint properties of the observer\nand the observed--relations between them, as we might put it. This\nis obvious enough in the case of the distance of Pittsburgh; it is\nhardly so obvious in the case of the position of the magnetic pole,\nvarying with the lapse of time. But if we reflect that the observation\nof 1900 and that of 1921 were both valid, and both represented the\ntrue position of the pole for the observer of the date in question, we\nmust see that this is the only explanation that shows us the way out.\n\nI do not wish to speak too definitely of the Einstein theories in\nthese introductory remarks, and so shall refrain from mentioning\nexplicitly in this place the situation which they bring up and upon\nwhich what I have just said has direct bearing. It will be recognized\nwhen it arises. What must be pointed out here, however, is that we\nare putting the thing which the scientist calls the \"observed value\"\non a footing of vastly greater consequence than we should have been\nwilling offhand to concede to it. So far as any single observer is\nconcerned, his own best observed values are themselves the external\nworld; he cannot properly go behind the conditions surrounding\nhis observations and speak of a real external world beyond these\nobservations. Any world which he may think of as so existing is purely\na conceptual world, one which for some reason he infers to exist\nbehind the deceptive observations. Provided he makes this reservation\nhe is quite privileged to speculate about this concealed world, to\nbestow upon it any characteristics that he pleases; but it can have\nno real existence for him until he becomes able to observe it. The\nonly reality he knows is the one he can directly observe.\n\n\n\n\nLAWS OF NATURE\n\nThe observations which we have been discussing, and which we have\nbeen trying to endow with characteristics of \"reality\" which they are\nfrequently not realized to possess, are the raw material of physical\nscience. The finished product is the result of bringing together a\nlarge number of these observations.]* [The whole underlying thought\nbehind the making of observations, in fact, is to correlate as many\nas possible of them, to obtain some generalization, and finally to\nexpress this in some simple mathematical form. This formulation is\nthen called a \"law of nature.\"]35\n\n[Much confusion exists because of a misunderstanding in the lay mind\nof what is meant by a \"law of nature.\" It is perhaps not a well chosen\nterm. One is accustomed to associate the word law with the idea of\nnecessity or compulsion. In the realm of nature the term carries\nno such meaning. The laws of nature are man's imperfect attempts to\nexplain natural phenomena; they are not inherent in matter and the\nuniverse, not an iron bar of necessity running through worlds, systems\nand suns. Laws of nature are little more than working hypotheses,\nsubject to change or alteration or enlargement or even abandonment,\nas man's vision widens and deepens. No sanctity attaches to them,\nand if any one, or all, of them fail to account for any part, or all,\nof the phenomena of the universe, then it or they must be supplemented\nor abandoned.]102\n\n[The test of one of these laws is that it can be shown to include\nall the related phenomena hitherto known and that it enables us to\npredict new phenomena which can then be verified. If new facts are\ndiscovered that are not in agreement with one of these generalized\nstatements, the assumptions on which the latter is based are examined,\nthose which are not in accordance with the new facts are given up,\nand the statement is modified so as to include the new facts.]10\n[And if one remembers that the laws of physics were formerly based\non a range of observations much narrower than at present available,\nit seems natural that in the light of this widening knowledge one law\nor another may be seen to be narrow and insufficient. New theories\nand laws do not necessarily disprove old ones, but explain certain\ndiscrepancies in them and penetrate more deeply into their underlying\nprinciples, thereby broadening our ideas of the universe. To follow the\nnew reasoning we must rid ourselves of the prejudice behind the old,\nnot because it is wrong but because it is insufficient. The universe\nwill not be distorted to fit our rules, but will teach us the rules\nof existence.]125\n\n[Always, however, we must guard against the too easy error of\nattributing to these rules anything like absolute truth.]* [The\nmodern scientist has attained a very business-like point of view\ntoward his \"laws of nature.\" To him a law is fundamentally nothing\nbut a short-hand way of expressing the results of a large number of\nexperiments in a single statement. And it is important to remember that\nthis mere shortening of the description of a lot of diverse occurrences\nis by no means any real explanation of how and why they happened. In\nother words, the aim of science is not ultimately to explain but only\nto discover the relations that hold good among physical quantities and\nto embody all these relations in as few and as simple physical laws\nas possible.]221 [This is inherently the method of relativism.]*\n[Under it a set of phenomena is observed. There are two or many\nobservers, and they write down their several findings. These are\nreviewed by a final observer or judge, who strains out the bias due\nto the different viewpoints of the original observers. He then writes\ndown, not any absolute law of nature governing the observed phenomena,\nbut a law as general as possible expressing their interrelations.]121\n[And through this procedure modern science and philosophy reveal\nwith increasing emphasis that we superimpose our human qualities\non external nature to such an extent that]106 [we have at once the\nstrongest practical justification, in addition to the arguments\nof reason, for our insistence that the contact between objective\nand subjective represented by the observation is the only thing\nwhich we shall ever be able to recognize as real. We may indulge in\nabstract metaphysical speculation to our heart's content, if we be\nmetaphysically inclined; we may not attempt to impose the dicta of\nmetaphysics upon the physical scientist.]*\n\n\n\n\nCONCEPTS AND REALITIES\n\n[From the inquiry and criticism which have gone on for centuries has\nemerged the following present-day attitude of mind toward the sum\ntotal of our knowledge. The conceptual universe in our minds in some\nmysterious way parallels the real universe, but is totally unlike\nit. Our conceptions (ideas) of matter, molecules, atoms, corpuscles,\nelectrons, the ether, motion, force, energy, space, and time stand\nin the same or similar relation to reality as the x's and y's of the\nmathematician do to the entities of his problem. Matter, molecules,\natoms, corpuscles, electrons, the ether, motion, force, energy, space,\nand time do not exist actually and really as we conceive them, nor do\nthey have actually and really the qualities and characteristics with\nwhich we endow them. The concepts are simply representations of things\noutside ourselves; things which, while real, have an essential nature\nnot known to us. Matter, molecules, atoms, corpuscles, electrons,\nthe ether, motion, force, energy, space and time are merely devices,\nsymbols, which enable us to reason about reality. They are parts of\na conceptual mechanism in our minds which operates, or enables our\nminds to operate, in the same sequence of events as the sequence of\nphenomena in the external universe, so that when we perceive by our\nsenses a group of phenomena in the external universe, we can reason\nout what result will flow from the interaction of the realities\ninvolved, and thus predict what the situation will be at a given\nstage in the sequence.\n\nBut while our conceptual universe has thus a mechanical aspect,\nwe do not regard the real universe as mechanical in its nature.]283\n[This may be illustrated by a little story. Entering his friend's\nhouse, a gentleman is seized unawares from behind. He turns his\nhead but sees nothing. His hat and coat are removed and deposited\nin their proper places by some invisible agent, seats and tables and\nrefreshments appear in due time where they are required, all without\nany apparent cause. The visitor shivers with horror and asks his\nhost for an explanation. He is then told that the ideas \"order\" and\n\"regularity\" are at work, and that it is they who acquit themselves so\nwell of their tasks. These ideas cannot be seen nor felt nor seized\nnor weighed; they reveal their existence only by their thoughtful\ncare for the welfare of mankind. I think the guest, coming home,\nwill relate that his friend's house is haunted. The ghosts may be\nkind, benevolent, even useful; yet ghosts they are. Now in Newtonian\nmechanics, absolute space and absolute time and force and inertia and\nall the other apparatus, altogether imperceptible, appearing only at\nthe proper time to make possible a proper building up of the theory,\nplay the same mysterious part as the ideas \"order\" and \"regularity\"\nin my story. Classical mechanics is haunted.]116\n\n[As a matter of fact, we realize this and do not allow ourselves to\nbe imposed upon with regard to the true nature of these agencies.]*\n[We use a mechanistic terminology and a mechanistic mode of reasoning\nonly because we have found by experience that they facilitate our\nreasoning. They are the tools which we find produce results. They\nare adapted to our minds, but perhaps it would be better to say that\nour minds are so constructed as to render our conceptual universe\nnecessarily mechanical in its aspect in order that our minds may reason\nat all. Two things antithetic are involved--subject (our perceiving\nmind which builds up concepts) and object (the external reality); and\nhaving neither complete nor absolute knowledge of either, we cannot\naffirm which is more truly to be said to be mechanistic in its nature,\nthough we may suspect that really neither is. We no longer think\nof cause and effect as dictated by inherent necessity, we simply\nregard them as sequences in the routine of our sense-impressions of\nphenomena. In a word, we have at length grasped the idea that our\nnotions of reality, at present at least, whatever they may become\nultimately, are not absolute, but simply relative. We see, too, that\nwe do not explain the universe, but only describe our perceptions of\nits contents.\n\nThe so-called laws of nature are simply statements of formul\u00e6 which\nresume or sum up the relationships and sequences of phenomena. Our\neffort is constantly to find formul\u00e6 which will describe the widest\npossible range of phenomena. As our knowledge increases, that is,\nas we perceive new phenomena, our laws or formul\u00e6 break down,\nthat is, they fail to afford a description in brief terms of all\nof our perceptions. It is not that the old laws are untrue, but\nsimply that they are not comprehensive enough to include all of our\nperceptions. The old laws are often particular or limiting instances\nof the new laws.]283\n\n[From what we have said of the reality of observations it follows that\nwe must support that school of psychology, and the parallel school\nof philosophy, which hold that concepts originate in perceptions. But\nthis does not impose so strong a restriction upon conceptions as might\nappear. The elements of all our concepts do come to us from outside;\nwe manufacture nothing out of whole cloth. But when perception\nhas supplied a sufficient volume of raw material, we may group its\nelements in ways foreign to actual occurrence in the perceptual world,\nand in so doing get conceptual results so entirely different from\nwhat we have consciously perceived that we are strongly tempted to\nlook upon them as having certainly been manufactured in our minds\nwithout reference to the externals. Of even more significance is our\nability to abstract from concrete objects and concrete incidents the\nessential features which make them alike and different. But unlike\nthe Greeks, we see that our concept of coldness is not something with\nwhich we were endowed from the beginning, but merely an abstraction\nfrom concrete experiences with concrete objects that have been cold.\n\n\n\n\nTHE CONCEPTS OF SPACE AND TIME\n\nWhen we have formed the abstract ideas of coldness and warmth, and\nhave had experience indicating that the occurrence of these properties\nvaries in degree, we are in a position to form the secondary abstract\nnotion covered by the word \"temperature.\" When we have formed the\nabstract ideas of size and position and separation, we are similarly\nin a position to form a secondary abstraction to which we give the name\n\"space.\" Not quite so easy to trace to its definite source but none the\nless clearly an abstraction based on experience, is our idea of what\nwe call \"time.\" None of us are deceived as to the reality of these\nabstractions.]* [We do not regard space as real in the sense that\nwe regard a chair as real; it is merely an abstract idea convenient\nfor the location of material objects like the chair.]198 [Nor do we\nregard time as real in this sense. Things occupy space, events occupy\ntime; space and time themselves we realize are immaterial and unreal;\nspace does not exist and time does not happen in the same sense that\nmaterial objects exist and events occur. But we find it absolutely\nnecessary to have, among the mental machinery mentioned above as\nthe apparatus by aid of which we keep track of the external world,\nthese vessels for that world to exist in and move in.\n\nSpace and time, then, are concepts.]* [It is not strange, however,\nthat when confronted with the vast and bewildering complexity of\nthe universe and the difficulty of keeping separate and distinct in\nour minds our perceptions and conceptions, we should at times and as\nrespects certain things project our conceptions illegitimately into the\nperpetual universe and mistake them for perceptions. The most notable\nexample perhaps of this projection has occurred in the very case of\nspace and time, most fundamental of all of our concepts. We got to\nthink of these as absolute, as independent of each other and of all\nother things, and as always existing and continuing to exist whether\nor not we or anything else existed--space as a three-dimensional,\nuniform continuum, having the same properties in all directions;\ntime as a one-dimensional, irreversible continuum, flowing in one\ndirection. It is difficult to get back to the idea that space and\ntime so described and defined are concepts merely, for the idea of\ntheir absolute existence is ingrained in us as the result probably\nof long ancestral experience.]283\n\n[Newton's definitions of course represent the classical idea of time\nand space. He tells us that \"absolute, true and mathematical time\nflows in virtue of its own nature, uniformly and without reference to\nany external object;\" and that \"absolute space, by virtue of its own\nnature and without reference to any external object, always remains\nthe same and is immovable.\" Of course from modern standpoints it is\nabsurd to call either of these pronouncements a definition; but they\nrepresent about as well as any words can the ideas which Newton had\nabout time and space, and they make it clear enough that he regarded\nboth as having real existence in the external world.\n\nIf space and time are to be the vessels of our universe, and if the\nonly thing that really matters is measured results, it is plain enough\nthat we must have, from the very beginning, means of measuring space\nand time. Whether we believe space and time to have real existence or\nnot, it is obvious that we can measure neither directly. We shall have\nto measure space by measuring from one material object to another;\nwe shall have to measure time by some similar convention based\non events. We shall later have something further to say about the\nmeasurement of time; for the present we need only point out that]*\n[Newtonian time is measured independently of space; and the existence\nis presupposed of a suitable timekeeper.]10\n\n[The space of Galileo and Newton was conceived of as empty, except in\nso far as certain parts of it were occupied by matter. Positions of\nbodies in this space were in general determined by reference to]283\n[a \"coordinate system\" of some kind. This is again something that\ndemands a certain amount of discussion.\n\n\n\n\nTHE REFERENCE FRAME FOR SPACE\n\nThe mathematician, following the lead of the great French all-around\ngenius, Descartes, shows us very clearly how to set up, for the\nmeasurement of space, the framework known as the Cartesian coordinate\nsystem. The person of most ordinary mathematical attainments\nwill realize that to locate a point in a plane we must have two\nmeasurements; and we could probably show this person, without too\nserious difficulty, that we can locate a point in any surface by\ntwo measurements. An example of this is the location of points on the\nearth's surface by means of their latitude and longitude. It is equally\nclear that if we add a third dimension and attempt to locate points\nin space, we must add a third measurement. In the case of points on\nthe earth's surface, this might be the elevation above sea level,\nwhich would define the point not as part of the spherical surface\nof the earth but as part of the solid sphere. Or we may fall back\non Dr. Slosson's suggestion that in order to define completely the\nposition of his laboratory, we must make a statement about Broadway,\nand one about 116th Street, and one telling how many flights of stairs\nthere are to climb. In any event, it should be clear enough that the\ncomplete definition of a point in space calls for three measurements.\n\nThe mathematician formulates all this with the utmost precision. He\nasks us to]* [pick out any point whatever in space and call it $O$. We\nthen draw or conceive to be drawn through this point three mutually\nperpendicular lines called coordinate axes, which we may designate\n$OX$, $OY$ and $OZ$, respectively. Finally, we consider the three\nplanes also mutually perpendicular like the two walls and the floor of\na room that meet in one common corner, which are formed by the lines\n$OX$ and $OY$, $OY$ and $OZ$, and $OZ$ and $OX$, respectively. These\nthree planes are called coordinate planes. And then any other point $P$\nin space can be represented with respect to $O$ by its perpendicular\ndistances from each of the three coordinate planes--the distances $x$,\n$y$, $z$ in the figure. These quantities are called the coordinates\nof the point.]272\n\n[To the layman there seems something altogether naive in this notion\nof the scientist's setting up the three sides of a box in space and\nusing them as the basis of all his work. The layman somehow feels\nthat while it is perfectly all right for him to tell us that he\nlives at 1065 (one coordinate) 156th Street (two coordinates) on\nthe third floor (three coordinates), it is rather trivial business\nfor the serious-minded scientist to consider the up-and-down, the\nforward-and-back, the right-and-left of every point with which he has\noccasion to deal. There seems to the layman something particularly\ninane and foolish and altogether puerile about a set of coordinate\naxes, and you simply can't make him believe that the serious-minded\nscientist has to monkey with any such funny business. He can't be\ninduced to take this coordinate-axis business seriously. Nevertheless,\nthe fact is that the scientist takes it with the utmost seriousness. It\nis necessary for him to define the positions of points; and he does\ndo it by means of a set of coordinate axes.\n\nThe scientist, however, is not interested in points of empty space. The\npoint is to him merely part again of the conceptual machinery which\nhe uses in his effort to run along with the external world. He knows\nthere are no real points, but it suits his convenience to keep track\nof certain things that are real by representing them as points. But\nthese things are in practically every instance material bodies; and\nin practically every instance, instead of staying put in one spot,\nthey insist upon moving about through space. The scientist has to use\nhis coordinate system, not merely to define a single position of such\na \"point,\" but to keep track of the path over which it moves and to\ndefine its position in that path at given moments.\n\n\n\n\nTIME AND THE COORDINATE SYSTEM\n\nThis introduces the concept of time into intimate relationship with\nthe spatial coordinate system. And at once we feel the lack of a\nconcrete, visualized fourth dimension.]* [If we want to fix objects\nin the floor alone, the edge of the room running toward the ceiling\nwould become unnecessary and could be dropped from our coordinate\nsystem. That is, we need only two coordinates to fix the position of a\npoint in a plane. Suppose instead of discarding the third coordinate,\nwe use it to represent units of time. It then enables us to record\nthe time it took a moving point in the floor to pass from position\nto position. Certain points in the room would be vertically above\nthe corresponding points occupied by the moving point in its path\nacross the floor; and the vertical height above the floor of such\npoints corresponds to a value of the time-coordinate which indicates\nthe time it took the point to move from position to position.]152\n[Just as the path of the point across the floor is a continuous curve\n(for the mathematician, it should be understood, this term \"curve\"\nincludes the straight line, as a special case in which the curvature\nhappens to be zero); so the series of points above these in the room\nforms a continuous curve which records for us, not merely the path of\nthe point across the floor, but in addition the time of its arrival\nat each of its successive positions. In the algebraic work connected\nwith such a problem, the third coordinate behaves exactly the same,\nregardless of whether we consider it to represent time or a third\nspatial dimension; we cannot even tell from the algebra what it\ndoes represent.\n\nWhen we come to the more general case of a point moving freely through\nspace, we have but three coordinates at our disposal; there is not\na fourth one by aid of which we can actually diagram its time-space\nrecord. Nevertheless, we can write down the numerical and algebraic\nrelations between its three space-coordinates and the time which it\ntakes to pass from one position to another; and by this means we can\nmake all necessary calculations. Its motion is completely defined\nwith regard both to space and to time. We are very apt to call\nattention to the fact that if we did have at our disposal a fourth,\nspace-coordinate, we could use it to represent the time graphically,\nas before, and actually construct a geometric picture of the path of\nour moving point with regard to space and time. And on this account\nwe are very apt to speak as though the time measurements constituted\na fourth coordinate, regardless of any question of our ability to\nconstruct a picture of this coordinate. The arrival of a point in\na given position constitutes an event; and this event is completely\ndefined by means of four coordinates--three in space, which we can\npicture on our coordinate axes, and one in time which we cannot.\n\nThe set of coordinate axes in space, together with the zero point from\nwhich we measure time, constitute what we call a frame of reference. If\nwe are not going to pay any attention to time, we can think of the\nspace coordinate system alone as constituting our reference frame. This\nexpression appears freely throughout the subsequent text, and always\nwith one or the other of these interpretations.\n\nWe see, then, how we can keep track of a moving point by keeping track\nof the successive positions which it occupies in our reference frame.]*\n[Now we have implied that these coordinate axes are fixed in space;\nbut there is nothing to prevent us from supposing that they move.]272\n[If they do, they carry with them all their points; and any motion\nof these points which we may speak about will be merely motion with\nreference to the coordinate system. If we find something outside\nour coordinate system that is not moving, the motion of points in\nour system with regard to those outside it will be a combination of\ntheir motion with regard to our coordinate axes and that of these axes\nwith regard to the external points. This will be a great nuisance;\nand it represents a state of affairs which we shall try to avoid. We\nshall avoid it, if at all, by selecting a coordinate system with\nreference to which we, ourselves, are not moving; one which partakes\nof any motion which we may have. Or perhaps we shall sometimes wish to\nreverse the process, in studying the behavior of some group of bodies,\nand seek a set of axes which is at rest with respect to these bodies;\none which partakes of any motion they may have.\n\n\n\n\nTHE CHOICE OF A COORDINATE FRAME\n\nAll this emphasizes the fact that our coordinate axes are not\npicked out for us in advance by nature, and set down in some one\nparticular spot. We select them for ourselves, and we select them in\nthe most convenient way. But different observers, or perhaps the same\nobserver studying different problems, will find it advantageous to\nutilize different coordinate systems.]* [The astronomer has found it\npossible, and highly convenient, to select a coordinate frame such\nthat the great majority of the stars have, on the whole, no motion\nwith respect to it.]283 [Such a system would be most unsuited for\ninvestigations confined to the earth; for these we naturally select\na framework attached to the earth, with its origin O at the earth's\ncenter if our investigation covers the entire globe and at some more\nconvenient point if it does not, and in either event accompanying the\nearth in its rotation and revolution. But such a framework, as well\nas the one attached to the fixed stars, would be highly inconvenient\nfor an investigator of the motions of the planets; he would doubtless\nattach his reference frame to the sun.]101\n\n[In this connection a vital question suggests itself. Is the\nexpression of natural law independent of or dependent upon the choice\nof a system of coordinates? And to what extent shall we be able to\nreconcile the results of one observer using one reference frame, and\na second observer using a different one? The answer to the second\nquestion is obvious.]* [True, if any series of events is described\nusing two different sets of axes, the descriptions will be different,\ndepending upon the time system adopted and the relative motion of the\naxes. But if the connection between the reference systems is known,\nit is possible by mathematical processes to deduce the quantities\nobserved in one system if those observed in the other are known.]35\n[This process of translating the results of one observer into those of\nanother is known as a transformation; and the mathematical statement\nof the rule governing the transformation is called the equation or the\nequations (there are usually several of them) of the transformation.]*\n[Transformations of this character constitute a well-developed branch\nof mathematics.]35\n\n[When we inquire about the invariance of natural law it is necessary to\nbe rather sure of just what we mean by this expression. The statement\nthat a given body is moving with a velocity of 75 miles per hour is of\ncourse not a natural law; it is a mere numerical observation. But aside\nfrom such numerical results, we have a large number of mathematical\nrelations which give us a more or less general statement of the\nrelations that exist between velocities, accelerations, masses,\nforces, times, lengths, temperatures, pressures, etc., etc. There\nare some of these which we would be prepared to state at once as\nuniversally valid--distance travelled equals velocity multiplied by\ntime, for instance. We do not believe that any conceivable change of\nreference systems could bring about a condition in which the product\nof velocity and time, as measured from a certain framework, would\nfail to equal distance as measured from this same framework. There\nare other relations more or less of the same sort which we probably\nbelieve to be in the same invariant category; there are others,\nperhaps, of which we might be doubtful; and presumably there are\nstill others which we should suspect of restricted validity, holding\nin certain reference systems only and not in others.\n\nThe question of invariance of natural law, then, may turn out to\nbe one which may be answered in the large by a single statement;\nit may equally turn out to be one that has to be answered in the\nsmall, by considering particular laws in connection with particular\ntransformations between particular reference systems. Or, perhaps,\nwe may find ourselves justified in taking the stand that an alleged\n\"law of nature\" is truly such a law only in the event that it is\nindependent of the change from one reference system to another. In\nany event, the question may be formulated as follows:\n\nObserver A, using the reference system R, measures certain quantities\n$t$, $w$, $x$, $y$, $z$. Observer B, using the reference system S,\nmeasures the same items and gets the values $t'$, $w'$, $x'$, $y'$,\n$z'$. The appropriate transformation equations for calculating the\none set of values from the other is found. If a mathematical relation\nof any sort is found to exist between the values $t$, $w$, $x$, $y$,\n$z$, will the same relation exist between the values $t'$, $w'$,\n$x'$, $y'$, $z'$? If it does not, are we justified in still calling\nit a law of nature? And if it does not, and we refrain from calling\nit such a law, may we expect in every case to find some relation that\nwill be invariant under the transformation, and that may therefore\nbe recognized as the natural law connecting $t$, $w$, $x$, $y$ and $z$?\n\nI have found it advisable to discuss this point in such detail because\nhere more than in any other single place the competing essayists\nbetray uncertainty of thought and sloppiness of expression. It doesn't\namount to much to talk about the invariance of natural laws and\ntheir persistence as we pass from one coordinate system to another,\nunless we are fairly well fortified with respect to just what we mean\nby invariance and by natural law. We don't expect the velocity of a\ntrain to be 60 miles per hour alike when we measure it with respect\nto a signal tower along the line and with respect to a moving train\non the other track. We don't expect the angular displacement of\nMars to change as rapidly when he is on the other side of the sun\nas when he is on our side. But we do, I think, rather expect that\nin any phenomenon which we may observe, we shall find a natural law\nof some sort which is dependent for its validity neither upon the\nunits we employ, nor the place from which we make our measurements,\nnor anything else external to the phenomenon itself. We shall see,\nlater, whether this expectation is justified, or whether it will have\nto be discarded in the final unravelling of the absolutist from the\nrelativistic philosophy which, with Einstein, we are to undertake.]*\n\n\n\n\n\n\n\n\n\nIII\n\nTHE RELATIVITY OF UNIFORM MOTION\n\nClassical Ideas on the Subject; the Ether and the Apparent Possibility\nof Absolute Motion; the Michelson-Morley Experiment and the Final\nNegation of This Possibility\n\nBY VARIOUS CONTRIBUTORS AND THE EDITOR\n\n\nWhen we speak of a body as being \"in motion,\" we mean that this\nbody is changing its position \"in space.\" Now it is clear that the\nposition of an object can only be determined with reference to other\nobjects: in order to describe the place of a material thing we must,\nfor example, state its distances from other things. If there were no\nsuch bodies of reference, the words \"position in space\" would have\nno definite meaning for us.]24 [The number of such external bodies of\nreference which it is necessary to cite in order to define completely\nthe position of a given body in space depends upon the character\nof the space dealt with. We have seen that when we visualize the\nspace of our experience as a surface of any character, two citations\nare sufficient; and that when we conceive of it as surrounding us\nin three dimensions we require three. It will be realized that the\nmathematician is merely meeting this requirement when he sets up his\nsystem of coordinate axes to serve as a reference frame.]*\n\n[What is true of \"place\" must be true also of \"motion,\" since the\nlatter is nothing but change of place. In fact, it would be impossible\nto ascribe a state of motion or of rest to a body poised all alone\nin empty space. Whether a body is to be regarded as resting or as\nmoving, and if the latter at what speed, depends entirely upon the\nobjects to which we refer its positions in space.]24 [As Einstein\nsits at his desk he appears to us to be at rest; but we know that\nhe is moving with the rotation of the earth on its axis, with the\nearth in its orbit about the sun, and with the solar system in its\npath through space--a complex motion of which the parts or the whole\ncan be detected only by reference to appropriately chosen ones of the\nheavenly bodies. No mechanical test has ever been devised which will\ndetect this motion,]182 [if we reserve for discussion in its proper\nplace the Foucault pendulum experiment which will reveal the axial\nrotation of our globe.]* [No savage, if he were to \"stand still,\"\ncould be convinced that he was moving with a very high velocity\nor in fact that he was moving at all.]30 [You drop a coin straight\ndown a ship's side: from the land its path appears parabolic; to a\npolar onlooker it whirls circle-wise; to dwellers on Mars it darts\nspirally about the sun; to a stellar observer it gyrates through\nthe sky]263 [in a path of many complications. To you it drops in a\nstraight line from the deck to the sea.]* [Yet its various tracks\nin ship-space, sea-space, earth-space, sun-space, star-space, are\nall equally real,]263 [and the one which will be singled out for\nattention depends entirely upon the observer, and the objects to\nwhich he refers the motion.]* [The earth moves in the solar system,\nwhich is itself approaching a distant star-cluster. But we cannot say\nwhether we are moving toward the cluster, or the cluster toward us,]18\n[or both, or whether we are conducting a successful stern chase of it,\nor it of us,]* [unless we have in mind some third body with reference\nto which the motions of earth and star-cluster are measured.]18 [And\nif we have this, the measurements made with reference to it are of\nsignificance with regard to it, rather than with regard to the earth\nand the star-cluster alone.]*\n\n[We can express all this by saying \"All motions are relative; there\nis no such thing as absolute motion.\" This line of argument has in\nfact been followed by many natural philosophers. But is its result\nin agreement with actual experience? Is it really impossible to\ndistinguish between rest and motion of a body if we do not take into\nconsideration its relations to other objects? In fact it can easily\nbe seen that, at least in many cases, no such distinction is possible.\n\n\n\n\nWHO IS MOVING?\n\nImagine yourself sitting in a railroad car with veiled windows and\nrunning on a perfectly straight track with unchanging velocity: you\nwould find it absolutely impossible to ascertain by any mechanical\nmeans whether the car were moving or not. All mechanical instruments\nbehave exactly the same, whether the car be standing still or in\nmotion.]24 [If you drop a ball you will see it fall to the floor in\na straight line, just as though you had dropped it while standing on\nthe station platform. Furthermore, if you drop the ball from the same\nheight in the two cases, and measure the velocities with which it\nstrikes the car floor and the station platform, or the times which\nit requires for the descent, you will find these identical in the\ntwo cases.]182\n\n[Any changes of speed or of direction (as when the car speeds up\nor slows down or rounds a curve) can be detected by observing the\nbehavior of bodies in the car, without apparent reference to any\noutside objects. This becomes particularly obvious with sudden\nirregularities of motion, which manifest themselves by shaking\neverything in the car. But a uniform motion in a straight line does\nnot reveal itself by any phenomenon within the vehicle.]24\n\n[Moreover, if we remove the veil from our window to the extent that\nwe may observe the train on the adjoining track, we shall be able to\nmake no decision as to whether we or it be moving. This is indeed an\nexperience which we have all had.]* [Often when seated in a train\nabout to leave the station, we have thought ourselves under way,\nonly to perceive as the motion becomes no longer uniform that another\ntrain has been backing into the station on the adjoining track. Again,\nas we were hurried on our journey, we have, raising suddenly our eyes,\nbeen puzzled to say whether the passing train were moving with us or\nagainst us or indeed standing still; or more rarely we have had the\nimpression that both it and we seemed to be at rest, when in truth\nboth were moving rapidly with the same speed.]82 [Even this phrase \"in\ntruth\" is a relative one, for it arises through using the earth as an\nabsolute reference body. We are indeed naive if we cannot appreciate\nthat there is no reason for doing this beyond convenience, and that\nto an observer detached from the earth it were just as reasonable\nto say that the rails are sliding under the train as that the train\nis advancing along the rails. One of my own most vivid childhood\nrecollections is of the terror with which, riding on a train that\npassed through a narrow cut, I hid my head in the maternal lap to shut\nout the horrid sight of the earth rushing past my window. The absence\nof a background in relatively slow retrograde motion was sufficient\nto prevent my consciousness from drawing the accustomed conclusion\nthat after all it was really the train that was moving.]*\n\n\n\n\nMECHANICAL RELATIVITY\n\n[So we can enunciate the following principle: When a body is in\nuniform rectilinear motion relatively to a second body, then all\nphenomena take place on the first in exactly the same manner as on\nthe second; the physical laws for the happenings on both bodies are\nidentical.]24 [And between a system of bodies, nothing but relative\nmotion may be detected by any mechanical means whatever; any attempt\nto discuss absolute motion presupposes a super-observer on some body\nexternal to the system. Even then, the \"absolute\" motion is nothing\nbut motion relative to this super-observer. By no mechanical means\nis uniform straight-line motion of any other than relative character\nto be detected. This is the Principle of Mechanical Relativity.\n\nThere is nothing new in this. It was known to Galileo, it was known\nto Newton, it has been known ever since. But the curious persistence\nof the human mind in habits of thought which confuse relativity with\nabsolutism brought about a state of affairs where we attempted to\nknow this and to ignore it at the same time. We shall have to return\nto the mathematical mode of reasoning to see how this happened. The\nmathematician has a way all his own of putting the statement of\nrelativity which we have made. He recalls, what we have already seen,\nthat the observer on the earth who is measuring his \"absolute\" motion\nwith respect to the earth has merely attached his reference framework\nto the earth; that the passenger in the train who measures all motion\nnaively with respect to his train is merely carrying his coordinate\naxes along with his baggage, instead of leaving them on the solid\nground; that the astronomer who deals with the motion of the earth\nabout the sun, or with that of the \"fixed\" stars against one another,\ndoes so simply by the artifice of hitching his frame of reference to\nthe sun or to one of the fixed stars. So the mathematician points out\nthat dispute as to which of two bodies is in motion comes right down\nto dispute as to which of two sets of coordinate axes is the better\none, the more nearly \"natural\" or \"absolute.\" He therefore phrases\nthe mechanical principle of relativity as follows:\n\nAmong all coordinate systems that are merely in uniform straight-line\nmotion to one another, no one occupies any position of unique natural\nadvantage; all such systems are equivalent for the investigation of\nnatural laws; all systems lead to the same laws and the same results.\n\nThe mathematician has thus removed the statement of relativity\nfrom its intimate association with the external observed phenomena,\nand transferred it to the observer and his reference frame. We must\neither accept the principle of relativity, or seek a set of coordinate\naxes that have been singled out by nature as an absolute reference\nframe. These axes must be in some way unique, so that when we refer\nphenomena to them, the laws of nature take a form of exceptional\nsimplicity not attained through reference to ordinary axes. Where\nshall we look for such a preferred coordinate system?]*\n\n\n\n\nTHE SEARCH FOR THE ABSOLUTE\n\n[Older theory clung to the belief that there was such a thing as\nabsolute motion in space.]197 [As the body of scientific law developed\nfrom the sixteenth century onward, the not unnatural hypothesis crept\nin, that these laws (that is to say, their mathematical formulations\nrather than their verbal statements) would reveal themselves in\nespecially simple forms, were it possible for experimenters to make\ntheir observations from some absolute standpoint; from an absolutely\nfixed position in space rather than from the moving earth.]264\n[Somewhere a set of coordinate axes incapable of motion was to be\nfound,]197 [a fixed set of axes for measuring absolute motion; and\nfor two hundred years the world of science strove to find it,]147 [in\nspite of what should have been assurance that it did not exist. But\nthe search failed, and gradually the universal applicability of the\nprinciple of relativity, so far as it concerned mechanical phenomena,\ngrew into general acceptance.]* [And after the development, by the\ngreat mathematicians of the eighteenth century, of Newton's laws of\nmotion into their most complete mathematical form, it was seen that so\nfar as these laws are concerned the absolutist hypothesis mentioned is\nquite unsupported. No complication is introduced into Newton's laws\nif the observer has to make his measurements in a frame of reference\nmoving uniformly through space; and for measurements in a frame like\nthe earth, which moves with changing speed and direction about the\nsun and rotates on its axis at the same time, the complication is\nnot of so decisive a nature as to give us any clue to the earth's\nabsolute motion in space.\n\nBut mechanics, albeit the oldest, is yet only one of the physical\nsciences. The great advance made in the mathematical formulation\nof optical and electromagnetic theory during the nineteenth century\nrevived the hope of discovering absolute motion in space by means of\nthe laws derived from this theory.]264 [Newton had supposed light\nto be a material emanation, and if it were so, its passage across\n\"empty space\" from sun and stars to the earth raised no problem. But\nagainst Newton's theory Huyghens, the Dutch astronomer, advanced the\nidea that light was a wave motion of some sort. During the Newtonian\nperiod and for many years after, the corpuscular theory prevailed;\nbut eventually the tables were turned.]* [Men made rays of light\ninterfere, producing darkness (see page 61). From this, and from\nother phenomena like polarization, they had deduced that light was a\nform of wave motion similar to water ripples; for these interfere,\nproducing level surfaces, or reinforce each other, producing waves\nof abnormal height. But if light were to be regarded as a form of\nwave motion--and the phenomena could apparently be explained on no\nother basis--then there must be some medium capable of undergoing\nthis form of motion.]135 [Transmission of waves across empty space\nwithout the aid of an intermediary material medium would be \"action\nat a distance,\" an idea repugnant to us. Trammeled by our tactual,\nwire-pulling conceptions of a material universe, we could not accustom\nourselves to the idea of something--even so immaterial a something\nas a wave--being transmitted by nothing. We needed a word--ether--to\ncarry light if not to shed it; just as we need a word--inertia--to\ncarry a projectile in its flight.]231 [It was necessary to invest this\nmedium with properties to account for the observed facts. On the whole\nit was regarded as the perfect fluid.]235 [The ether was imagined as\nan all-pervading, imponderable substance filling the vast emptiness\nthrough which light reaches us, and as well the intermolecular spaces\nof all matter. Nothing more was known definitely, yet this much served\nas a good working hypothesis on the basis of which Maxwell was enabled\nto predict the possibility of radio communication. By its fruits the\nether hypothesis justified itself; but does the ether exist?]231\n\n\n\n\nTHE ETHER AND ABSOLUTE MOTION\n\n[If it does exist, it seems quite necessary, on mere philosophical\ngrounds, that it shall be eligible to serve as the long-sought\nreference frame for absolute motion. Surely it does not make sense to\nspeak of a homogeneous medium filling all space, sufficiently material\nto serve as a means of communication between remote worlds, and in\nthe next breath to deny that motion with respect to this medium is a\nconcept of significance.]* [Such a system of reference as was offered\nby the ether, coextensive with the entire known region of the universe,\nmust necessarily serve for all motions within our perceptions.]186\n[The conclusion seems inescapable that motion with respect to the\nether ought to be of a sufficiently unique character to stand out\nabove all other motion. In particular, we ought to be able to use the\nether to define, somewhere, a system of axes fixed with respect to\nthe ether, the use of which would lead to natural laws of a uniquely\nsimply description.\n\nMaxwell's work added fuel to this hope.]* [During the last century,\nafter the units of electricity had been defined, one set for static\nelectrical calculations and one for electromagnetic calculations, it\nwas found that the ratio of the metric units of capacity for the two\nsystems was numerically equal to what had already been found as the\nvelocity with which light is transmitted through the hypothetical\nether. One definition refers to electricity at rest, the other\nto electricity in motion. Maxwell, with little more working basis\nthan this, undertook to prove that electrical and optical phenomena\nwere merely two aspects of a common cause,]235 [to which the general\ndesignation of \"electromagnetic waves\" was applied. Maxwell treated\nthis topic in great fullness and with complete success. In particular,\nhe derived certain equations giving the relations between the various\nelectrical quantities involved in a given phenomenon. But it was\nfound, extraordinarily enough, that these relations were of such\ncharacter that, when we subject the quantities involved to a change\nof coordinate axes, the transformed quantities did not preserve these\nrelations if the new axes happened to be in motion with respect to\nthe original ones. This, of course, was taken to indicate that motion\nreally is absolute when we come to deal with electromagnetic phenomena,\nand that the ether which carries the electromagnetic waves really may\nbe looked to to display the properties of an absolute reference frame.\n\nReference to the phenomenon of aberration, which Dr. Pickering has\ndiscussed adequately in his essay and which I need therefore mention\nhere only by name, indicated that the ether was not dragged along\nby material bodies over and through which it might pass. It seemed\nthat it must filter through such bodies, presumably via the molecular\ninterstices, without appreciable opposition. Were this not the case,\nwe should be in some doubt as to the possibility of observing the\nvelocity through the ether of material bodies; if the ether adjacent\nto such bodies is not dragged along or thrown into eddies, but \"stands\nstill\" while the bodies pass, there seems no imaginable reason for\nanything other than the complete success of such observations. And\nof course these are of the utmost importance, the moment we assign\nto the ether the r\u00f4le of absolute reference frame.\n\n\n\n\nTHE EARTH AND THE ETHER\n\nOne body in motion with respect to the ether is our earth itself. We\ndo not know in advance in what direction to expect this motion or what\nmagnitude to anticipate that it will have. But one thing is clear.]*\n[In its motion around the sun, the earth has, at opposite points on its\norbit, a difference in velocity with respect to the surrounding medium\nwhich is double its orbital velocity with respect to the sun. This\ndifference comes to 37 miles per second. The earth should therefore,\nat some time in the year, show a velocity equal to or greater than\n18 1/2 miles per second, with reference to the universal medium. The\nfamous Michelson-Morley experiment of 1887 was carried out with the\nexpectation of observing this velocity.]267\n\n[The ether, of course, and hence velocities through it, cannot be\nobserved directly. But it acts as the medium for the transmission of\nlight.]* [If the velocity of light through the ether is $C$ and that\nof the earth through the ether is $v$, then the velocity of light\npast the earth, so the argument runs, must vary from $C-v$ to $C+v$,\naccording as the light is moving exactly in the same direction as the\nearth, or in the opposite direction,]182 [or diagonally across the\nearth's path so as to get the influence only of a part of the earth's\nmotion. This of course assumes that $C$ has always the same value;\nan assumption that impresses one as inherently probable, and one that\nis at the same time in accord with ordinary astronomical observation.\n\nIt is not possible to measure directly the velocity of light (186,330\nmiles per second, more or less) with sufficient accuracy to give any\nmeaning to the variation in this velocity which might be effected by\nadding or subtracting that of the earth in its orbit (a mere 18 1/2\nmiles per second). It is, however, possible to play a trick on the\nlight by sending it back and forth over several paths, and comparing\n(not measuring absolutely, but merely comparing) with great minuteness\nthe times consumed in these several round trips.\n\n\n\n\nA JOURNEY UPSTREAM AND BACK\n\nThe number of letters the Scientific American has received questioning\nthe Michelson-Morley experiment indicates that many people are not\nacquainted with the fundamental principle on which it is based. So\nlet us look at a simple analogous case. Suppose a swimmer or a rower\nmake a return trip upstream and down, contending with the current\nas he goes up and getting its benefit when he comes down. Obviously,\nsays snap judgment, since the two legs of the journey are equal, he\nderives exactly as much benefit from the current when he goes with it\nas he suffers handicap from it when he goes against it; so the round\ntrip must take exactly the same time as a journey of the same length\nin still water, the argument applying equally in the case where the\n\"swimmer\" is a wave of light in the ether stream.\n\nBut let us look now at a numerical case. A man can row in still water\nat four miles per hour. He rows twelve miles upstream and back, in a\ncurrent of two miles per hour. At a net speed of two miles per hour\nhe arrives at his turning point in six hours. At a net speed of six\nmiles per hour he makes the down-stream leg in two hours. The elapsed\ntime for the journey is eight hours; in still water he would row the\ntwenty-four miles in six hours.\n\nIf we were to attempt an explanation of this result in words we should\nsay that by virtue of the very fact that it does delay him, the adverse\ncurrent prolongs the time during which it operates; while by virtue of\nthe very fact that it accelerates his progress, the favoring current\nshortens its venue. The careless observer realizes that distances are\nequal between the two legs of the journey, and unconsciously assumes\nthat times are equal.\n\nIf the journey be made directly with and directly against the stream\nof water or ether or what not, retardation is effected to its fullest\nextent. If the course be a diagonal one, retardation is felt to an\nextent measurable as a component, and depending for its exact value\nupon the exact angle of the path. Felt, however, it must always be.\n\nHere is where we begin to get a grip on the problem of the earth\nand the ether. In any problem involving the return-trip principle,\nthere will enter two velocities--that of the swimmer and that of the\nmedium; and the time of retardation. If we know any two of these items\nwe can calculate the third. When the swimmer is a ray of light and the\nvelocity of the medium is that of the ether as it flows past the earth,\nwe know the first of these two; we hope to observe the retardation\nso that we may calculate the second velocity. The apparatus for the\nexperiment is ingenious and demands description.\n\n\n\n\nTHE MICHELSON-MORLEY EXPERIMENT\n\nThe machine is of structural steel, weighing 1,900 pounds. It has two\narms which form a Greek cross. Each arm is 14 feet in length. The whole\napparatus is floated in a trough containing 800 pounds of mercury.\n\nFour mirrors are arranged on the end of each arm, sixteen in all,\nwith a seventeenth mirror, M, set at one of the inside corners of the\ncross, as diagrammed. A source of light (in this case a calcium flame)\nis provided, and its rays directed by a lens toward the mirror M. Part\nof the light is allowed to pass straight through M to the opposite\narm of the cross, where it strikes mirror 1. It is reflected back\nacross the arm to mirror 2, thence to 3, and so on until it reaches\nmirror 8. Thence it is reflected back to mirror 7, to 6, and so on,\nretracing its former path, and finally is caught by the reverse side\nof the mirror M and is sent to an observer at O. In retracing its\npath the light sets up an interference phenomenon (see below) and the\ninterference bands are visible to the observer, who is provided with\na telescope to magnify the results.\n\nA second part of the original light-beam is reflected off at right\nangles by the mirror M, and is passed to and fro on the adjacent arms\nof the machine, in exactly the same manner and over a similar path,\nby means of the mirrors I, II, III, ... VIII. This light finally\nreaches the observer at the telescope, setting up a second set of\ninterference bands, parallel to the first.\n\nA word now about this business of light interference. Light is a wave\nmotion. The length of a wave is but a few millionths of an inch,\nand the amplitude is correspondingly minute; but none the less,\nthese waves behave in a thoroughly wave-like manner. In particular,\nif the crests of two waves are superposed, there is a double effect;\nwhile if a crest of one wave falls with a trough of another, there\nis a killing-off or \"interference\".\n\nUnder ordinary circumstances interference of light waves does not\noccur. This is simply because under ordinary circumstances light waves\nare not piled up on one another. But sometimes this piling up occurs;\nand then, just so sure as the piled-up waves are in the same phase\nthey reinforce one another, while if they are in opposite phase they\ninterfere. And the conditions which we have outlined above, with the\ntelescope and the mirrors and the ray of light retracing the path\nover which it went out, are conditions under which interference does\noccur. If the returning wave is in exact phase with the outgoing one,\nthe effect is that of uniform double illumination; if it is in exactly\nopposite phase the effect is that of complete extinguishing of the\nlight, the reversed wave exactly cancelling out the original one. If\nthe two rays are partly in phase, there is partial reinforcement or\npartial cancelling out, according to whether they are nearly in phase\nor nearly out of phase. Finally, if the mirrors are not set absolutely\nparallel--as must in practice be the case when we attempt to measure\ntheir parallelism in terms of the wave-length of light--adjacent\nparts of the light ray will vary in the extent to which they are out\nof phase, since they will have travelled a fraction of a wave-length\nfurther to get to and from this, that or the other mirror. There\nwill then appear in the telescope alternate bands of illumination\nand darkness, whose width and spacing depend upon all the factors\nentering into the problem.\n\nIf it were possible for us to make the apparatus with such a degree\nof refinement that the path from mirror M via mirrors 1, 2, 3, etc.,\nback through M and into the telescope, were exactly the same length\nas that from flame to telescope by way of the mirrors I, II, III,\netc.--exactly the same to a margin of error materially less than a\nsingle wave-length of light--why, then, the two sets of interference\nfringes would come out exactly superposed provided the motion of\nthe earth through the \"ether\" turn out to have no influence upon the\nvelocity of light; or, if such influence exist, these fringes would\nbe displaced from one another to an extent measuring the influence in\nquestion. But our ability to set up this complicated pattern of mirrors\nat predetermined distances falls far short of the wave-length as a\nmeasure of error. So in practice all that we can say is that having\nonce set the instrument up, and passed a beam of light through it,\nthere will be produced two sets of parallel interference fringes. These\nsets will fail of superposition--each fringe of one set will be removed\nfrom the corresponding fringe of the other set--by some definite\ndistance. Then, any subsequent variation in the speed of light along\nthe two arms will at once be detected by a shifting of the interference\nbands through a distance which we shall be able to measure.\n\n\n\n\nTHE VERDICT\n\nUnder the theories and assumptions governing at the time of the\noriginal performance of this experiment, it will be readily seen that\nif this machine be set up in an \"ether stream\" with one arm parallel\nto the direction of the stream and the other at right angles thereto,\nthere will be a difference in the speed of the light along the two\narms. Then if the apparatus be shifted to a position oblique to the\nether stream, the excess velocity of the light in the one arm would\nbe diminished, and gradually come to zero at the 45-degree angle,\nafter which the light traveling along the other arm would assume the\ngreater speed. In making observations, therefore, the entire apparatus\nwas slowly rotated, the observers walking with it, so that changes\nof the sort anticipated would be observed.\n\nThe investigators were, however, ignorant of the position in which the\napparatus ought to be set to insure that one of the arms lie across\nthe ether drift; and they were ignorant of the time of year at which\nthe earth's maximum velocity through the ether was to be looked for. In\nparticular, it is plain that if the solar system as a whole is moving\nthrough the ether at a rate less than the earth's orbital velocity,\nthere is a point in our orbit where our velocity through the ether\nand that around the sun just cancel out and leave us temporarily in\na state of \"absolute rest.\" So it was anticipated that the experiment\nmight have to be repeated in many orientations of the machine and at\nmany seasons of the year in order to give a series of readings from\nwhich the true motion of the earth through the ether might be deduced.\n\nFor those who have a little algebra the demonstration which Dr. Russell\ngives on a subsequent page will be interesting as showing the\nsituation in perfectly general terms. It will be realized that the\nmore complicated arrangement of mirrors in the experiment as just\ndescribed is simply an eightfold repetition of the simple experiment\nas outlined by Dr. Russell, and that it was done so for the mere\nsake of multiplying by eight the distances travelled and hence the\ndifference in time and in phase.\n\nAnd now for the grand climax. The experiment was repeated many times,\nwith the original and with other apparatus, indoors and outdoors, at\nall seasons of the year, with variation of every condition that could\nimaginably affect the result. The apparatus was ordinarily such that a\nshift in the fringes of anywhere from one-tenth to one one-hundredth\nof that which would have followed from any reasonable value for\nthe earth's motion through the ether would have been systematically\napparent. The result was uniformly negative. At all times and in all\ndirections the velocity of light past the earth-bound observer was\nthe same. The earth has no motion with reference to the ether!\n\n[The amazing character of this result is not by any possibility to\nbe exaggerated.]* [According to one experiment the ether was carried\nalong by a rapidly moving body and according to another equally\nwell-planned and well-executed experiment a rapidly moving body did\nnot disturb the ether at all. This was the blind alley into which\nscience had been led.]232\n\n\n\n\nTHE \"CONTRACTION\" HYPOTHESIS\n\n[Numerous efforts were made to explain the contradiction.]* [It\nis indeed a very puzzling one, and it gave physicists no end of\ntrouble. However Lorentz and Fitzgerald finally put forward an\ningenious explanation, to the effect that the actual motion of\nthe earth through the ether is balanced, as far as the ability of\nour measuring instruments is concerned, by a contraction of these\nsame instruments in the direction of their motion. This contraction\nobviously cannot be observed directly because all bodies, including\nthe measuring instruments themselves (which after all are only\narbitrary guides), will suffer the contraction equally. According to\nthis theory, called the Lorentz-Fitzgerald contraction theory,]272\n[all bodies in motion suffer such contraction of their length in the\ndirection of their motion;]283 [the contraction being made evident\nby our inability to observe the absolute motion of the earth, which\nit is assumed must exist.]272 [This would suffice to show why the\nMichelson-Morley experiment gave a negative result, and would preserve\nthe concept of absolute motion with reference to the ether.]283\n\n[This proposal of Lorentz and Fitzgerald loses its startling aspect\nwhen we consider that all matter appears to be an electrical structure,\nand that the dimensions of the electric and magnetic fields which\naccompany the electrons of which it is constituted change with the\nvelocity of motion.]267 [The forces of cohesion which determine the\nform of a rigid body are held to be electromagnetic in nature; the\ncontraction may be regarded as due to a change in the electromagnetic\nforces between the molecules.]10 [As one writer has put it, the\norientation, in the electromagnetic medium, of a body depending for\nits very existence upon electromagnetic forces is not necessarily a\nmatter of indifference.]*\n\n[Granting the plausibility of all this, on the basis of an\nelectromagnetic theory of matter, it leaves us in an unsatisfactory\nposition. We are left with a fixed ether with reference to which\nabsolute motion has a meaning, but that motion remains undetected and\napparently undetectable. Further, if we on shore measure the length\nof a moving ship, using a yard-stick which is stationary on shore,\nwe shall obtain one result. If we take our stick aboard it contracts,\nand so we obtain a greater length for the ship. Not knowing our\n\"real\" motion through the ether, we cannot say which is the \"true\"\nlength. Is it not, then, more satisfactory to discard all notion\nof true length as an inherent quality of bodies, and, by regarding\nlength as the measure of a relation between a particular object and\na particular observer, to make one length as true as the other?]182\n[The opponents of such a viewpoint contend that Michelson's result\nwas due to a fluke; some mysterious counterbalancing influence was\nfor some reason at work, concealing the result which should normally\nhave been expected. Einstein refuses to accept this explanation;]192\n[he refuses to believe that all nature is in a contemptible conspiracy\nto delude us.]*\n\n[The Fitzgerald suggestion is further unsatisfactory because it\nassumes all substances, of whatever density, to undergo the same\ncontraction; and above all for the reason that it sheds no light upon\nother phenomena.]194 [It is indeed a very special explanation; that\nis, it applies only to the particular experiment in question. And\nindeed it is only one of many possible explanations. Einstein\nconceived the notion that it might be infinitely more valuable to\ntake the most general explanation possible, and then try to find\nfrom this its logical consequences. This \"most general explanation\"\nis, of course, simply that it is impossible in any way whatever to\nmeasure the absolute motion of a body in space.]272 [Accordingly\nEinstein enunciated, first the Special Theory of Relativity, and\nlater the General Theory of Relativity. The special theory was so\ncalled because it was, limited to uniform rectilinear and non-rotary\nmotions. The general theory, on the other hand, dealt not only with\nuniform rectilinear motions, but with any arbitrary motion whatever.\n\n\n\n\nTAKING THE BULL BY THE HORNS\n\nThe hypothesis of relativity asserts that there can be no such concept\nas absolute position, absolute motion, absolute time; that space and\ntime are inter-dependent, not independent; that everything is relative\nto something else. It thus accords with the philosophical notion of\nthe relativity of all knowledge.]283 [Knowledge is based, ultimately,\nupon measurement; and clearly all measurement is relative, consisting\nmerely in the application of a standard to the magnitude measured. All\nmetric numbers are relative; dividing the unit multiplies the metric\nnumber. Moreover, if measure and measured change proportionately,\nthe measuring number is unchanged. Should space with all its contents\nswell in fixed ratio throughout, no measurement could detect this;\nnor even should it pulse uniformly throughout. Furthermore, were\nspace and space-contents in any way systematically transformed (as by\nreflection in curved mirrors) point for point, continuously, without\nrending, no measurement could reveal this distortion; experience\nwould proceed undisturbed.]263\n\n[Mark Twain said that the street in Damascus \"which is called\nstraight,\" is so called because while it is not as straight as a\nrainbow it is straighter than a corkscrew. This expresses the basic\nidea of relativity--the idea of comparison. All our knowledge is\nrelative, not absolute. Things are big or little, long or short,\nlight or heavy, fast or slow, only by comparison. An atom may be\nas large, compared to an electron, as is a cathedral compared to\na fly. The relativity theory of Einstein emphasizes two cases of\nrelative knowledge; our knowledge of time and space, and our knowledge\nof motion.]216 [And in each case, instead of allowing the notions of\nrelativity to guide us only so far as it pleases us to follow them,\nthere abandoning them for ideas more in accord with what we find\nit easy to take for granted, Einstein builds his structure on the\nthesis that relativity must be admitted, must be followed out to the\nbitter end, in spite of anything that it may do to our preconceived\nnotions. If relativity is to be admitted at all, it must be admitted\nin toto; no matter what else it contradicts, we have no appeal from\nits conclusions so long as it refrains from contradicting itself.]*\n\n[The hypothesis of relativity was developed by Einstein through\na priori methods, not the more usual a posteriori ones. That is,\ncertain principles were enunciated as probably true, the consequences\nof these were developed, and these deductions tested by comparison of\nthe predicted and the observed phenomena. It was in no sense attained\nby the more usual procedure of observing groups of phenomena and\nformulating a law or formula which would embrace them and correctly\ndescribe the routine or sequence of phenomena.\n\nThe first principle thus enunciated is that it is impossible to\nmeasure or detect absolute translatory motion through space, under\nany circumstances or by any means. The second is that the velocity of\nlight in free space appears the same to all observers regardless of\nthe relative motion of the source of light and the observer. This\nvelocity is not affected by motion of the source toward or away\nfrom the observer,]283 [if we may for the moment use this expression\nwith its implication of absolute motion.]* [But universal relativity\ninsists that motion of the source toward the observer is identical\nwith motion of the observer toward the source.]283\n\n[It will be seen that we are at once on the horns of a dilemma. Either\nwe must give up relativity before we get fairly started on it, or we\nmust overturn the foundations of common sense by admitting that time\nand space are so constituted that when we go to meet an advancing\nlight-impulse, or when we retreat from it, it still reaches us with\nthe same velocity as though we stood still waiting for it. We shall\nfind when we are through with our investigation that common sense is\nat fault; that our fixed impression of the absurdity of the state of\naffairs just outlined springs from a confusion between relativism\nand absolutism which has heretofore dominated our thought and gone\nunquestioned. The impression of absurdity will vanish when we have\nresolved this confusion.]*\n\n\n\n\nQUESTIONS OF COMMON SENSE\n\n[But it is obvious from what has just been said that if we are to\nadopt Einstein's theory, we must make very radical changes in some of\nour fundamental notions, changes that seem in violent conflict with\ncommon sense. It is unfortunate that many popularizers of relativity\nhave been more concerned to astonish their readers with incredible\nparadoxes than to give an account such as would appeal to sound\njudgment. Many of these paradoxes do not belong essentially to the\ntheory at all. There is nothing in the latter that an enlarged and\nenlightened common sense would not readily endorse. But common sense\nmust be educated up to the necessary level.]141\n\n[There was a time when it was believed, as a result of centuries of\nexperience, that the world was flat. This belief checked up with the\nknown facts, and it could be used as the basis for a system of science\nwhich would account for things that had happened and that were to\nhappen. It was entirely sufficient for the time in which it prevailed.\n\nThen one day a man arose to point out that all the known facts were\nequally accounted for on the theory that the earth was a sphere. It\nwas in order for his contemporaries to admit this, to say that so far\nas the facts in hand were concerned they could not tell whether the\nearth was flat or round--that new facts would have to be sought that\nwould contradict one or the other hypothesis. Instead of this the\nworld laughed and insisted that the earth could not be round because\nit was flat; that it could not be round because then the people would\nfall off the other side.\n\nBut the field of experimentation widened, and men were able to observe\nfacts that had been hidden from them. Presently a man sailed west and\narrived east; and it became clear that in spite of previously accepted\n\"facts\" to the contrary, the earth was really round. The previously\naccepted \"facts\" were then revised to fit the newly discovered truth;\nand finally a new system of science came into being, which accounted\nfor all the old facts and all the new ones.\n\nAt intervals this sort of thing has been repeated. A Galileo shows\nthat preconceived ideas with regard to the heavens are wrong, and\nmust be revised to accord with his newly promulgated principles. A\nNewton does the same for physics--and people unlearn the \"fact\"\nthat motion has to be supported by continued application of force,\nsubstituting the new idea that it actually requires force to stop a\nmoving body. A Harvey shows that the things which have been \"known\"\nfor generations about the human body are not so. A Lyell and a Darwin\nforce men to throw overboard the things they have always believed about\nthe way in which the earth and its creatures came into being. Every\nscience we possess has passed through one or more of these periods\nof readjustment to new facts.\n\n\n\n\nSHIFTING THE MENTAL GEARS\n\nNow we are apt to lose sight of the true significance of this. It\nis not alone our opinions that are altered; it is our fundamental\nconcepts. We get concepts wholly from our perceptions, making them\nto fit those perceptions. Whenever a new vista is opened to our\nperceptions, we find facts that we never could have suspected from the\nrestricted viewpoint. We must then actually alter our concepts to make\nthe new facts fit in with the greatest degree of harmony. And we must\nnot hesitate to undertake this alteration, through any feeling that\nfundamental concepts are more sacred and less freely to be tampered\nwith than derived facts.]* [We do, to be sure, want fundamental\nconcepts that are easy for a human mind to conceive; but we also\nwant our laws of nature to be simple. If the laws begin to become,\nintricate, why not reshape, somewhat, the fundamental concepts, in\norder to simplify the scientific laws? Ultimately it is the simplicity\nof the scientific system as a whole that is our principal aim.]178\n\n[As a fair example, see what the acceptance of the earth's sphericity\ndid to the idea represented by the word \"down.\" With a flat earth,\n\"down\" is a single direction, the same throughout the universe; with\na round earth, \"down\" becomes merely the direction leading toward\nthe center of the particular heavenly body on which we happen to be\nlocated. It is so with every concept we have. No matter how intrinsic\na part of nature and of our being a certain notion may seem, we can\nnever know that new facts will not develop which will show it to be a\nmistaken one. Today we are merely confronted by a gigantic example of\nthis sort of thing. Einstein tells us that when velocities are attained\nwhich have just now come within the range of our close investigation,\nextraordinary things happen--things quite irreconcilable with our\npresent concepts of time and space and mass and dimension. We are\ntempted to laugh at him, to tell him that the phenomena he suggests\nare absurd because they contradict these concepts. Nothing could be\nmore rash than this.\n\nWhen we consider the results which follow from physical velocities\ncomparable with that of light, we must confess that here are\nconditions which have never before been carefully investigated. We\nmust be quite as well prepared to have these conditions reveal some\nepoch-making fact as was Galileo when he turned the first telescope\nupon the skies. And if this fact requires that we discard present\nideas of time and space and mass and dimension, we must be prepared\nto do so quite as thoroughly as our medieval fathers had to discard\ntheir notions of celestial \"perfection\" which demanded that there\nbe but seven major heavenly bodies and that everything center about\nthe earth as a common universal hub. We must be prepared to revise\nour concepts of these or any other fundamentals quite as severely as\ndid the first philosopher who realized that \"down\" in London was not\nparallel to \"down\" in Bagdad or on Mars.]*\n\n[In all ordinary terrestrial matters we take the earth as a fixed\nbody, light as instantaneous. This is perfectly proper, for such\nmatters. But we carry our earth-acquired habits with us into the\ncelestial regions. Though we have no longer the earth to stand on,\nyet we assume, as on the earth, that all measurements and movements\nmust be referred to some fixed body, and are only then valid. We cling\nto our earth-bound notion that there is an absolute up-and-down,\nback-and-forth, right-and-left, in space. We may admit that we can\nnever find it, but we still think it is there, and seek to approach\nit as nearly as possible. And similarly from our earth experiences,\nwhich are sufficiently in a single place to make possible this\nsimplifying assumption, we get the idea that there is one universal\ntime, applicable at once to the entire universe.]141 [The difficulty\nin accepting Einstein is entirely the difficulty in getting away from\nthese earth-bound habits of thought.]*\n\n\n\n\n\n\n\n\n\nIV\n\nTHE SPECIAL THEORY OF RELATIVITY\n\nWhat Einstein's Study of Uniform Motion Tells Us About Time and Space\nand the Nature of the External Reality\n\nBY VARIOUS CONTRIBUTORS AND THE EDITOR\n\n\nWhatever the explanation adopted for the negative result of the\nMichelson-Morley experiment, one thing stands out clearly: the attempt\nto isolate absolute motion has again failed.]* [Einstein generalizes\nthis with all the other and older negative results of similar sort into\na negative deduction to the effect that no experiment is possible upon\ntwo systems which will determine that one of them is in motion and\nthe other at rest.]121 [He elevates the repeated failure to detect\nabsolute motion through space into the principle that experiment\nwill never reveal anything in the nature of absolute velocities. He\npostulates that all laws of nature can and should be enunciated in\nsuch forms that they are as true in these forms for one observer\nas for another, even though these observers with their frames of\nreference be in motion relative to one another.]264\n\n[There are various ways of stating the principle of the relativity\nof uniform motion which has been thus arrived at, and which forms\nthe basis of the Special Theory of Einstein. If we care to emphasize\nthe r\u00f4le of mathematics and the reference frame we may say that]*\n[any coordinate system having a uniform rectilinear motion with\nrespect to the bodies under observation may be interchangeably used\nwith any other such system in describing their motions;]232 [or that\nthe unaccelerated motion of a system of reference cannot be detected\nby observations made on this system alone.]194 [Or we can let this\naspect of the matter go, and state the relativity postulate in a form\nmore intelligible to the non-mathematician by simply insisting that\nit is impossible by any means whatever to distinguish any other than\nthe relative motion between two systems that are moving uniformly. As\nDr. Russell puts it on a later page, we can assume boldly that the\nuniverse is so constituted that uniform straight-ahead motion of an\nobserver and all his apparatus will not produce any difference whatever\nin the result of any physical process or experiment of any kind.\n\nAs we have seen, this is entirely reasonable, on philosophical grounds,\nuntil we come to consider the assumptions of the past century with\nregard to light and its propagation. On the basis of these assumptions\nwe had expected the Michelson-Morley experiment to produce a result\nnegativing the notion of universal relativity. It refused to do this,\nand we agree with Einstein that the best explanation is to return\nto the notion of relativity, rather than to invent a forced and\nspecial hypothesis to account for the experiment's failure. But we\nmust now investigate the assumptions underlying the theory of light,\nand remove the one that requires the ether to serve as a universal\nstandard of absolute motion.\n\n\n\n\nLIGHT AND THE ETHER\n\nIt is among the possibilities that the wave theory of light itself\nwill in the end be more or less seriously modified. It is even more\ndefinitely among the possibilities that the ether will be discarded.]*\n[Certainly when Lord Kelvin estimates that its mass per cubic\ncentimeter is .000,000,000,000,000,001 gram, while Sir Oliver Lodge\ninsists that the correct figure is 1,000,000,000,000,000 grams, it is\nquite evident that we know so little about it that it is better to\nget along without it if we can.]216 [But to avoid confusion we must\nemphasize that Einstein makes no mention whatsoever of the ether;\nhis theory is absolutely independent of any theory of the ether.]139\n[Save as he forbids us to employ the ether as a standard of absolute\nmotion, Einstein does not in the least care what qualities we assign\nto it, or whether we retain it at all. His demands are going to be made\nupon light itself, not upon the alleged medium of light transmission.\n\nWhen two observers in relative motion to one another measure their\nvelocities with respect to a third material object, they expect to get\ndifferent results. Their velocities with regard to this object properly\ndiffer, for it is no more to be taken as a universal super-observer\nthan either of them. But if they get different results when they come\nto measure the velocity with which light passes their respective\nsystems, relativity is challenged. Light is with some propriety\nto be regarded as a universal observer; and if it will measure our\nvelocities against each other we cannot deny it rank as an absolute\nstandard. If we are not prepared to abandon universal relativity,\nand adopt one of the \"fluke\" explanations for the Michelson-Morley\nresult, we must boldly postulate that in free space light presents the\nsame velocity $C$ to all observers--whatever the source of the light,\nwhatever the relative motion between source and observer, whatever the\nrelative motion between the several observers. The departure here from\nthe old assumption lies in the circumstance that the old physics with\nits ether assigned to light a velocity universally constant in this\nether; we have stopped talking about the medium and have made the\nconstant $C$ refer to the observer's measured value of the velocity\nof light with regard to himself.\n\nWe are fortified in this assumption by the Michelson-Morley\nresult and by all other observations bearing directly upon the\nmatter. Nevertheless, as Mr. Francis says in his essay, we feel\ninstinctively that space and time are not so constituted as to\nmake it possible, if I pass you at 100 miles per hour, for the\nsame light-impulse to pass us both at the same speed $C$.]* [The\nimplicit assumptions underlying this feeling, be they true or false,\nare now so interwoven with the commonly received notions of space and\ntime that any theory which questions them has all the appearance of\na fantastic and unthinkable thing.]115 [We cannot, however, go back\non our relativity; so when]* [Einstein shows us that an entirely new\nset of time and space concepts is necessary to reconcile universe\nrelativity with this fundamental fact of the absolute constancy of\nthe observed velocity of light in vacuo,]18 [all that is left for us\nto do is to inquire what revisions are necessary, and submit to them.]*\n\n[The conceptual difficulties of the theory arise principally from\nattributing to space and time the properties of things. No portion of\nspace can be compared with another, save by convention; it is things\nwhich we compare. No interval of time can be compared with another,\nsave by convention. The first has gone when the second becomes\n\"now\".]149 [It is events that we compare, through the intervention\nof things. Our measurements are never of space or of time, but only\nof the things and the events that occupy space and time. And since\nthe measurements which we deal with as though they were of space and\nof time lie at the foundation of all physical science, while at the\nsame time themselves constituting, as we have seen, the only reality\nof which we are entitled to speak, it is in order to examine with\nthe utmost care the assumptions underlying them. That there are such\nassumptions is clear--the very possibility of making measurements is\nitself an assumption, and every technique for carrying them out rests\non an assumption. Let us inquire which of these it is that relativity\nasks us to revise.]*\n\n\n\n\nTHE MEASUREMENT OF TIME AND SPACE\n\n[Time is generally conceived as perfectly uniform. How do we judge\nabout it? What tells us that the second just elapsed is equal to the\none following? By the very nature of time the superposition of its\nsuccessive intervals is impossible. How then can we talk about the\nrelative duration of these intervals? It is clear that any relationship\nbetween them can only be conventional.]178 [As a matter of fact, we\nhabitually measure time in terms of moving bodies. The simplest method\nis to agree that some entity moves with uniform velocity. It will be\nconsidered as travelling equal distances in equal intervals of time,\nthe distances to be measured as may be specified by our assumptions\ngoverning this department of investigation.]179 [The motions of the\nearth through which we ultimately define the length of day and year,\nthe division of the former into 86,400 \"equal\" intervals as defined by\nthe motions of pendulum or balance wheel through equal distances, are\nexamples of this convention of time measurement. Even when we correct\nthe motions of the earth, on the basis of what our clocks tell us of\nthese motions, we are following this lead; the earth and the clocks\nfall out, it is plain that one of them does not satisfy our assumption\nof equal lengths in equal times, and we decide to believe the clock.]*\n\n[The foregoing concerning time may be accepted as inherent in\ntime itself. But concerning lengths it may be thought that we\nare able to verify absolutely their equality and especially their\ninvariability. Let us have the audacity to verify this statement. We\nhave two lengths, in the shape of two rods, which coincide perfectly\nwhen brought together. What may we conclude from this coincidence? Only\nthat the two rods so considered have equal lengths at the same\nplace in space and at the same moment. It may very well be that each\nrod has a different length at different locations in space and at\ndifferent times; that their equality is purely a local matter. Such\nchanges could never be detected if they affected all objects in the\nuniverse. We cannot even ascertain that both rods remain straight\nwhen we transport them to another location, for both can very well\ntake the same curvature and we shall have no means of detecting it.\n\nEuclidean geometry assumes that geometrical objects have sizes and\nshapes independent of position and of orientation in space, and\nequally invariable in time. But the properties thus presupposed are\nonly conventional and in no way subject to direct verification. We\ncannot even ascertain space to be independent of time, because when\ncomparing geometrical objects we have to conceive them as brought to\nthe same place in space and in time.]178 [Even the statement that\nwhen they are made to coincide their lengths are equal is, after\nall, itself an assumption inherent in our ideas of what constitutes\nlength. And certainly the notion that we can shift them from place\nto place and from moment to moment, for purposes of comparison, is\nan assumption; even Euclid, loose as he was from modern standards\nin this business of \"axioms,\" knew this and included a superposition\naxiom among his assumptions.\n\nAs a matter of fact, this procedure for determining equality of lengths\nis not always available. It assumes, it will be noted, that we have\nfree access to the object which is to be measured--which is to say,\nit assumes that this object is at rest with respect to us. If it\nis not so at rest, we must employ at least a modification of this\nmethod; a modification that will in some manner involve the sending\nof signals. Even when we employ the Euclidean method of superposition\ndirectly, we must be assured that the respective ends of the lengths\nunder comparison coincide at the same time. The observer cannot be\npresent at both ends simultaneously; at best he can only be present\nat one end and receive a signal from the other end.\n\n\n\n\nTHE PROBLEM OF COMMUNICATION\n\nAccordingly, in making the necessary assumptions to cover the matter\nof measuring lengths, we must make one with regard to the character\nof the signals which are to be employed for this purpose. If we\ncould assume a system of signalling that would consume no time in\ntransmission all would be simple enough. But we have no experience with\nsuch a system. Even if we believe that it ought to be possible thus\nto transmit signals at infinite velocity, we may not, in the absence\nof our present ability to do this, assume that it is possible. So we\nmay only assume, with Einstein, that for our signals we shall employ\nthe speediest messenger with which we are at present acquainted. This\nof course is light, the term including any of the electromagnetic\nimpulses that travel at the speed $C$.\n\nOf course in the vast majority of cases the distance that any light\nsignal in which we are interested must go to reach us is so small that\nthe time taken by its transmission can by no means be measured. We\nare then, to all intents and purposes, at both places--the point of\norigin of the signal and the point of receipt--simultaneously. But\nthis is not the question at all. Waiving the fact that in astronomical\ninvestigations this approximation no longer holds, the fact remains\nthat it is, in every case, merely an approximation. Approximations are\nall right in observations, where we know that they are approximations\nand act accordingly. But in the conceptual universe that parallels the\nexternal reality, computation is as good an agent of observation as\nvisual or auditory or tactile sensation; if we can compute the error\ninvolved in a wrong procedure the error is there, regardless of whether\nwe can see it or not. We must have methods which are conceptually\nfree from error; and if we attempt to ignore the velocity of our\nlight signals we do not meet this condition.\n\nThe measurement of lengths demands that we have a criterion of\nsimultaneity between two remote points--remote in inches or remote\nin light-years, it does not matter which. There is no difficulty in\ndefining simultaneity of two events that fall in the same point--or\nrather, in agreeing that we know what we mean by such simultaneity. But\nwith regard to two events that occur in remote places there may be a\nquestion. A scientific definition differs from a mere description in\nthat it must afford us a means of testing whether a given item comes\nunder the definition or not. There is some difficulty in setting up\na definition of simultaneity between distant events that satisfies\nthis requirement. If we try simply to fall back upon our inherent\nideas of what we mean by \"the same instant\" we see that this is not\nadequate. We must lay down a procedure for determining whether two\nevents at remote points occur at \"the same instant,\" and check up\nalleged simultaneity by means of this procedure.\n\nEinstein says, and we must agree with him, that he can find but one\nreasonable definition to cover this ground. An observer can tell\nwhether he is located half way between two points of his observation;\nhe can have mirrors set up at these points, send out light-signals,\nand note the time at which he gets back the reflection. He knows\nthat the velocity of both signals, going and coming, is the same;\nif he observes that they return to him together so that their time of\ntransit for the round trip is the same, he must accept the distances as\nequal. He is then at the mid-point of the line joining the two points\nunder observation; and he may define simultaneity as follows, without\nintroducing anything new or indeterminate: Two events are simultaneous\nif an observer midway between them sees them at the same instant, by\nmeans, of course, of light originating at the points of occurrence.]*\n\n[It is this definition of simultaneity, coupled with the assumption\nthat all observers, on whatever uniformly moving systems, would obtain\nthe same experimental value for the velocity of light, that leads to\nthe apparent paradoxes of the Special Theory of Relativity. If it be\nasked why we adopt it, we must in turn ask the inquirer to propose\na better system for defining simultaneous events on different moving\nbodies.]198\n\n[There is nothing in this definition to indicate, directly, whether\nsimultaneity persists for all observers, or whether it is relative,\nso that events simultaneous to one observer are not so to another. The\nquestion must then be investigated; and the answer, of course, will\nhinge upon the possibility of making proper allowances for the time of\ntransit of the light signals that may be involved. It seems as though\nthis ought to be possible; but a simple experiment will indicate that\nit is not, unless the observers involved are at rest with respect to\none another.\n\n\n\n\nAN EINSTEINIAN EXPERIMENT\n\nLet us imagine an indefinitely long, straight railroad track, with\nan observer located somewhere along it at the point $M$. According to\nthe convention suggested above, he has determined points $A$ and $B$\nin opposite directions from him along the track, and equally distant\nfrom him. We shall imagine, further, than a beneficent Providence\nsupplies two lightning flashes, one striking at $A$ and one at $B$,\nin such a way that observer $M$ finds them to be simultaneous.\n\nWhile all this is going on, a train is passing--a very long train,\namply long enough to overlap the section $AMB$ of the track. Among\nthe passengers there is one, whom we may call $M'$, who is directly\nopposite $M$ at the instant when, according to $M$, the lightning\nstrikes. Observe he is not opposite $M$ when $M$ sees the flashes,\nbut a brief time earlier--at the instant when, according to $M$'s\ncomputation, the simultaneous flashes occurred. At this instant there\nare definitely determined the points $A'$ and $B'$, on the train;\nand since we may quite well think of the two systems--train-system\nand track-system--as in coincidence at this instant, $M'$ is midway\nbetween $A'$ and $B'$, and likewise is midway between $A$ and $B$.\n\nNow if we think of the train as moving over the track in the direction\nof the arrow, we see very easily that $M'$ is running away from the\nlight from $A$ and toward that from $B$, and that, despite--or if\nyou prefer because of--the uniform velocity of these light signals,\nthe one from $B$ reaches him, over a slightly shorter course, sooner\nthan the one from $A$, over the slightly longer course. When the light\nsignals reach $M$, $M'$ is no longer abreast of him but has moved along\na wee bit, so that at this instant when $M$ has the two signals, one\nof these has passed $M'$ and the other has yet to reach him. The upshot\nis that the events which were simultaneous to $M$ are not so to $M'$.\n\nIt will probably be felt that this result is due to our having,\nsomewhat unjustifiably and inconsistently, localized on the train\nthe relative motion between train and track. But if we think of\nthe track as sliding back under the train in the direction opposite\nto the arrow, and carrying with it the points $A$ and $B$; and if\nwe remember that this in no way affects $M$'s observed velocity of\nlight or the distances $AM$ and $BM$ as he observes them: we can\nstill accept his claim that the flashes were simultaneous. Then we\nhave again the same situation: when the flashes from $A$ and from\n$B$ reach $M$ at the same moment, in his new position a trifle to\nthe left of his initial position of the diagram, the flash from $A$\nhas not yet reached $M'$ in his original position while that from\n$B$ has passed him. Regardless of what assumption we make concerning\nthe motion between train-system and track-system, or more elegantly\nregardless of what coordinate system we use to define that motion,\nthe event at $B$ precedes that at $A$ in the observation of $M'$. If\nwe introduce a second train moving on the other track in the opposite\ndirection, the observer on it will of course find that the flash at $A$\nprecedes that at $B$--a disagreement not merely as to simultaneity but\nactually as to the order of two events! If we conceive the lightning\nas striking at the points $A'$ and $B'$ on the train, these points\ntravel with $M'$ instead of with $M$; they are fixed to his coordinate\nsystem instead of to the other. If you carry out the argument now,\nyou will find that when the flashes are simultaneous to $M'$, the\none at $A$ precedes that at $B$ in $M$'s observation.\n\nA large number of experiments more or less similar in outline to\nthis one can be set up to demonstrate the consequences, with regard\nto measured values of time and space, of relative motion between two\nobservers. I do not believe that a multiplicity of such demonstrations\ncontributes to the intelligibility of the subject, and it is for\nthis reason that I have cut loose from immediate dependence upon the\nessayists in this part of the discussion, concentrating upon the single\nexperiment to which Einstein himself gives the place of importance.\n\n\n\n\nWHO IS RIGHT?\n\nWe may permit Mr. Francis to remind us here that neither $M$ nor\n$M'$ may correct his observation to make it accord with the other\nfellow's. The one who does this is admitting that the other is at\nabsolute rest and that he is himself in absolute motion; and this\ncannot be. They are simply in disagreement as to the simultaneity of\ntwo events, just as two observers might be in disagreement about the\ndistance or the direction of a single event. This can mean nothing\nelse than that, under the assumptions we have made, simultaneity is\nnot an absolute characteristic as we had supposed it to be, but, like\ndistance and direction, is in fact merely a relation between observer\nand objective, and therefore depends upon the particular observer\nwho happens to be operating and upon the reference frame he is using.\n\nBut this is serious. My time measurements depend ultimately upon\nmy space measurements; the latter, and hence both, depend closely\nupon my ideas of simultaneity. Yours depend upon your reading of\nsimultaneity in precisely the same way.]* [Suppose the observer on\nthe track, in the above experiment, wants to measure the length of\nsomething on the car, or the observer on the car something on the\ntrack. The observer, or his assistant, must be at both ends of the\nlength to be measured at the same time, or get simultaneous reports\nin some way from these ends; else they will obtain false results. It\nis plain, then, that with different criteria of what the \"same time\"\nis, the observers in the two systems may get different values for\nthe measured lengths in question.]220\n\n[Who is right? According to the principle of relativity a decision on\nthis question is absolutely impossible. Both parties are right from\ntheir own points of view; and we must admit that two events in two\ndifferent places may be simultaneous for certain observers, and yet\nnot simultaneous for other observers who move with respect to the first\nones. There is no contradiction in this statement, although it is not\nin accordance with common opinion, which believes simultaneousness to\nbe something absolute. But this common opinion lacks foundation. It\ncannot be proved by direct perception, for simultaneity of events\ncan be perceived directly,]24 [and in a manner involving none of our\narbitrary assumptions,]* [only if they happen at the same place;\nif the events are distant from each other, their simultaneity or\nsuccession can be stated only through some method of communicating by\nsignals. There is no logical reason why such a method should not lead\nto different results for observers who move with regard to one another.\n\nFrom what we have said, it follows immediately that in the new theory\nnot only the concept of simultaneousness but also that of duration is\nrevealed as dependent on the motion of the observer.]24 [Demonstration\nof this should be superfluous; it ought to be plain without argument\nthat if two observers cannot agree whether two instants are the same\ninstant or not, they cannot agree on the interval of time between\ninstants. In the very example which we have already examined, one\nobserver says that a certain time-interval is zero, and another gives\nit a value different from zero. The same thing happens whenever the\nobservers are in relative motion.]* [Two physicists who measure the\nduration of a physical process will not obtain the same result if\nthey are in relative motion with regard to one another.\n\nThey will also find different results for the length of a body. An\nobserver who wants to measure the length of a body which is moving\npast him must in one way or another hold a measuring rod parallel to\nits motion and mark those points on his rod with which the ends of\nthe body come into simultaneous coincidence. The distance between the\ntwo marks will then indicate the length of the body. But if the two\nmarkings are simultaneous for one observer, they will not be so for\nanother one who moves with a different velocity, or who is at rest,\nwith regard to the body under observation. He will have to ascribe\na different length to it. And there will be no sense in asking which\nof them is right: length is a purely relative concept, just as well\nas duration.]24\n\n\n\n\nTHE RELATIVITY OF TIME AND SPACE\n\n[The degree to which distance and time become relative instead of\nabsolute quantities under the Special Theory of Relativity can be\nstated very definitely. In the first place, we must point out that the\nrelativity of lengths applies with full force only to lengths that lie\nparallel to the direction of relative motion. Those that lie exactly\nperpendicular to that direction come out the same for both observers;\nthose that lie obliquely to it show an effect, depending upon the\nangle, which of course becomes greater and greater as the direction\nof parallelism is approached.\n\nThe magnitude of the effect is easily demonstrated, but with this\ndemonstration we do not need to be concerned here. It turns out\nthat if an observer moving with a system finds that a certain time\ninterval in the system is $T$ seconds and that a certain length in\nthe system is $L$ inches, then an observer moving parallel with $L$\nand with a velocity $v$ relative to the system will find for these\nthe respective values $T \\div K$ and $L \\times K$, where\n\n    $$K = \\sqrt{1 - v^2/C^2}.$$\n\n$C$ in this expression of course represents the velocity of light. It\nwill be noted that the fraction $v^2/C^2$ is ordinarily very small;\nthat the expression under the radical is therefore less than 1 but\nby a very slight margin; and that the entire expression $K$ is itself\ntherefore less than 1 but by an even slighter margin. This means, then,\nthat the observer outside the system finds the lengths in the system\nto be a wee bit shorter and the time intervals a wee bit longer than\ndoes the observer in the system. Another way of putting the matter\nis based, ultimately, upon the fact that in order for the observer\nin the system to get the larger value for distance and the smaller\nvalue for time, his measuring rod must go into the distance under\nmeasurement more times than that of the moving observer, while his\nclock must beat a longer second in order that less of them shall be\nrecorded in a given interval between two events. So it is often said\nthat the measuring rod as observed from without is contracted and\nthe clock runs slow. This does not impress me as a happy statement,\neither in form or in content.]*\n\n[The argument that these formulae are contradicted by human experience\ncan be refuted by examining a concrete instance. If a train is\n1,000 feet long at rest, how long will it be when running a mile a\nminute?]232 [I have quoted this question exactly as it appears in the\nessay from which it is taken, because it is such a capital example of\nthe objectionable way in which this business is customarily put. For\nthe statement that lengths decrease and time-intervals increase \"with\nvelocity\" is not true in just this form. The velocity, to have meaning,\nmust be relative to some external system; and it is the observations\nfrom that external system that are affected. So long as we confine\nourselves to the system in which the alleged modifications of size\nare stated as having taken place, there is nothing to observe that\nis any different from what is usual; there is no way to establish\nthat we are enjoying a velocity, and in fact within the intent of\nthe relativity theory we are not enjoying a velocity, for we are\nmoving with the objects which we are observing. It is inter-systemic\nobservations, and these alone, that show the effect. When we travel\nwith the system under observation, we get the same results as any\nother observer on this system; when we do not so travel, we must\nconduct our observations from our own system, in relative motion to\nthe other, and refer our results to our system.\n\nNow when no particular observer is specified, we must of course\nassume an observer connected with the train, or with whatever the body\nmentioned. To that observer it doesn't make the slightest difference\nwhat the train does; it may stand at rest with respect to some external\nsystem or it may move at any velocity whatsoever; its length remains\nalways 1,000 feet. In order for this question to have the significance\nwhich its propounder means it to have, I must restate it as follows:\nA train is 1,000 feet long as measured by an observer travelling with\nit. If it passes a second observer at 60 miles per hour, what is its\nlength as observed by him? The answer is now easy.]* [According to the\nformula the length of the moving train as seen from the ground will be\n\n    $$1,000 \\times \\sqrt{1 - (88)^2 / (186,000 \\times 5,280)^2} =\n    999.999,999,999,996$$\n\nfeet, a change entirely too small for detection by the most delicate\ninstruments. Examination of the expression K shows that in so far\nas terrestrial movements of material objects are concerned it is\nequal to 1]232 [within a far smaller margin than we can ever hope to\nmake our observations. Even the diameter of the earth, as many of the\nessayists point out, will be shortened only 2 1/2 inches for an outside\nobserver past whom it rushes with its orbital speed of 18.5 miles per\nsecond. But slight as the difference may be in these familiar cases,\nits scientific importance remains the same.]*\n\n\n\n\nRELATIVITY AND REALITY\n\n[A simple computation shows that this effect is exactly the amount\nsuggested by Lorentz and Fitzgerald to explain the Michelson-Morley\nexperiment.]188 [This ought not to surprise us, since both that\nexplanation and the present one are got up with the same purpose. If\nthey both achieve that purpose they must, numerically, come to the\nsame thing in any numerical case. It is, however, most emphatically\nto be insisted that the present \"shortening\" of lengths]* [no longer\nappears as a \"physical\" shortening caused by absolute motion through\nthe ether but is simply a result of our methods of measuring space\nand time.]188 [Where Fitzgerald and Lorentz had assumed that a body in\nmotion has its dimensions shortened in the direction of its motion,]220\n[this very form of statement ceases to possess significance under\nthe relativity assumption.]* [For if we cannot tell which of two\nbodies is moving, which one is shortened? The answer is, both--for\nthe other fellow. For each frame of reference there is a scale of\nlength and a scale of time, and these scales for different frames\nare related in a manner involving both the length and the time.]220\n[But we must not yield to the temptation to say that all this is not\nreal; the confinement of a certain scale of length and of time to\na single observation system does not in the least make it unreal.]*\n[The situation is real--as real as any other physical event.]165\n\n[The word physical is used in two senses in the above paragraph. It\nis denied that the observed variability in lengths indicates any\n\"physical\" contraction or shrinkage; and on the heels of this it\nis asserted that this observed variability is of itself an actual\n\"physical\" event. It is difficult to express in words the distinction\nbetween the two senses in which the term physical is employed in these\ntwo statements, but I think this distinction ought to be clear once\nits existence is emphasized. There is no material contraction; it is\nnot right to say that objects in motion contract or are shorter; they\nare not shorter to an observer in motion with them. The whole thing is\na phenomenon of observation. The definitions which we are obliged to\nlay down and the assumptions which we are obliged to make in order,\nfirst, that we shall be able to measure at all, and second, that we\nshall be able to escape the inadmissible concept of absolute motion,\nare such that certain realities which we had supposed ought to be the\nsame for all observers turn out not to be the same for observers who\nare in relative motion with respect to one another. We have found this\nout, and we have found out the numerical relation which holds between\nthe reality of the one observer and that of the other. We have found\nthat this relation depends upon nothing save the relative velocity\nof the two observers. As good a way of emphasizing this as any is to\npoint out that two observers who have the same velocity with respect\nto the system under examination (and whose mutual relative velocity\nis therefore zero) will always get the same results when measuring\nlengths and times on that system. The object does not go through any\nprocess of contraction; it is simply shorter because it is observed\nfrom a station with respect to which it is moving. Similar remarks\nmight be made about the time effect; but the time-interval is not so\neasily visualized as a concrete thing and hence does not offer such\ntemptation for loose statement.\n\nThe purely relative aspect of the matter is further brought out if\nwe consider a single example both backwards and forwards. Systems $S$\nand $S'$ are in relative motion. An object in $S$ which to an observer\nin $S$ is $L$ units long, is shorter for an observer in $S'$--shorter\nby an amount indicated through the \"correction factor\" $K$. Now if\nwe have, in the first instance, made the objectionable statement\nthat objects are shorter in system $S'$ than they are in $S$, it\nwill be quite natural for us to infer from this that objects in $S$\nmust be longer than those in $S'$; and from this to assert that when\nthe observer in $S$ measures objects lying in $S'$, he gets for them\ngreater lengths than does the home observer in $S'$. But if we have,\nin the first instance, avoided the objectionable statement referred\nto, we shall be much better able to realize that the whole business is\nquite reciprocal; that the phenomena are symmetric with respect to the\ntwo systems, to the extent that we can interchange the systems in any\nof our statements without modifying the statements in any other way.\n\nObjects in $S$ appear shorter and times in $S$ appear longer to the\nexternal \"moving\" observer in $S'$ than they do to the domestic\nobserver in $S$. Exactly in the same way, objects in $S'$ appear\nshorter to observers in the foreign system $S$ than to the home\nobserver in $S'$, who remains at rest with respect to them. I think\nthat when we get the right angle upon this situation, it loses the\nalleged startling character which has been imposed upon it by many\nwriters. The \"apparent size\" of the astronomer is an analogy in\npoint. Objects on the moon, by virtue of their great distance, look\nsmaller to observers on the earth than to observers on the moon. Do\nobjects on the earth, on this account, look larger to a moon observer\nthan they do to us? They do not; any suggestion that they do we should\nreceive with appropriate scorn. The variation in size introduced by\ndistance is reciprocal, and this reciprocity does not in the least\npuzzle us. Why, then, should that introduced by relative motion\npuzzle us?\n\n\n\n\nTIME AND SPACE IN A SINGLE PACKAGE\n\nOur old, accustomed concepts of time and space, which have grown\nup through countless generations of our ancestors, and been handed\ndown to us in the form in which we are familiar with them, leave no\nroom for a condition where time intervals and space intervals are\nnot universally fixed and invariant. They leave no room for us to\nsay that]* [one cannot know the time until he knows where he is,\nnor where he is until he knows the time,]220 [nor either time or\nplace until he knows something about velocity. But in this concise\nformulation of the difference between what we have always believed\nand what we have seen to be among the consequences of Einstein's\npostulates of the universal relativity of uniform motion, we may\nat once locate the assumption which, underlying all the old ideas,\nis the root of all the trouble. The fact is we have always supposed\ntime and space to be absolutely distinct and independent entities.]*\n\n[The concept of time has ever been one of the most absolute of all\nthe categories. It is true that there is much of the mysterious about\ntime; and philosophers have spent much effort trying to clear up\nthe mystery--with unsatisfactory results. However, to most persons\nit has seemed possible to adopt an arbitrary measure or unit of\nduration and to say that this is absolute, independent of the state\nof the body or bodies on which it is used for practical purposes.]272\n[Time has thus been regarded as something which of itself flows on\nregularly and continuously, regardless of physical events concerning\nmatter.]150 [In other words, according to this view, time is not\naffected by conditions or motions in space.]272 [We have deliberately\nchosen to ignore the obvious fact that time can never appear to us,\nbe measured by us, or have the least significance for us, save as\na measure of something that is closely tied up with space and with\nmaterial space-dimensions. Not merely have we supposed that time and\nspace are separated in nature as in our easiest perceptions, but we\nhave supposed that they are of such fundamentally distinct character\nthat they can never be tied up together. In no way whatever, assumes\nthe Euclidean and Newtonian intellect, may space ever depend upon\ntime or time upon space. This is the assumption which we must remove\nin order to attain universal relativity; and while it may come hard,\nit will not come so hard as the alternative. For this alternative is\nnothing other than to abandon universal relativity. This course would\nleave us with logical contradictions and discrepancies that could not\nbe resolved by any revision of fundamental concepts or by any cleaning\nout of the Augean stables of old assumptions; whereas the relativity\ndoctrine as built up by Einstein requires only such a cleaning out in\norder to leave us with a strictly logical and consistent whole. The\nr\u00f4le of Hercules is a very difficult one for us to play. Einstein\nhas played it for the race at large, but each of us must follow him\nin playing it for himself.\n\n\n\n\nSOME FURTHER CONSEQUENCES\n\nI need not trespass upon the subject matter of those essays which\nappear in full by going here into any details with regard to the\nmanner in which time and space are finally found to depend upon one\nanother and to form the parts of a single universal whole. But I\nmay appropriately point out that if time and space are found to be\nrelative, we may surely expect some of the less fundamental concepts\nthat depend upon them to be relative also. In this expectation we are\nnot disappointed. For one thing,]* [mass has always been assumed to\nbe a constant, independent of any motion or energy which it might\npossess. Just as lengths and times depend upon relative motion,\nhowever, it is found that mass, which is the remaining factor in\nthe expression for energy due to motion, also depends upon relative\nvelocities. The dependence is such that if a body takes up an amount\nof energy $E$ with respect to a certain system, the body behaves,\nto measurements made from that system, as though its mass had been\nincreased by an amount $E/C^2$, where $C$ is as usual the velocity\nof light.]194\n\n[This should not startle us. The key to the situation lies in the\nitalicized words above, which indicate that the answer to the query\nwhether a body has taken up energy or not depends upon the seat of\nobservation. If I take up my location on the system $S$, and you\non the system $S'$, and if we find that we are in relative motion,\nwe must make some assumption about the energy which was necessary,\ninitially, to get us into this condition. Suppose we are on two\npassing trains.]* [The chances are that either of us will assume that\nhe is at rest and that it is the other train which moves, although\nif sufficiently sophisticated one of us may assume that he is moving\nand that the other train is at rest.]272 [Whatever our assumption,\nwhatever the system, the localization of the energy that is carried\nin latent form by our systems depends upon this assumption. Indeed,\nif our systems are of differing mass, our assumptions will even govern\nour ideas of the amount of energy which is represented by our relative\nmotion; if your system be the more massive, more energy would have to\nbe localized in it than in mine to produce our relative motion. If\nwe did not have the universal principle of relativity to forbid,\nwe might make an arbitrary assumption about our motions and hence\nabout our respective latent energies; in the presence of this veto,\nthe only chance of adjustment lies in our masses, which must differ\naccording to whether you or I observe them.]*\n\n[For most of the velocities with which we are familiar $E/C^2$ is,\nlike the difference between $K$ and unity, such an extremely small\nquantity that the most delicate measurements fail to detect it. But\nthe electrons in a highly evacuated tube and the particles shot\nout from radioactive materials attain in some cases velocities as\nhigh as eight-tenths that of light. When we measure the mass of such\nparticles at different velocities we find that it actually increases\nwith the velocity, and in accordance with the foregoing law.]194\n[This observation, in fact, antedates Einstein's explanation, which\nis far more satisfactory than the earlier differentiation between\n\"normal mass\" and \"electrical mass\" which was called upon to account\nfor the increase.]*\n\n[But if the quantity $E/C^2$ is to be considered as an actual increase\nin mass, may it not be possible that all mass is energy? This would\nlead to the conclusion that the energy stored up in any mass is\n$mC^2$. The value is very great, since $C$ is so large; but it is in\ngood agreement with the internal energy of the atom as calculated\nfrom other considerations. It is obvious that conservation of mass\nand of momentum cannot both hold good under a theory that translates\nthe one into the other. Mass is then not considered by Einstein as\nconservative in the ordinary sense, but it is the total quantity of\nmass plus energy in any closed system that remains constant. Small\namounts of energy may be transformed into mass, and vice versa.]194\n\n[Other features of the theory which are often displayed as consequences\nare really more in the nature of assumptions. It will be recalled that\nwhen we had agreed upon the necessity of employing signals of some\nsort, we selected as the means of signalling the speediest messenger\nwith which we happened to be acquainted. Our subsequent difficulties\nwere largely due to the impossibility of making a proper allowance\nfor this messenger's speed, even though we knew its numerical value;\nand as a consequence, this speed enters into our formulae. Now we have\nnot said in so many words that $C$ is the greatest speed attainable,\nbut we have tacitly assumed that it is. We need not, therefore, be\nsurprised if our formulae give us absurd results for speeds higher than\n$C$, and indicate the impossibility of ever attaining these. Whatever\nwe put into a problem the algebra is bound to give us back. If we look\nat our formula for $K$, we see that in the event of $v$ equalling $C$,\nlengths become zero and times infinite. The light messenger itself,\nthen, has no dimension; and for it time stands still.\n\nIf we suppose $v$ to be greater than $C$, we get even more bizarre\nresults, for then the factor $K$ is the square root of a negative\nnumber, or as the mathematician calls it an \"imaginary\" quantity;\nand with it, lengths and times become imaginary too.\n\nThe fact that time stops for it, and the fact that it is the limiting\nvelocity, give to $C$ certain of the attributes of the mathematician's\ninfinity. Certainly if it can never be exceeded, we must have a new\nformula for the composition of velocities. Otherwise when my system\npasses yours at a speed of 100,000 miles per second, while yours\npasses a third in the same direction at the same velocity, I shall\nbe passing this third framework at the forbidden velocity of 200,000\nmiles per second--greater than $C$. In fact Einstein is able to show\nthat an old formula, which had already been found to connect the speed\nof light in a material medium with the speed of that medium, will now\nserve universally for the composition of velocities. When we combine\nthe velocities $v$ and $u$, instead of getting the resultant $v+u$\nas we would have supposed, we get the resultant\n\n    $$(v + u) / (1 + (uv / C^2))\\,,$$ or $$C^2(v + u) / (C^2 + uv)$$\n\nThis need not surprise us either, if we will but reflect that\nthe second velocity effects a second revision of length and time\nmeasurements between the systems involved. And now, if we let either\n$v$, or $u$, or even both of them, take the value $C$, the resultant\nstill is $C$. In another way we have found $C$ to behave like the\nmathematician's infinity, to which, in the words of the blind poet,\nif we add untold thousands, we effect no real increment.\n\n\n\n\nASSUMPTION AND CONSEQUENCE\n\nA good many correspondents who have given the subject sufficient\nthought to realize that the limiting character of the velocity $C$ is\nreally read into Einstein's system by assumption have written, in more\nor less perturbed inquiry, to know whether this does not invalidate\nthe whole structure. The answer, of course, is yes--provided you can\nshow this assumption to be invalid. The same answer may be made of\nany scientific doctrine whatever, and in reference to any one of the\nmultitudinous assumptions underlying it. If we were to discover,\ntomorrow, a way of sending signals absolutely instantaneously,\nEinstein's whole structure would collapse as soon as we had agreed to\nuse this new method. If we were to discover a signalling agent with\nfinite velocity greater than that of light, relativity would persist\nwith this velocity written in its formulae in the place of $C$.\n\nIt is a mistake to quote Einstein's theory in support of the\nstatement that such a velocity can never be. An assumption proves its\nconsequences, but never can prove itself; it must remain always an\nassumption. But in the presence of long human experience supporting\nEinstein's assumption that no velocity in excess of $C$ can be found,\nit is fair to demand that it be disputed not with argument but with\ndemonstration. The one line of argument that would hold out a priori\nhope of reducing the assumption to an absurdity would be one based\non the familiar idea of adding velocities; but Einstein has spiked\nthis argument before it is started by replacing the direct addition\nof velocities with another method of combining them that fits in\nwith his assumption and as well with the observed facts. The burden\nof proof is then on the prosecution; anyone who would contradict our\nassertion that $C$ is the greatest velocity attainable may do so only\nby showing us a greater one. Until this has been done, the admission\nthat it may properly be attempted can in no way be construed as a\nconfession of weakness on the part of Einstein.\n\nIt may be well to point out that in no event may analogy be drawn with\nsound, as many have tried to do. In the first place sound requires\na material medium and its velocity with regard to this rather than\nrelative to the observer we know to be fixed; in the second place,\nrequiring a material medium, sound is not a universal signalling\nagent; in the third place, we know definitely that its velocity can\nbe exceeded, and are therefore barred from making the assumption\nnecessary to establish the analogy. The very extraordinary behavior\nof light in presenting a velocity that is the same for all observers,\nand in refusing to betray the least material evidence of any medium\nfor its transmission, rather fortifies us in believing that Einstein's\nassumption regarding the ultimate character of this velocity is in\naccord with the nature of things.\n\n\n\n\nRELATIVITY AND THE LAYMAN\n\nA great deal can be said in the direction of general comment making the\nSpecial Theory and its surprising accompaniments easier of acceptance,\nand we shall conclude the present discussion by saying some of these\nthings.]* [It has been objected that the various effects catalogued\nabove are only apparent, due to the finite velocity of light--that\nthe real shape and size of a body or the real time of an event cannot\nbe affected by the point of view or the motion of an observer. This\nargument would be perfectly valid, if there were real times and\ndistances; but there are not. These are earth-bound notions, due to\nour experience on an apparently motionless platform, with slow-moving\nbodies. Under these circumstances different observations of the same\nthing or of the same event agree. But when we no longer have the solid\nearth to stand on, and are dealing with velocities so high that the\nrelativity effects become appreciable, there is no standard by which\nto resolve the disagreements. No one of the observations can claim\nto be nearer reality than any other. To demand the real size of a\nthing is to demand a stationary observer or an instantaneous means\nof information. Both are impossible.\n\nWhen relativity asks us to give up our earth-bound notions of absolute\nspace and absolute time the sensation, at first, is that we have\nnothing left to stand on. So must the contemporaries of Columbus have\nfelt when told that the earth rested on--nothing. The remedy too is\nsimilar. Just as they had to be taught that falling is a local affair,\nthat the earth is self-contained, and needs no external support--so we\nmust be taught that space and time standards are local affairs. Each\nmoving body carries its own space and time standards with it; it is\nself-contained. It does not need to reach out for eternal support,\nfor an absolute space and time that can never quite be attained. All\nwe ever need to know is the relation of the other fellow's space\nand time standards to our own. This is the first thing relativity\nteaches us.]141\n\n[The consequences of Einstein's assumptions have led many to reject the\ntheory of relativity, on the ground that its conclusions are contrary\nto common sense--as they undoubtedly are. But to the contemporaries\nof Copernicus and Galileo the theory that the earth rotates on its\naxis and revolves around the sun was contrary to common sense; yet\nthis theory prevailed. There is nothing sacred about common sense;\nin the last analysis its judgments are based on the accumulated\nexperience of the human race. From the beginning of the world up to\nthe present generation, no bodies were known whose velocities were\nnot extremely small compared with that of light. The development of\nmodern physics has led to discovery of very much larger velocities,\nsome as high as 165,000 miles per second. It is not to be wondered at\nthat such an enlargement of our experience requires a corresponding\nenlargement or generalization of the concepts of space and time. Just\nas the presupposition of primitive man that the earth was flat had\nto be given up in the light of advancing knowledge, so we are now\ncalled upon to give up our presupposition that space and time are\nabsolute and independent in their nature.\n\nThe reader must not expect to understand the theory of relativity in\nthe sense of making it fit in with his previous ideas. If the theory\nbe right these ideas are wrong and must be modified, a process apt\nto be painful.]223 [All the reader can do is to become familiar with\nthe new concepts, just as a child gets used to the simple relations\nand quantities he meets until he \"understands\" them.]221 [Mr. Francis\nhas said something of the utmost significance when he points out that\n\"understanding\" really means nothing in the world except familiarity\nand accustomedness.]* [The one thing about the relativity doctrine\nthat we can hope thus to understand at once and without pain is the\nlogical process used in arriving at our results.]221 [Particularly is\nit hard to give a satisfactory explanation of the theory in popular\nlanguage, because the language itself is based on the old concepts;\nthe only language which is really adequate is that of mathematics.]223\n[Unless we have, in addition to the terms of our ordinary knowledge,\na set of definitions that comes with a wide knowledge of mathematics\nand a lively sense of the reality of mathematical constructions, we\nare likely to view the theory of relativity through a fog of familiar\nterms suddenly become self-contradictory and deceptive. Not that we\nare unfamiliar with the idea that some of our habitual notions may\nbe wrong; but knowledge of their illusory nature arises and becomes\nconvincing only with time. We may now be ready to grant that the earth,\nseemingly so solid, is really a whirling globe rushing through space;\nbut we are no more ready immediately to accept the bald assertion that\nthis space is not what it seems than our ancestors were to accept the\nidea that the earth was round or that it moved.]156 [What we must have,\nif we are to comprehend relativity with any degree of thoroughness,\nis the mathematician's attitude toward his assumptions, and his\ncomplete readiness to swap one set of assumptions for another as a\nmere part of the day's work, the spirit of which I have endeavored\nto convey in the chapter on non-Euclidean geometry.]*\n\n\n\n\nPHYSICS VS. METAPHYSICS\n\n[The ideas of relativity may seem, at first sight, to be giving us\na new and metaphysical theory of time and space. New, doubtless; but\ncertainly the theory was meant by its author to be quite the opposite\nof metaphysical. Our actual perception of space is by measurement,\nreal and imagined, of distances between objects, just as our actual\nperception of time is by measurement. Is it not less metaphysical\nto accept space and time as our measurements present them to us,\nthan to invent hypotheses to force our perceptual space into an\nabsolute space that is forever hidden from us?]182 [In order not to\nbe metaphysical, we must eliminate our preconceived notions of space\nand time and motion, and focus our attention upon the indications\nof our instruments of observation, as affording the only objective\nmanifestations of these qualities and therefore the only attributes\nwhich we can consider as functions of observed phenomena.]47 [Einstein\nhas consistently followed the teachings of experience, and completely\nfreed himself from metaphysics.]114 [That this is not always easy\nto do is clear, I think, if we will recall the highly metaphysical\ncharacter often taken by the objections to action-at-a-distance\ntheories and concepts; and if we will remind ourselves that it was\non purely metaphysical grounds that Newton refused to countenance\nHuyghens' wave theory of light. Whether, as in the one case, it\nleads us to valid conclusions, or, as in the other, to false ones,\nmetaphysical reasoning is something to avoid. Einstein, I think,\nhas avoided it about as thoroughly as anyone ever did.]*\n\n\n\n\n\n\n\n\n\nV\n\nTHAT PARALLEL POSTULATE\n\nModern Geometric Methods; the Dividing Line Between Euclidean and\nNon-Euclidean; and the Significance of the Latter\n\nBY THE EDITOR\n\n\nThe science of geometry has undergone a revolution of which the\noutsider is not informed, but which it is necessary to understand\nif we are to attain any comprehension of the geometric formulation\nof Einstein's results; and especially if we are to appreciate why\nit is proper and desirable to formulate these results geometrically\nat all. The classical geometer regarded his science from a narrow\nviewpoint, as the study of a certain set of observed phenomena--those\nof the space about us, considered as an entity in itself and divorced\nfrom everything in it. It is clear that some things about that space\nare not as they appear (optical illusions), and that other things\nabout it are true but by no means apparent (the sum-of-squares\nproperty of a right triangle, the formul\u00e6 for surface and volume\nof a sphere, etc.). While many things about space are \"obvious,\"\nthese need in the one case disproof and in the other discovery and\nproof. With all their love of mental processes for their own sake,\nit is then not surprising that the Greeks should have set themselves\nthe task of proving by logical process the properties of space,\nwhich a less thoughtful folk would have regarded as a subject only\nfor observational and experimental determination.\n\nBut, abstract or concrete, the logical structure must have a starting\npoint; and it is fair to demand that this consist in a statement\nof the terms we are going to use and the meanings we are going to\nattach to them. In other words, the first thing on the program will\nbe a definition, or more probably, several definitions.\n\nNow the modern scientist has a somewhat iconoclastic viewpoint toward\ndefinitions, and especially toward the definition of his very most\nfundamental ideas.\n\nWe do not speak here in terms of dictionary definitions. These have\nfor object the eminent necessity of explaining the meaning and use\nof a word to some one who has just met it for the first time. It\nis easy enough to do this, if the doer possesses a good command\nof the language. It is not even a matter of grave concern that the\nwords used in the definition be themselves known to the reader; if\nthey are not, he must make their acquaintance too. Dr. Johnson's\ncelebrated definition of a needle stands as perpetual evidence\nthat when he cannot define a simple thing in terms of things still\nsimpler, the lexicographer is forced to define it in terms of things\nmore complex. Or we might demonstrate this by noting that the best\ndictionaries are driven to define such words as \"and\" and \"but\"\nby using such complex notions as are embodied in \"connective,\"\n\"continuative,\" \"adversative,\" and \"particle.\"\n\nIt is otherwise with the scientist who undertakes to lay down a\ndefinition as the basis of further procedure in building up the\ntissue of his science. Here a degree of rigorous logic is called for\nwhich would be as superfluous in the dictionary as the effort there\nto attain it would be out of place. The scientist, in building up\na logical structure that will withstand every assault, must define\neverything, not in terms of something which he is more or less\nwarranted in supposing his audience to know about, but actually in\nterms of things that have already been defined. This really means that\nhe must explain what he is talking about in terms of simpler ideas\nand simpler things, which is precisely what the lexicographer does\nnot have to worry about. This is why it is quite trivial to quote a\ndictionary definition of time or space or matter or force or motion\nin settlement of a controversy of scientific or semi-scientific nature.\n\n\n\n\nTERMS WE CANNOT DEFINE\n\nBut the scientist who attempts to carry out this ideal system of\ndefining everything in terms of what precedes meets one obstacle\nwhich he cannot surmount directly. Even a layman can construct a\npassable definition of a complex thing like a parallelopiped, in\nterms of simpler concepts like point, line, plane and parallel. But\nwho shall define point in terms of something simpler and something\nwhich precedes point in the formulation of geometry? The scientist is\nembarrassed, not in handling the complicated later parts of his work,\nbut in the very beginning, in dealing with the simplest concepts with\nwhich he has to deal.\n\nSuppose a dictionary were to be compiled with the definitions\narranged in logical rather than alphabetical order: every word to be\ndefined by the use only of words that have already been defined. The\nfurther back toward the beginning we push this project, the harder it\ngets. Obviously we can never define the first word, or the second,\nsave as synonymous with the first. In fact we should need a dozen\nwords, more or less, to start with--God-given words which we cannot\ndefine and shall not try to define, but of which we must agree that\nwe know the significance. Then we have tools for further procedure;\nwe can start with, say, the thirteenth word and define all the rest\nof the words in the language, in strictly logical fashion.\n\nWhat we have said about definitions applies equally to statements\nof fact, of the sort which are going to constitute the body of our\nscience. In the absence of simpler facts to cite as authority, we shall\nnever be able to prove anything, however simple this may itself be;\nand in fact the simpler it be, the harder it is to find something\nsimpler to underlie it. If we are to have a logical structure of any\nsort, we must begin by laying down certain terms which we shall not\nattempt to define, and certain statements which we shall not try to\nprove. Mathematics, physics, chemistry--in the large and in all their\nmany minor fields--all these must start somewhere. Instead of deceiving\nourselves as to the circumstances surrounding their start, we prefer to\nbe quite frank in recognizing that they start where we decide to start\nthem. If we don't like one set of undefined terms as the foundation,\nby all means let us try another. But always we must have such a set.\n\nThe classical geometer sensed the difficulty of defining his first\nterms. But he supposed that he had met it when he defined these in\nwords free of technical significance. \"A point is that which has\nposition without size\" seemed to him an adequate definition, because\n\"position\" and \"size\" are words of the ordinary language with which\nwe may all be assumed familiar. But today we feel that \"position\"\nand \"size\" represent ideas that are not necessarily more fundamental\nthan those of \"line\" and \"point,\" and that such a definition begs\nthe question. We get nowhere by replacing the undefined terms \"point\"\nand \"line\" and \"plane,\" which really everybody understands, by other\nundefined terms which nobody understands any better.\n\nIn handling the facts that it was inconvenient to prove, the classical\ngeometer came closer to modern practice. He laid down at the beginning\na few statements which he called \"axioms,\" and which he considered to\nbe so self-evident that demonstration was superfluous. That the term\n\"self-evident\" left room for a vast amount of ambiguity appears to\nhave escaped him altogether. His axioms were axioms solely because\nthey were obviously true.\n\n\n\n\nLAYING THE FOUNDATION\n\nThe modern geometer falls in with Euclid when he writes an elementary\ntext, satisfying the beginner's demand for apparent rigor by defining\npoint and line in some fashion. But when he addresses to his peers an\neffort to clarify the foundations of geometry to a further degree of\nrigor and lucidity than has ever before been attained, he meets these\ndifficulties from another quarter. In the first place he is always\nin search of the utmost possible generality, for he has found this to\nbe his most effective tool, enabling him as it does to make a single\ngeneral statement take the place and do the work of many particular\nstatements. The classical geometer attained generality of a sort,\nfor all his statements were of any point or line or plane. But the\nmodern geometer, confronted with a relation that holds among points\nor between points and lines, at once goes to speculating whether there\nare not other elements among or between which it holds. The classical\ngeometer isn't interested in this question at all, because he is\nseeking the absolute truth about the points and lines and planes which\nhe sees as the elements of space; to him it is actually an object so to\ncircumscribe his statements that they may by no possibility refer to\nanything other than these elements. Whereas the modern geometer feels\nthat his primary concern is with the fabric of logical propositions\nthat he is building up, and not at all with the elements about which\nthose propositions revolve.\n\nIt is of obvious value if the mathematician can lay down a proposition\ntrue of points, lines and planes. But he would much rather lay down\na proposition true at once of these and of numerous other things;\nfor such a proposition will group more phenomena under a single\nprinciple. He feels that on pure scientific grounds there is quite\nas much interest in any one set of elements to which his proposition\napplies as there is in any other; that if any person is to confine\nhis attention to the set that stands for the physicist's space,\nthat person ought to be the physicist, not the geometer. If he has\nproduced a tool which the physicist can use, the physicist is welcome\nto use it; but the geometer cannot understand why, on that ground,\nhe should be asked to confine his attention to the materials on which\nthe physicist employs that tool.\n\nIt will be alleged that points and lines and planes lie in the\nmathematician's domain, and that the other things to which his\npropositions may apply may not so lie--and especially that if he will\nnot name them in advance he cannot expect that they will so lie. But\nthe mathematician will not admit this. If mathematics is defined on\nnarrow grounds as the science of number, even the point and line\nand plane may be excluded from its field. If any wider definition\nbe sought--and of course one must be--there is just one definition\nthat the mathematician will accept: Dr. Keyser's statement that\n\"mathematics is the art or science of rigorous thinking.\"\n\nThe immediate concern of this science is the means of rigorous\nthinking--undefined terms and definitions, axioms and propositions. Its\ncollateral concern is the things to which these may apply, the things\nwhich may be thought about rigorously--everything. But now the\nmathematician's domain is so vastly extended that it becomes more\nthan ever important for him to attain the utmost generality in all\nhis pronouncements.\n\nOne barrier to such generalization is the very name \"geometry,\"\nwith the restricted significance which its derivation and long usage\ncarry. The geometer therefore must have it distinctly understood\nthat for him \"geometry\" means simply the process of deducing a set\nof propositions from a set of undefined primitive terms and axioms;\nand that when he speaks of \"a geometry\" he means some particular set\nof propositions so deduced, together with the axioms, etc., on which\nthey are based. If you take a new set of axioms you get a new geometry.\n\nThe geometer will, if you insist, go on calling his undefined terms by\nthe familiar names \"point,\" \"line,\" \"plane.\" But you must distinctly\nunderstand that this is a concession to usage, and that you are not\nfor a moment to restrict the application of his statements in any\nway. He would much prefer, however, to be allowed new names for his\nelements, to say \"We start with three elements of different sorts,\nwhich we assume to exist, and to which we attach the names A, B and\nC--or if you prefer, primary, secondary and tertiary elements--or\nyet again, names possessing no intrinsic significance at all, such as\nching, chang and chung.\" He will then lay down whatever statements he\nrequires to serve the purposes of the ancient axioms, all of these\nreferring to some one or more of his elements. Then he is ready for\nthe serious business of proving that, all his hypotheses being granted,\nhis elements A, B and C, or I, II and III, or ching, chang and chung,\nare subject to this and that and the other propositions.\n\nThe objection will be urged that the mathematician who does all\nthis usurps the place of the logician. A little reflection will\nshow this not to be the case. The logician in fact occupies the same\nposition with reference to the geometer that the geometer occupies\nwith reference to the physicist, the chemist, the arithmetician,\nthe engineer, or anybody else whose primary interest lies with some\nparticular set of elements to which the geometer's system applies. The\nmathematician is the tool-maker of all science, but he does not make\nhis own tools--these the logician supplies. The logician in turn\nnever descends to the actual practice of rigorous thinking, save as\nhe must necessarily do this in laying down the general procedures\nwhich govern rigorous thinking. He is interested in processes, not\nin their application. He tells us that if a proposition is true its\nconverse may be true or false or ambiguous, but its contrapositive\nis always true, while its negative is always false. But he never,\nfrom a particular proposition \"If A is B then C is D,\" draws the\nparticular contrapositive inference \"If C is not D then A is not\nB.\" That is the mathematician's business.\n\n\n\n\nTHE R\u00d4LE OF GEOMETRY\n\nThe mathematician is the quantity-production man of science. In\nhis absence, the worker in each narrower field where the elements\nunder discussion take particular concrete forms could work out,\nfor himself, the propositions of the logical structure that applies\nto those elements. But it would then be found that the engineer had\nduplicated the work of the physicist, and so for many other cases;\nfor the whole trend of modern science is toward showing that the\nsame background of principles lies at the root of all things. So the\nmathematician develops the fabric of propositions that follows from\nthis, that and the other group of assumptions, and does this without\nin the least concerning himself as to the nature of the elements of\nwhich these propositions may be true. He knows only that they are true\nfor any elements of which his assumptions are true, and that is all he\nneeds to know. Whenever the worker in some particular field finds that\na certain group of the geometer's assumptions is true for his elements,\nthe geometry of those elements is ready at hand for him to use.\n\nNow it is all right purposely to avoid knowing what it is that we\nare talking about, so that the names of these things shall constitute\nmere blank forms which may be filled in, when and if we wish, by the\nnames of any things in the universe of which our \"axioms\" turn out\nto be true. But what about these axioms themselves? When we lay them\ndown in ignorance of the identity of the elements to which they may\neventually apply, they cannot by any possibility be \"self-evident.\" We\nmay, at pleasure, accept as self-evident a statement about points and\nlines and planes; or one about electrons, centimeters and seconds;\nor one about integers, fractions, and irrational numbers; or one about\nany other concrete thing or things whatever. But we cannot accept as\nself-evident a statement about chings, changs and chungs. So we must\nbase our \"axioms\" on some other ground than this; and our modern\ngeometer has his ground ready and waiting. He accepts his axioms\non the ground that it pleases him to do so. To avoid all suggestion\nthat they are supposed to be self-evident, or even necessarily true,\nhe drops the term \"axiom\" and substitutes for it the more color-less\nword \"postulate.\" A postulate is merely something that we agreed to\naccept, for the time being, as a basis of further argument. If it\nturns out to be true, or if we can find circumstances under which and\nelements to which it applies, any conclusions which we deduce from it\nby trustworthy processes are valid within the same limitations. And\nthe propositions which tell us that, if our postulates are true, such\nand such conclusions are true--they, too, are valid, but without any\nreservation at all!\n\nPerhaps an illustration of just what this means will not be out of\nplace. Let it be admitted, as a postulate, that $7+20$ is greater, by\n1, than $7+19$. Let us then consider the statement: \"If $7 + 19 = 65$,\nthen $7 + 20 = 66$.\" We know--at least we are quite certain--that $7 +\n19$ is not equal to 65, if by \"7\" and \"19\" and \"65\" we mean what you\nthink we mean. We are equally sure, on the same grounds, that $7 +\n20$ is not equal to 66. But, under the one assumption that we have\npermitted ourselves, it is unquestionable that if $7 + 19$ were equal\nto 65, then $7 + 20$ certainly would be equal to 66. So, while the\nconclusion of the proposition which I have put in quotation marks is\naltogether false, the proposition itself, under our assumption, is\nentirely true. I have taken an illustration designed to be striking\nrather than to possess scientific interest; I could just as easily\nhave shown a true proposition leading to a false conclusion, but of\nsuch sort that it would be of decided scientific interest as telling\nus one of the consequences of a certain assumption.\n\n\n\n\nWHAT MAY WE TAKE FOR GRANTED?\n\nThis is all very fine; but how does the geometer know what postulates\nto lay down? One is tempted to say that he is at liberty to postulate\nanything he pleases, and investigate the results; and that whether or\nnot his postulate ever be realized, the propositions that he deduces\nfrom it, being true, are of scientific interest. Actually, however,\nit is not quite as simple as all that. If it were sufficient to make\na single postulate it would be as simple as all that; but it turns\nout that this is not sufficient any more than it is sufficient to\nhave a single undefined term. We must have several postulates; and\nthey must be such, as a whole, that a geometry flows out of them. The\nrequirements are three.\n\nIn the first place, the system of postulates must be \"categorical\"\nor complete--there must be enough of them, and they must cover enough\nground, for the support of a complete system of geometry. In practice\nthe test for this is direct. If we got to a point in the building up\nof a geometry where we could not prove whether a certain thing was\none way always, or always the other way, or sometimes one way and\nsometimes the other, we should conclude that we needed an additional\npostulate covering this ground directly or indirectly. And we should\nmake that postulate--because it is precisely the things that we can't\nprove which, in practical work, we agree to assume. Even Euclid had\nto adopt this philosophy.\n\nIn the second place, the system of postulates must be consistent--no\none or more of them may lead, individually or collectively, to\nconsequences that contradict the results or any other or others. If\nin the course of building up a geometry we find we have proved\ntwo propositions that deny one another, we search out the implied\ncontradiction in our postulates and remedy it.\n\nFinally, the postulates ought to be independent. It should not be\npossible to prove any one of them as a consequence of the others. If\nthis property fails, the geometry does not fail with it; but it is\nseriously disfigured by the superfluity of assumptions, and one of\nthem should be eliminated. If we are to assume anything unnecessarily,\nwe may as well assume the whole geometry and be done with it.\n\nThe geometer's business then is to draw up a set of postulates. This\nhe may do on any basis whatever. They may be suggested to him by\nthe behavior of points, lines and planes, or by some other concrete\nphenomena; they may with equal propriety be the product of an\ninventive imagination. On proceeding to deduce their consequences,\nhe will discover and remedy any lack of categoricity or consistence\nor independence which his original system of postulates may have\nlacked. In the end he will have so large a body of propositions without\ncontradiction or failure that he will conclude the propriety of his\npostulates to have been established, and the geometry based on them\nto be a valid one.\n\n\n\n\nAND WHAT IS IT ALL ABOUT?\n\nIs this geometry ever realized? Strictly it is not the geometer's\nbusiness to ask or answer this question. But research develops two\nviewpoints. There is always the man who indulges in the pursuit of\nfacts for their sake alone, and equally the man who wants to see his\nnew facts lead to something else. One great mathematician is quoted\nas enunciating a new theory of surpassing mathematical beauty with\nthe climacteric remark \"And, thank God, no one will ever be able to\nfind any use for it.\" An equally distinguished contemporary, on being\ninterrogated concerning possible applications for one of his most\nabstruse theorems, replied that he knew no present use for it; but that\nlong experience had made him confident that the mathematician would\nnever develop any tool, however remote from immediate utility, for\nwhich the delvers in other fields would not presently find some use.\n\nIf we wish, however, we may inquire with perfect propriety, from the\nside lines, whether a given geometry is ever realized. We may learn\nthat so far as has yet been discovered there are no elements for\nwhich all its postulates are verified, and that there is therefore no\nrealization known. On the other hand, we may more likely find that\nmany different sets of elements are such that the postulates can be\ninterpreted as applying to them, and that we therefore have numerous\nrealizations of the geometry. As a human being the geometer may be\ninterested in all this, but as a geometer it really makes little\ndifference to him.\n\nWhen we look at space about us, we see it, for some reason grounded\nin the psychological history of the human race, as made up in the\nsmall of points, which go to make up lines, which in turn constitute\nplanes. Or we can start at the other end and break space down first\ninto planes, then into lines, finally into points. Our perceptions\nand conceptions of these points, lines and planes are very definite\nindeed; it seems indeed, as the Greeks thought, that certain things\nabout them are self-evident. If we wish to take these self-evident\nproperties of point, line and plane, and combine with them enough\nadditional hair-splitting specifications to assure the modern geometer\nthat we have really a categorical system of assumptions, we shall\nhave the basis of a perfectly good system of geometry. This will be\nwhat we unavoidably think of as the absolute truth with regard to\nthe space about us; but you mustn't say so in the presence of the\ngeometer. It will also be what we call the Euclidean geometry. It\nhas been satisfactory in the last degree, because not only space,\nbut pretty much every other system of two or three elements bearing\nany relations to one another can be made, by employing as a means\nof interpretation the Cartesian scheme of plotting, to fit into the\nframework of Euclidean geometry. But it is not the only thing in the\nworld of conceptual possibilities, and it begins to appear that it\nmay not even be the only thing in the world of cold hard fact that\nsurrounds us. To see just how this is so we must return to Euclid,\nand survey the historical development of geometry from his day to\nthe present time.\n\n\n\n\nEUCLID'S GEOMETRY\n\nPoint, line and plane Euclid attempts to define. Modern objection\nto these efforts was made clear above. Against Euclid's specific\nperformance we urge the further specific fault that his \"definitions\"\nare really assumptions bestowing certain properties upon points, lines\nand planes. These assumptions Euclid supplements in his axioms; and in\nthe process of proving propositions he unconsciously supplements them\nstill further. This is to be expected from one whose justification for\nlaying down an axiom was the alleged obvious character of the statement\nmade. If some things are too obvious to require demonstration, others\nmay be admitted as too obvious to demand explicit statement at all.\n\nThus, if Euclid has two points A and B in a plane, on opposite sides of\na line M, he will draw the line AB and without further formality speak\nof the point C in which it intersects M. That it does so intercept M,\nrather than in some way dodges it, is really an assumption as to the\nnature of lines and planes. Or again, Euclid will speak of a point D\non the line AB, between or outside the points A and B, without making\nthe formal assumption necessary to insure that the line is \"full\" of\npoints so that such a point as D must exist. That such assumptions as\nthese are necessary follows from our previous remarks. If we think of\nour geometry as dealing with \"chings,\" \"changs,\" and \"chungs,\" or with\nelements I, II and III, it is no longer in the least degree obvious\nthat the simplest property in the world applies to these elements. If\nwe wish any property to prevail we must state it explicitly.\n\nWith the postulates embodied in his definitions, those stated in his\naxioms, and those which he reads into his structure by his methods of\nproof, Euclid has a categorical set--enough to serve as foundation\nfor a geometry. We may then climb into Euclid's shoes and take the\nnext step with him. We follow him while he proves a number of things\nabout intersecting lines and about triangles. To be sure, when he\nproves that two triangles are identically constituted by moving one of\nthem over on top of the other, we may protest on the ground that the\nadmission of motion, especially of motion thus imposed from without,\ninto a geometry of things is not beyond dispute. If Euclid has caught\nour modern viewpoint, he will rejoin that if we have any doubts as to\nthe admissibility of motion he will lay down a postulate admitting it,\nand we shall be silenced.\n\nHaving exhausted for the present the interest of intersecting lines,\nour guide now passes to a consideration of lines in the same plane\nthat never meet. He defines such lines as parallel. If we object that\nhe should show the existence of a derived concept like this before\nlaying down a definition that calls for it to exist, he can show that\ntwo lines drawn perpendicular to the same line never meet. He will\nexecute this proof by a special sort of superposition, which requires\nthat the plane be folded over on itself, through the third dimension\nof surrounding space, rather than merely slid along upon itself.\n\nWe remain quiet while Euclid demonstrates that if two lines are cut by\nany transversal in such a way as to make corresponding angles at the\ntwo intersections equal, the lines are parallel. It is then in order\nto investigate the converse: if the lines are parallel to begin with,\nare the angles equal?\n\n\n\n\nAXIOMS MADE TO ORDER\n\nThis sounds innocent enough; but in no way was Euclid able to devise a\nproof--or, for that matter, a disproof. So he took the only way out,\nand said that if the lines were parallel, obviously they extended in\nthe same direction and made the angles equal. The thing was so obvious,\nhe argued, that it was really an axiom and he didn't have to prove\nit; so he stated it as an axiom and proceeded. He didn't state it\nin precisely the form I have used; he apparently cast about for the\nform in which it would appear most obvious, and found a statement that\nsuited him better than this one, and that comes to the same thing. This\nstatement tells us that if the transversal makes two corresponding\nangles unequal, the lines that it cuts are not parallel and do meet\nif sufficiently prolonged. But wisely enough, he did not transplant\nthis axiom, once he had arrived at it, to the beginning of the book\nwhere the other axioms were grouped; he left it right where it was,\nfollowing the proposition that if the angles were equal the lines were\nparallel. This of course was so that it might appeal back, for its\nclaim to obviousness, to its demonstrated converse of the proposition.\n\nEuclid must have been dissatisfied with this cutting of the Gordian\nknot; his successors were acutely so. For twenty centuries the parallel\naxiom was regarded as the one blemish in an otherwise perfect work;\nevery respectable mathematician had his shot at removing the defect by\n\"proving\" the objectionable axiom. The procedure was always the same:\nexpunge the parallel axiom, in its place write another more or less\n\"obvious\" assumption, and from this derive the parallel statement more\nor less directly. Thus if we may assume that the sum of the angles\nof a triangle is always exactly 180 degrees, or that there can be\ndrawn only one line through a given point parallel to a given line,\nwe can prove Euclid's axiom. Sometimes the substitute assumption was\nopenly made and stated, as in the two instances cited; as often it\nwas admitted into the demonstration implicitly, as when it is quietly\nassumed that we can draw a triangle similar to any given triangle and\nwith area as great as we please, or when parallels are \"defined\" as\neverywhere equidistant. But such \"proofs\" never satisfied anyone other\nthan the man who made them; the search went merrily on for a valid\n\"proof\" that should not in substance assume the thing to be proved.\n\n\n\n\nLOCATING THE DISCREPANCY\n\nSaccheri, an Italian Jesuit, would have struck bottom if he had had a\nlittle more imagination. He gave an exhaustive reductio ad absurdum,\non the basis of the angle-sum theorem. This sum must be (a) greater\nthan or (b) equal to or (c) less than 180 degrees. Saccheri showed\nthat if one of these alternatives occurs in a single triangle, it must\noccur in every triangle. The first case gave little trouble; admitting\nthe possibility of superposing in the special manner mentioned above,\nwhich he did implicitly, he showed that this \"obtuse-angled hypothesis\"\ncontradicted itself. He pursued the \"acute-angled hypothesis\" for\na long time before he satisfied himself that he had caught it, too,\nin an inconsistency. This left only the \"right-angled hypothesis,\"\nproving the Euclidean angle-sum theory and through it the parallel\npostulate. But Saccheri was wrong: he had found no actual contradiction\nin the acute-angled hypothesis--for none exists therein.\n\nThe full facts were probably first known to Gauss, who had a finger in\nevery mathematical pie that had to do with the transition to modern\ntimes. They were first published by Lobatchewsky, the Russian, who\nanticipated the Hungarian John Bolyai by a narrow margin. All three\nworked independently of Saccheri, whose book, though theoretically\navailable in Italian libraries, was actually lost to sight and had\nto be rediscovered in recent years.\n\nLike Saccheri, Lobatchewsky investigated alternative possibilities. But\nhe chose another point of attack: through a given point it must be\npossible to draw, in the same plane with a given line (a) no lines or\n(b) one line or (c) a plurality of lines, which shall not meet the\ngiven line. The word parallel is defined only in terms of the second\nof these hypotheses, so we avoid it here. These three cases correspond,\nrespectively, to those of Saccheri.\n\nThe first case Lobatchewsky ruled out just as did Saccheri, but\naccepting consciously the proviso attached to its elimination;\nthe third he could not rule out. He developed the consequences of\nthis hypothesis as far as Euclid develops those of the second one,\nsketching in a full outline for a system of geometry and trigonometry\nbased on a plurality of \"non-cutters.\" This geometry constitutes a\ncoherent whole, without a logical flaw.\n\nThis made it plain what was the matter with Euclid's parallel\naxiom. Nobody could prove it from his other assumptions because it\nis not a consequence of these. True or false, it is independent of\nthem. Trinity Church is in New York, Faneuil Hall is in Boston, but\nFaneuil Hall is not in Boston because Trinity is in New York; and we\ncould not prove that Faneuil Hall was in Boston if we knew nothing\nabout America save that Trinity is in New York. The mathematicians\nof 2,000 years had been pursuing, on a gigantic scale, a delusion of\npost hoc, ergo propter hoc.\n\n\n\n\nWHAT THE POSTULATE REALLY DOES\n\nMoreover, in the absence of an assumption covering the ground, we\nshall not know which of the alternatives (a), (b), (c) holds. But\nwhen one holds in a single case it holds permanently, as Saccheri and\nLobatchewsky both showed. So we cannot proceed on this indefinite\nbasis; we must know which one is to hold. Without the parallel\npostulate or a substitute therefor that shall tell us the same thing\nor tell us something different, we have not got a categorical set of\nassumptions--we cannot build a geometry at all. That is why Euclid\nhad to have his parallel postulate before he could proceed. That is\nwhy his successors had to have an assumption equivalent to his.\n\nThe reason why it took so long for this to percolate into the\nunderstanding of the mathematicians was that they were thinking,\nnot in terms of the modern geometry and about undefined elements;\nbut in terms of the old geometry and about strictly defined and\ncircumscribed elements. If we understand what is meant by Euclidean\nline and plane, of course the parallel postulate, to use the old\ngeometer's word, is true--of course, to adopt the modern viewpoint,\nif we agree to employ an element to which that assumption applies,\nthe assumption is realized. The very fact of accepting the \"straight\"\nline and the \"flat\" plane of Euclid constitutes acceptance of his\nparallel postulate--the only thing that can separate his geometry\nfrom other geometries. But of course we can't prove it; the prior\npostulates which we would have to use in such an attempt apply where\nit does not apply, and hence it cannot possibly be consequences of.\n\nTo all this the classical Euclidean rejoins that we seem to have\nin mind elements of some sort to which, with one reservation, his\npostulates apply. He wants to know what these elements look like. We\ncan, and must, produce them--else our talk about generality is mere\ndrivel. But we must take care that the Euclidean geometer does not\ntry to apply to our elements the notions of straightness and flatness\nwhich inhere in the parallel postulate. We cannot satisfy and defy\nthat postulate at the same time. If we do not insist on this point, we\nshall find that we are reading non-Euclidean properties into Euclidean\ngeometry, and interpreting the elements of the latter as straight\nlines that are not straight, flat planes that are not flat. It is not\nthe mission of non-Euclidean geometry thus to deny the possibility\nof Euclidean geometry; it merely demands a place of equal honor.\n\n\n\n\nTHE GEOMETRY OF SURFACES\n\nLet us ask the Euclidean geometer whether he can recognize his plane\nafter we have crumpled it up like a piece of paper en route to the\nwaste basket. He will hesitate only long enough to recall that in\nthe special case of superposition he has reserved for himself the\nprivilege of deforming his own plane, and to realize that he can\nalways iron his plane out smooth again after we are through with\nit. This emphasizes the true nature of the two-dimensionality which\nis the fundamental characteristic of the plane (and of other things,\nas we shall directly see). The plane is two-dimensional in points not\nbecause two sets of mutually perpendicular Euclidean straight lines\ncan be drawn in it defining directions of north-south and east-west,\nbut because a point in it can be located by means of two measures. The\nsame statement may be made of anything whatever to which the term\n\"surface\" is applicable; anything, however crumpled or irregular it\nbe, that possesses length and breadth without thickness. The surface\nof a sphere, of a cylinder, of an ellipsoid, of a cone, of a doughnut\n(mathematically known as a torus), of a gear wheel, of a French horn,\nall these possess two-dimensionality in points; on all of them we can\ndraw lines and curves and derive a geometry of these figures. If we\nget away from the notion that geometry of two dimensions must deal\nwith planes, and adopt in place of this idea the broader restriction\nthat it shall deal with surfaces, we shall have the generalization\nwhich the Euclidean has demanded that we produce, and the one which\nin the hands of the modern geometer has shown results.\n\nIn this two-dimensional geometry of surfaces in general, that of the\nplane is merely one special case. Certain of the features met in that\ncase are general. If we agree that we know what we mean by distance,\nwe find that on every surface there is a shortest distance between\ntwo points, together with a series of lines or curves along which\nsuch distances are taken. These lines or curves we call geodesics. On\nthe plane the geodesic is the straight line. On surfaces in general\nthe geodesic, whatever its particular and peculiar shape, plays\nthe same r\u00f4le that is played by the straight line in the plane;\nit is the secondary element of the geometry, the surface itself\nand all other surfaces of its type are the tertiary elements. And\nit is a fact that we can take all the possible spheres, or all the\npossible French-horn surfaces, and conceive of space as we know it\nbeing broken down by analysis into these surfaces instead of into\nplanes. The only reason we habitually decompose space into planes is\nbecause it comes natural to us to think that way. But geometric points,\nlines and surfaces must be recognized as abstractions without actual\nexistence, for all of them lack one or more of the three dimensions\nwhich such existence implies. These figures exist in our minds but\nnot in the external world about us. So any decomposition of space\ninto geometric elements is a phenomenon of the mind only; it has no\nparallel and no significance in the external world, and is made in\none way or in another purely at our pleasure. There isn't a true,\nhonest-to-goodness geometrical plane in existence any more than there\nis an honest-to-goodness spherical surface: so on intrinsic grounds\none decomposition is as reasonable as another.\n\nCertain of the most fundamental postulates are obeyed by all\nsurfaces. As we attempt to discriminate between surfaces of different\ntypes, and get, for instance, a geometry that shall be valid for\nspheres and ellipsoids but not for conicoids in general, we must do\nso by bringing in additional postulates that embody the necessary\nrestrictions. A characteristic shared by planes, spheres, and various\nother surfaces is that the geodesics can be freely slid along upon\nthemselves and will coincide with themselves in all positions when\nthus slid; with a similar arrangement for the surface itself. But\nthe plane stands almost unique among surfaces in that it does not\nforce us to distinguish between its two sides; we can turn it over\nand still it will coincide with itself; and this property belongs\nalso to the straight line. It does not belong to the sphere, or to\nthe great circles which are the geodesics of spherical geometry;\nwhen we turn one of these over, through the three-dimensional space\nthat surrounds it, we find that the curvature lies in the wrong way\nto make superposition possible. If we postulate that superposition be\npossible under such treatment, we throw out the sphere and spherical\ngeometry; if we postulate that superposition be only by sliding the\nsurface upon itself we admit that geometry--as Saccheri failed to\nsee, as Lobatchewsky realized, and as Riemann showed at great length\nin rehabilitating the \"obtuse-angled hypothesis.\" Lobatchewsky's\nacute-angled geometry is realized on a surface of the proper sort,\nwhich admits of unrestricted superposition; but it is not the sort\nof a surface that I care to discuss in an article of this scope.\n\nEuclidean geometry is the natural and easy one, I suppose, because it\nmakes it easy to stop with three dimensions. If we take a secondary\nelement, a geodesic, which is \"curved\" in the Euclidean sense, we\nget a tertiary element, a surface, which is likewise curved. Then\nunless we are to make an altogether abrupt and unreasonable break,\nwe shall find that just as the curved geodesic generated a curved\nsurface, the curved surface must give rise to a \"curved space\"; and\njust as the curved geodesic needed a second dimension to curve into,\nand the curved surface a third, so the curved three-space requires\na fourth. Once started on this sort of thing, there doesn't really\nseem to be any end.\n\n\n\n\nEUCLIDEAN OR NON-EUCLIDEAN\n\nNevertheless, we must face the possibility that the space we live\nin, or any other manifold of any sort whatever with which we deal on\ngeometric principles, may turn out to be non-Euclidean. How shall we\nfinally determine this? By measures--the Euclidean measures the angles\nof an actual triangle and finds the sum to be exactly 180 degrees;\nor he draws parallel lines of indefinite extent and finds them to\nbe everywhere equally distant; and from these data he concludes that\nour space is really Euclidean. But he is not necessarily right.\n\nWe ask him to level off a plot of ground by means of a plumb\nline. Since the line always points to the earth's center, the \"level\"\nplot is actually a very small piece of a spherical surface. Any test\nconducted on this plot will exhibit the numerical characteristics of\nthe Euclidean geometry; yet we know the geometry of this surface is\nRiemannian. The angle-sum is really greater than 180 degrees; lines\nthat are everywhere equidistant are not both geodesics.\n\nThe trouble, of course, is that on this plot we deal with\nso minute a fraction of the whole sphere that we cannot make\nmeasurements sufficiently refined to detect the departure from\nEuclidean standards. So it is altogether sensible for us to ask:\n\"Is the universe of space about us really Euclidean in whatever of\nrealized geometry it presents to us? Or is it really non-Euclidean,\nbut so vast in size that we have never yet been able to extend our\nmeasures to a sufficiently large portion of it to make the divergence\nfrom the Euclidean standard discernible to us?\"\n\nThis discussion is necessarily fragmentary, leaving out much that\nthe writer would prefer to include. But it is hoped that it will\nnevertheless make it clear that when the contestants in the Einstein\ncompetition speak of a non-Euclidean universe as apparently having\nbeen revealed by Einstein, they mean simply that to Einstein has\noccurred a happy expedient for testing Euclideanism on a smaller\nscale than has heretofore been supposed possible. He has devised a\nnew and ingenious sort of measure which, if his results be valid,\nenables us to operate in a smaller region while yet anticipating that\nany non-Euclidean characteristics of the manifold with which we deal\nwill rise above the threshold of measurement. This does not mean that\nEuclidean lines and planes, as we picture them in our mind, are no\nlonger non-Euclidean, but merely that these concepts do not quite so\nclosely correspond with the external reality as we had supposed.\n\nAs to the precise character of the non-Euclideanism which is revealed,\nwe may leave this to later chapters and to the competing essayists. We\nneed only point out here that it will not necessarily be restricted\nto the matter of parallelism. The parallel postulate is of extreme\ninterest to us for two reasons; first because historically it was the\nmeans by which the possibilities and the importance of non-Euclidean\ngeometry were forced upon our attention; and second because it happens\nto be the immediate ground of distinction between Euclidean geometry\nand two of the most interesting alternatives. But Euclidean geometry\nis characterized, not by a single postulate, but by a considerable\nnumber of postulates. We may attempt to omit any one of these so that\nits ground is not specifically covered at all, or to replace any one\nof them by a direct alternative. We might conceivably do away with the\nsuperposition postulate entirely, and demand that figures be proved\nequivalent, if at all, by some more drastic test. We might do away\nwith the postulate, first properly formulated by Hilbert, on which our\nideas of the property represented in the word \"between\" depend. We\nmight do away with any single one of the Euclidean postulates, or\nwith any combination of two or more of them. In some cases this would\nlead to a lack of categoricity and we should get no geometry at all;\nin most cases, provided we brought a proper degree of astuteness to\nthe formulation of alternatives for the rejected postulates, we should\nget a perfectly good system of non-Euclidean geometry: one realized,\nif at all, by other elements than the Euclidean point, line and plane,\nand one whose elements behave toward one another differently from\nthe Euclidean point, line and plane.\n\nMerely to add definiteness to this chapter, I annex here the\nstatement that in the geometry which Einstein builds up as more\nnearly representing the true external world than does Euclid's, we\nshall dispense with Euclid's (implicit) assumption, underlying his\n(explicitly stated) superposition postulate, to the effect that the\nact of moving things about does not affect their lengths. We shall\nat the same time dispense with his parallel postulate. And we shall\nadd a fourth dimension to his three--not, of course, anything in the\nnature of a fourth Euclidean straight line perpendicular, in Euclidean\nspace, to three lines that are already perpendicular to each other,\nbut something quite distinct from this, whose nature we shall see\nmore exactly in the next chapter. If the present chapter has made it\nclear that it is proper for us to do this, and has prevented anyone\nfrom supposing that the results of doing it must be visualized in a\nEuclidean space of three dimensions or of any number of dimensions,\nit will have served its purpose.\n\n\n\n\n\n\n\n\n\nVI\n\nTHE SPACE-TIME CONTINUUM\n\nMinkowski's World of Events, and the Way It Fits Into Einstein's\nStructure\n\nBY THE EDITOR, EXCEPT AS NOTED\n\n\nSeeking a basis for the secure formulation of his results, and\nespecially a means for expressing mathematically the facts of the\ndependence which he had found to exist between time and space, Einstein\nfell back upon the prior work of Minkowski. It may be stated right\nhere that the idea of time as a fourth dimension is not particularly\na new one. It has been a topic of abstract speculation for the best\npart of a century, even on the part of those whose notions of the\nfourth dimension were pretty closely tied down to the idea of a\nfourth dimension of Euclidean point-space, which would be marked by\na fourth real line, perpendicular to the other three, and visible\nto us if we were only able to see it. Moreover, every mathematician,\nwhether or not he be inclined to this sort of mental exercise, knows\nwell that whenever time enters his equations at all, it does so on an\nabsolutely equal footing with each of his space coordinates, so that\nas far as his algebra is concerned he could never distinguish between\nthem. When the variables $x$, $y$, $z$, $t$ come to the mathematician\nin connection with some physical investigation, he knows before he\nstarts that the first three represent the dimensions of Euclidean\nthree-space and that the last stands for time. But if the algebraic\nexpressions of such a problem were handed to him independently of\nall physical tie-up, he would never be able to tell, from them alone,\nwhether one of the four variables represented time, or if so, which\none to pick out for this distinction.\n\nIt was Minkowski who first formulated all this in a form susceptible\nof use in connection with the theory of relativity. His starting point\nlies in the distinction between the point and the event. Mr. Francis\nhas brought this out rather well in his essay, being the only\ncompetitor to present the Euclidean geometry as a real predecessor\nof Newtonian science, rather than as a mere part of the Newtonian\nsystem. I think his point here is very well taken. As he says,\nEuclid looked into the world about him and saw it composed of\npoints. Ignoring all dynamic considerations, he built up in his\nmind a static world of points, and constructed his geometry as a\nscientific machine for dealing with this world in which motion played\nno part. It could to be sure be introduced by the observer for his\nown purposes, but when so introduced it was specifically postulated\nto be a matter of no moment at all to the points or lines or figures\nthat were moved. It was purely an observational device, intended\nfor the observer's convenience, and in the bargain a mental device,\ncalling for no physical action and the play of no force. So far as\nEuclid in his daily life was obliged to take cognizance of the fact\nthat in the world of work-a-day realities motion existed, he must,\nas a true Greek, have looked upon this as a most unfortunate deviation\nof the reality from his beautiful world of intellectual abstraction,\nand as something to be deplored and ignored. Even in their statuary\nthe Greeks clung to this idea. A group of marvelous action, like\nthe Laocoon, they held to be distinctly a second rate production,\na prostitution of the noble art; their ideal was a figure like the\nmajestic Zeus--not necessarily a mere bust, be it understood, but\nalways a figure in repose without action. Their statuary stood for\nthings, not for action, just as their geometry stood for points,\nnot for events.\n\nGalileo and Newton took a different viewpoint. They were interested\nin the world as it is, not as it ought to be; and if motion appears\nto be a fundamental part of that world, they were bound to include\nit in their scheme. This made it necessary for them to pay much more\nattention to the concept of time and its place in the world than did\nthe Greeks. In the superposition process, and even when he allowed\na curve to be generated by a moving point, the sole interest which\nEuclid had in the motion was the effect which was to be observed upon\nhis static figures after its completion. In this effect the rate of\nthe motion did not enter. So all questions of velocity and time are\ncompletely ignored, and we have in fact the curious spectacle of\nmotion without time.\n\nTo Galileo and Newton, on the other hand, the time which it took a\nbody to pass from one point of its path to another was of paramount\nimportance. The motion itself was the object of their study, and they\nrecognized the part played by velocity. But Galileo and Newton were\nstill sufficiently under the influence of Euclid to fit the observed\nphenomena of motion, so far as they could, upon Euclid's static world\nof points. This they effected by falling in with the age-old procedure\nof regarding time and space as something entirely disassociated and\ndistinct. The motion of an object--in theory, of a point--was to be\nrecorded by observing its successive positions. With each of these\npositions a time was to be associated, marking the instant at which\nthe point attained that position. But in the face of this association,\nspace and time were to be maintained as entirely separate entities.\n\n\n\n\nTHE FOUR-DIMENSIONAL WORLD OF EVENTS\n\nThis severe separation of time and space Minkowski has now questioned,\nwith the statement that the elements of which the external\nworld is composed, and which we observe, are not points at all,\nbut are events. This calls for a revision of our whole habit of\nthought. It means that the perceptual world is four-dimensioned,\nnot three-dimensioned as we have always supposed; and it means, at\nthe very least, that the distinction between time and space is not\nso fundamental as we had supposed.\n\n[This should not impress us as strange or incomprehensible. What\ndo we mean when we say that a plane is two-dimensional? Simply that\ntwo coordinates, two numbers, must be given to specify the position\nof any point of the plane. Similarly for a point in the space of our\naccustomed concepts we must give three numbers to fix the position--as\nby giving the latitude and longitude of a point on the earth and its\nheight above sea-level. So we say this space is three-dimensional. But\na material body is not merely somewhere; it is somewhere now,]182 or\nwas somewhere yesterday, or will be somewhere tomorrow. The statement\nof position for a material object is meaningless unless we at the\nsame time specify the time at which it held that position. [If I\nam considering the life-history of an object on a moving train,\nI must give three space-coordinates and one time-coordinate to fix\neach of its positions.]182 And each of its positions, with the time\npertaining to that position, constitutes an event. The dynamic,\never-changing world about us, that shows the same aspect at no two\ndifferent moments, is a world of events; and since four measures or\ncoordinates are required to fix an event, we say this world of events\nis four-dimensional. If we wish to test out the soundness of this\nviewpoint, we may well do so by asking whether the naming of values\nfor the four coordinates fixes the event uniquely, as the naming of\nthree under the old system fixes the point uniquely.\n\nSuppose we take some particular event as the one from which to measure,\nand agree upon the directions to be taken by our space axes, and make\nany convention about our time-axis which subsequent investigation\nmay show to be necessary. Certainly then the act of measuring so many\nmiles north, and so many west, and so many down, and so many seconds\nbackward, brings us to a definite time and place--which is to say,\nto a definite event. Perhaps nothing \"happened\" there, in the sense\nin which we usually employ the word; but that is no more serious\nthan if we were to locate a point with reference to our familiar\nspace coordinate system, and find it to lie in the empty void of\ninterstellar space, with no material body occupying it. In this\nsecond case we still have a point, which requires, to insure its\nexistence and location, three coordinates and nothing more; in the\nfirst case we still have an event, which requires for its existence\nand definition four coordinates and nothing more. It is not an event\nabout which the headline writers are likely to get greatly excited;\nbut what of that? It is there, ready and waiting to define any physical\nhappening that falls upon it, just as the geometer's point is ready\nand waiting to define any physical body that chances to fall upon it.\n\n\n\n\nA CONTINUUM OF POINTS\n\nIt is now in order to introduce a word, which I shall have to confess\nthe great majority of the essayists introduce, somewhat improperly,\nwithout explanation. But when I attempt to explain it, I realize\nquite well why they did this. They had to have it; and they didn't\nhave space in their three thousand words to talk adequately about\nit and about anything else besides. The mathematician knows very\nwell indeed what he means by a continuum; but it is far from easy\nto explain it in ordinary language. I think I may do best by talking\nfirst at some length about a straight line, and the points on it.\n\nIf the line contains only the points corresponding to the integral\ndistances 1, 2, 3, etc., from the starting point, it is obviously\nnot continuous--there are gaps in it vastly more inclusive than the\nfew (comparatively speaking) points that are present. If we extend\nthe limitations so that the line includes all points corresponding\nto ordinary proper and improper fractions like 1/4 and 17/29 and\n1633/7--what the mathematician calls the rational numbers--we shall\napparently fill in these gaps; and I think the layman's first impulse\nwould be to say that the line is now continuous. Certainly we cannot\nstand now at one point on the line and name the \"next\" point, as we\ncould a moment ago. There is no \"next\" rational number to 116/125,\nfor instance; 115/124 comes before it and 117/126 comes after it,\nbut between it and either, or between it and any other rational\nnumber we might name, lie many others of the same sort. Yet in spite\nof the fact that the line containing all these rational points is now\n\"dense\" (the technical term for the property I have just indicated),\nit is still not continuous; for I can easily define numbers that\nare not contained in it--irrational numbers in infinite variety\nlike $\\sqrt{2}$; or, even worse, the number pi = 3.141592 ... which\ndefines the ratio of the circumference of a circle to the diameter,\nand many other numbers of similar sort.\n\nIf the line is to be continuous, there may be no holes in it at all;\nit must have a point corresponding to every number I can possibly\nname. Similarly for the plane, and for our three-space; if they are\nto be continuous, the one must contain a point for every possible\npair of numbers $x$ and $y$, and the other for every possible set\nof three numbers $x$, $y$ and $z$, that I can name. There may be no\nholes in them at all.\n\nA line is a continuum of points. A plane is a continuum of points. A\nthree-space is a continuum of points. These three cases differ only in\ntheir dimensionality; it requires but one number to determine a point\nof the first continuum, two and three respectively in the second and\nthird cases. But the essential feature is not that a continuum shall\nconsist of points, or that we shall be able to visualize a pseudo-real\nexistence for it of just the sort that we can visualize in the case\nof line, plane and point. The essential thing is merely that it shall\nbe an aggregate of elements numerically determined in such a way as to\nleave no holes, but to be just as continuous as the real number system\nitself. Examples, however, aside from the three which I have used,\nare difficult to construct of such sort that the layman shall grasp\nthem readily; so perhaps, fortified with the background of example\nalready presented, I may venture first upon a general statement.\n\n\n\n\nTHE CONTINUUM IN GENERAL\n\nSuppose we have a set of \"elements\" of some sort--any sort. Suppose\nthat these elements possess one or more fundamental identifying\ncharacteristics, analogous to the coordinates of a point, and\nwhich, like these coordinates, are capable of being given numerical\nvalues. Suppose we find that no two elements of the set possess\nidentically the same set of defining values. Suppose finally--and\nthis is the critical test--that the elements of the set are such that,\nno matter what numerical values we may specify, it we do specify the\nproper number of defining magnitudes we define by these an actual\nelement of the set, that corresponds to this particular collection\nof values. Our elements then share with the real number system\nthe property of leaving no holes, of constituting a continuous\nsuccession in every dimension which they possess. We have then a\ncontinuum. Whatever its elements, whatever the character of their\nnumerical identifiers, whatever the number n of these which stands for\nits dimension, there may be no holes or we have no continuum. There\nmust be an element for every possible combination of n numbers\nwe can name, and no two of these combinations may give the same\nelement. Granted this condition, our elements constitute a continuum.\n\nAs I have remarked, it is not easy to cite examples of continua which\nshall mean anything to the person unaccustomed to the term. The\ntotality of carbon-oxygen-nitrogen-hydrogen compounds suggested\nby one essayist as an example is not a continuum at all, for the\nset contains elements corresponding only to integer values of the\nnumbers which tell us how many atoms of each substance occur in the\nmolecule. We cannot have a compound containing $\\sqrt2$ carbon atoms,\nor $3\\pi$ oxygen atoms. Perhaps the most satisfactory of the continua,\noutside the three Euclidean space-continua already cited, [is the\nmanifold of music notes. This is four-dimensional; each note has\nfour distinctions--length, pitch, intensity, timbre--to distinguish\nit perfectly, to tell how long, how high, how loud, how rich.]263\nWe might have a little difficulty in reducing the characteristic of\nrichness to numerical expression, but presumably it could be done;\nand we should then be satisfied that every possible combination of four\nvalues, $l$, $p$, $i$, $t$ for these four identifying characteristics\nwould give us a musical effect, and one to be confused with no other.\n\nThere is in the physical world a vast quantity of continua of one\nsort or another. The music-note continuum brings attention to the\nfact that not all of these are such that their elements make their\nappeal to the visual sense. This remark is a pertinent one; for we are\nby every right of heritage an eye-minded race, and it is frequently\nnecessary for us to be reminded that so far as the external world is\nconcerned, the verdict of every other sense is entirely on a par with\nthat of sight. The things which we really see, like matter, and the\nthings which we abstract from these visual impressions, like space,\nare by no means all there is to the world.\n\n\n\n\nEUCLIDEAN AND NON-EUCLIDEAN CONTINUA\n\nIf we are dealing with a continuum of any sort whatever having one\nor two or three dimensions, we are able to represent it graphically\nby means of the line, the plane, or the three-space. The same set of\nnumbers that defines an element of the given continuum likewise defines\nan element of the Euclidean continuum of the same dimensionality;\nso the one continuum corresponds to the other, element for element,\nand either may stand for the other. But if we have a continuum of four\nor more dimensions, this representation breaks down in the absence\nof a real, four-dimensional Euclidean point-space to serve as a\npicture. This does not in the least detract from the reality of the\ncontinuum which we are thus prevented from representing graphically\nin the accustomed fashion.\n\nThe Euclidean representation, in fact, may in some cases be\nunfortunate--it may be so entirely without significance as to be\nactually misleading. For in the Euclidean continuum of points, be\nit line, plane or three-space, there are certain things which we\nordinarily regard as secondary derived properties, but which possess\na great deal of significance none the less.\n\nIn particular, in the Euclidean plane and in Euclidean three-space,\nthere is the distance between two points. I have indicated, in\nthe chapter on non-Euclidean geometry, that the parallel postulate\nof Euclid, which distinguishes his geometry from others, could be\nreplaced by any one of numerous other postulates. Grant Euclid's\npostulate and you can prove any of these substitutes; grant any of\nthe substitutes and you can prove Euclid's postulate. Now it happens\nthat there is one of these substitutes to which modern analysis has\ngiven a position of considerable importance. It is merely our good\nold friend the Pythagorean theorem, that the square on the hypotenuse\nequals the sum of the squares on the sides; but it is dressed in new\nclothes for the present occasion.\n\nMr. Francis' discussion of this part of the subject, and especially his\nfigure, ought to make it clear that this theorem can be considered\nas dealing with the distance between any two points. When we so\nconsider it, and take it as the fundamental, defining postulate of\nEuclidean geometry which distinguishes this geometry from others,\nwe have a statement of considerable content. We have, first, that the\ncharacteristic property of Euclidean space is that the distance between\ntwo points is given by the square root of the sum of the squares of\nthe coordinate-differences for these points--by the expression\n\n    $$D = \\sqrt{(X - x)^2 + (Y - y)^2 + (Z - z)^2}\\,,$$\n\nwhere the large letters represent the coordinates of the one point\nand the small ones those of the other. We have more than this,\nhowever; we have that this distance is the same for all observers,\nno matter how different their values for the individual coordinates\nof the individual points. And we have, finally, as a direct result of\nlooking upon the thing from this viewpoint, that the expression for $D$\nis an \"invariant\"; which simply means that every observer may use the\nsame expression in calculating the value of $D$ in terms of his own\nvalues for the coordinates involved. The distance between two points\nin our space is given numerically by the square root of the sum of the\nsquares of my coordinate-differences for the two points involved; it\nis given equally by the square root of the sum of the squares of your\ncoordinate-differences, or those of any other observer whatsoever. We\nhave then a natural law--the fundamental natural law characterizing\nEuclidean space. If we wish to apply it to the Euclidean two-space (the\nplane) we have only to drop out the superfluous coordinate-difference;\nif we wish to see by analogy what would be the fundamental natural law\nfor a four-dimensional Euclidean space, we have only to introduce under\nthe radical a fourth coordinate-difference for the fourth dimension.\n\nIf we were not able to attach any concrete meaning to the expression\nfor $D$ the value of all this would be materially lessened. Consider,\nfor instance, the continuum of music notes. There is no distance\nbetween different notes. There is of course significance in talking\nabout the difference in pitch, in intensity, in duration, in timbre,\nbetween two notes; but there is none in a mode of speech that implies\na composite expression indicating how far one note escapes being\nidentical with another in all four respects at once. The trouble,\nof course, is that the four dimensions of the music-note continuum\nare not measurable in terms of a common unit. If they were, we should\nexpect to measure their combination more or less absolutely in terms\nof this same unit. We can make measurements in all three dimensions\nof Euclidean space with the same unit, with the same measuring\nrod in fact. [This presents a peculiarity of our three-space which\nis not possessed by all three-dimensional manifolds. Riemann has\ngiven another illustration in the system of all possible colors,\ncomposed of arbitrary proportions of the three primaries, red, green\nand violet. This system forms a three-dimensional continuum; but we\ncannot measure the \"distance\" or difference between two colors in\nterms of the difference between two others.]130\n\nAccordingly, in spite of the fact that the Euclidean three-space\ngives us a formal representation of the color continuum, and in\nspite of the fact that the hypothetical four-dimensional Euclidean\nspace would perform a like office for the music-note continuum, this\nrepresentation would be without significance. We should not say that\nthe geometry of these two manifolds is Euclidean. We should realize\nthat any set of numerical elements can be plotted in a Euclidean\nspace of the appropriate dimensionality; and that accordingly, before\nallowing such a plot to influence us to classify the geometry of the\ngiven manifold as Euclidean, we must pause long enough to ask whether\nthe rest of the Euclidean system fits into the picture. If the square\nroot of the sum of the squares of the coordinate-differences between\ntwo elements possesses significance in the given continuum, and if it\nis invariant between observers of that continuum who employ different\nbases of reference, then and only then may we allege the Euclidean\ncharacter of the given continuum.\n\nIf under this test the given continuum fails of Euclideanism, it is\nin order to ask what type of geometry it does present. If it is of\nsuch character that the \"distance\" between two elements possesses\nsignificance, we should answer this question by investigating that\ndistance in the hope of discovering a non-Euclidean expression for it\nwhich will be invariant. If it is not of such character, we should seek\nsome other characteristic of single elements or groups of elements,\nof real physical significance and of such sort that the numerical\nexpression for it would be invariant.\n\nIf the continuum with which we have to do is one in which the\n\"distance\" between two elements possesses significance, and if it\nturns out that the invariant expression for this distance is not\nthe Pythagorean one, but one indicating the non-Euclideanism of\nour continuum, we say that this continuum has a \"curvature.\" This\nmeans that, if we interpret the elements of our continuum as points\nin space (which of course we may properly do) and if we then try to\nsuperpose this point-continuum upon a Euclidean continuum, it will not\n\"go\"; we shall be caught in some such absurdity as trying to force a\nsphere into coincidence with a plane. And of course if it won't go,\nthe only possible reason is that it is curved or distorted, like\nthe sphere, in such a way as to prevent its going. It is unfortunate\nthat the visualizing of such curvature requires the visualizing of\nan additional dimension for the curved continuum to curve into; so\nthat while we can picture a curved surface easily enough, we can't\npicture a curved three-space or four-space. But that is a barrier to\nvisualization alone, and in no sense to understanding.\n\n\n\n\nOUR WORLD OF FOUR DIMENSIONS\n\nIt will be observed that we have now a much broader definition of\nnon-Euclideanism than the one which served us for the investigation\nof Euclid's parallel postulate. If we may at pleasure accept\nthis postulate or replace it by another and different one, we\nmay presumably do the same for any other or any others of Euclid's\npostulates. The very statement that the distance between elements of\nthe continuum shall possess significance, and shall be measurable by\nconsidering a path in the continuum which involves other elements,\nis an assumption. If we discard it altogether, or replace it by one\npostulating that some other joint property of the elements than\ntheir distance be the center of interest, we get a non-Euclidean\ngeometry. So for any other of Euclid's postulates; they are all\nnecessary for a Euclidean system, and in the absence of any one of\nthem we get a non-Euclidean system.\n\nNow the four-dimensional time-space continuum of Minkowski is plainly\nof a sort which ought to make susceptible of measurement the separation\nbetween two of its events. We can pass from one element to another\nin this continuum--from one event to another--by traversing a path\ninvolving \"successive\" events. Our very lives consist in doing just\nthis: we pass from the initial event of our career to the final\nevent by traversing a path leading us from event to event, changing\nour time and space coordinates continuously and simultaneously in the\nprocess. And while we have not been in the habit of measuring anything\nexcept the space interval between two events and the time interval\nbetween two events, separately, I think it is clear enough that,\nconsidered as events, as elements in the world of four dimensions,\nthere is a less separation between two events that occur in my office\non the same day than between two which occur in my office a year\napart; or between two events occurring 10 minutes apart when both\ntake place in my office than when one takes place there and one in\nLondon or on Betelgeuse.\n\nIt is not at all unreasonable, a priori, then, to seek a numerical\nmeasure for the separation, in space-time of four-dimensions, of\ntwo events. If we find it, we shall doubtless be asked just what\nits subjective significance to us is. This must be answered with\nsome circumspection. It will presumably be something which we cannot\nobserve with the visual sense alone, or it would have forced itself\nupon our attention thousands of years ago. It ought, I should think,\nto be something that we would sense by employing at the same time\nthe visual sense and the sense of time-passage. In fact, I might\nvery plausibly insist that, by my very remarks about it in the above\nparagraph, I have sensed it.\n\nMinkowski, however, was not worried about this phase of the matter. He\nhad only to identify the invariant expression for distance; sensing\nit could wait. He found, of course, that this expression was not the\nEuclidean expression for a four-dimensional interval. He had discarded\nseveral of the Euclidean assumptions and could not expect that the\npostulate governing the metric properties of Euclid's space would\npersist. Especially had he violated the Euclidean canons in discarding,\nwith Einstein, the notion that nothing which may happen to a measuring\nrod in the way of uniform translation at high velocity can affect\nits measures. So he had to be prepared to find that his geometry was\nnon-Euclidean; yet it is surprising to learn how slightly it deviates\nfrom that of Euclid. Without any extended discussion to support the\nstatement, we may say that he found that when two observers measure the\ntime- and the space-coordinates of two events, using the assumptions\nand therefore the methods of Einstein and hence subjecting themselves\nto the condition that their measures of the pure time-interval and of\nthe pure space-interval between these events will not necessarily be\nthe same, they will discover that they both get the same value for\nthe expression\n\n    $$S = \\sqrt{(X - x)^2 + (Y - y)^2 + (Z - z)^2 - (CT - Ct)^2}\\,.$$\n\nIf our acceptance of this as the numerical measure of the separation\nin space-time between the two events should lead to contradiction we\ncould not so accept it. No contradiction arises however and we may\ntherefore accept it. And at once the mathematician is ready with some\ninterpretative remarks.\n\n\n\n\nTHE CURVATURE OF SPACE-TIME\n\nThe invariant expression for separation, it will be seen, is in\nthe same form as that of the Euclidean four-dimensional invariant\nsave for the minus sign before the time-difference (the appearance\nof the constant $C$ in connection with the time coordinate $t$ is\nmerely an adjustment of units; see page 153). This tells us that not\nalone is the geometry of the time-space continuum non-Euclidean in its\nmethods of measurement, but also in its results, to the extent that it\npossesses a curvature. It compares with the Euclidean four-dimensional\ncontinuum in much the same way that a spherical surface compares with\na plane. As a matter of fact, a more illuminating analogy here would\nbe that between the cylindrical surface and the plane, though neither\nis quite exact. To make this clear requires a little discussion of\nan elementary notion which we have not yet had to consider.\n\nOur three-dimensional existence often reduces, for all practical\npurposes, to a two-dimensional one. The objects and the events of\na certain room may quite satisfactorily be defined by thinking\nof them, not as located in space, but as lying in the floor of\nthe room. Mathematically the justification for this viewpoint\nis got by saying that we have elected to consider a slice of our\nthree-dimensional world of the sort which we know as a plane. When we\nconsider this plane and the points in it, we find that we have taken\na cross-section of the three-dimensional world. A line in that world\nis now reduced, for us, to a single point--the point where it cuts\nour plane; a plane is reduced to a line--the line where it cuts our\nplane; the three-dimensional world itself is reduced to our plane\nitself. Everything three-dimensional falls down into its shadow in\nour plane, losing in the process that one of the three dimensions\nwhich is not present in our plane.\n\nFor simplicity's sake it is usual to take a cross-section of space\nparallel to one of our coordinate axes. We think of our three\ndimensions as extending in the directions of those axes; and it is\neasier to take a horizontal or vertical section which shall simply\nwipe out one of these dimensions than to take an oblique section which\nshall wipe out a dimension that consists partly of our original length,\nand partly of our original width, and partly of our original height.\n\nIf we have a four-dimensional manifold to begin with, we may equally\nshake out one of the four dimensions, one of the four coordinates,\nand consider the three-dimensional result of this process as a\ncross-section of the original four-dimensional continuum. And where, in\ncross-sectioning a three-dimensioned world, we have but three choices\nof a coordinate to eliminate, in cross-sectioning a world of four\ndimensions we have four choices. By dropping out either the $x$, or the\n$y$, or the $z$, or the $t$, we get a three-dimensioned cross-section.\n\nNow our accustomed three-dimensional space is strictly Euclidean. When\nwe cross-section it, we get a Euclidean plane no matter what the\ndirection in which we make the cut. Likewise the Euclidean plane is\nwholly Euclidean, because when we cross-section it in any direction\nwhatever we get a Euclidean line. A cylindrical surface, on the other\nhand, is neither wholly Euclidean nor wholly non-Euclidean in this\nmatter of cross-sectioning. If we take a section in one direction we\nget a Euclidean line and if we take a section in the other direction\nwe get a circle (if the cylindrical surface be a circular one). And\nof course if we take an oblique section of any sort, it is neither\nline nor circle, but a compromise between the two--the significant\nthing being that it is still not a Euclidean line.\n\nThe space-time continuum presents an analogous situation. When\nwe cross-section it by dropping out any one of the three space\ndimensions, we get a three-dimensional complex in which the distance\nformula is still non-Euclidean, retaining the minus sign before\nthe time-difference and therefore retaining the geometric character\nof its parent. But if we take our cross-section in such a way as to\neliminate the time coordinate, this peculiarity disappears. The signs\nin the invariant expression are then all plus, and the cross-section\nis in fact our familiar Euclidean three-space.\n\nIf we set up a surface geometry on a sphere, we find that the\nelimination of one dimension leaves us with a line-geometry that is\nstill non-Euclidean since it pertains to the great circles of the\nsphere rather than to Euclidean straight lines. In shaking Minkowski's\ncontinuum down into a three-dimensional one by eliminating any one of\nhis coordinates, if we eliminate either the $x$, the $y$ or the $z$,\nwe have left a three-dimensional geometry in which the disturbing minus\nsign still occurs in the distance-formula, and which is therefore still\nnon-Euclidean. If we omit the $t$, this does not occur. We see, then,\nthat the time dimension is the disturbing factor, the one which gives\nto space-time its non-Euclidean character so far as the possession\nof curvature is concerned. And we see that this curvature is not the\nsame in all directions, and in one direction is actually zero--whence\nthe attempted analogy with a cylinder instead of with a sphere.\n\nMany writers on relativity try to give the space-time continuum an\nappeal to our reason and a character of inevitableness by insisting on\nthe lack of any fundamental distinction between space and time. The\nvery expression for the space-time invariant denies this. Time is\ndistinguishable from space. The three dimensions of space are quite\nindistinguishable--we can interchange them without affecting the\nformula, we can drop one out and never know which is gone. But the\nvery formula singles out time as distinct from space, as inherently\ndifferent in some way. It is not so inherently different as we have\nalways supposed; it is not sufficiently different to offer any obstacle\nto our thinking in terms of the four-dimensional continuum. But while\nwe can group space and time together in this way, [this does not mean\nat all that space and time cease to differ. A cook may combine meat\nwith potatoes and call the product hash, but meat and potatoes do\nnot thereby become identical.]223\n\n\n\n\nTHE QUESTION OF VISUALIZATION\n\nTo the layman there is a great temptation to say that while,\nmathematically speaking, the space-time continuum may be a great\nsimplification, it does not really represent the external world. To\nbe sure, you can't see the space-time continuum in precisely the same\nway that you can the three-dimensional space continuum, but this is\nonly because Einstein finds the time dimension to be not quite freely\ninterchangeable with the space dimension. Yet you do perceive this\nspace-time continuum, in the manner appropriate for its perception;\nand it would be just as sensible to throw out the space continuum\nitself on the ground that perception of the two is not of exactly\nthe same sort, as to throw out the space-time continuum on this\nground. With appropriate conventions, either may stand as the mental\npicture of the external world; it is for us to choose which is the\nmore convenient and useful image. Einstein tells us that his image\nis the better, and tells us why.\n\nBefore we look into this, we must let him tell us something more\nabout the geometry of his continuum. What he tells us is, in its\nessentials, just this. The observer in a pure space continuum of three\ndimensions finds that as he changes his position, his right-and-left,\nhis backward-and-forward, and his up-and-down are not fixed directions\ninherent in nature, but are fully interchangeable. The observers, in\nthe adjoined sketch, whose verticals are as indicated by the arrows,\nfind very different vertical and horizontal components for the distance\nbetween the points $O$ and $P$; a similar situation would prevail if\nwe used all three space directions. The statement analogous to this\nfor Einstein's four-dimensional continuum of space and time combined\nis that, as observers change their relative motion, their time axes\ntake slightly different directions, so that what is purely space or\npurely time for the one becomes space with a small component in the\ntime direction, or time with a small component in the space direction,\nfor the other. This it will be seen explains fully why observers in\nrelative motion can differ about space and time measurements. We\nshould not be at all surprised if the two observers of the figure\nreported different values for horizontals and verticals; we should\nrealize that what was vertical for one had become partly horizontal\nfor the other. It is just so, says Einstein, with his observers of\ntime and space who are in relative motion to one another; what one\nsees as space the other sees as partly time, because their time axes\ndo not run quite parallel.\n\nThe natural question here, of course, is \"Well, where are their\ntime axes?\" If you know what to look for, of course, you ought to\nbe able to perceive them in just the way you perceive ordinary time\nintervals--with the reservation that they are imaginary, after all,\njust like your space axes, and that you must only expect to see them\nin imagination. If you look for a fourth axis in Euclidean three-space\nto represent your time axis, you will of course not find it. But you\nwill by all means agree with me that your time runs in a definite\ndirection; and this it is that defines your time axis. Einstein adds\nthat if you and I are in relative motion, my time does not run in\nquite the same direction as yours.\n\nHow shall we prove it? Well, how would we prove it if he told us that\nour space axes did not run in precisely the same direction? Of course\nwe could not proceed through direct measures upon the axes themselves;\nwe know these are imaginary. What we should do would be to strike out,\neach of us, a very long line indeed in what seemed the true horizontal\ndirection; and we should hope that if we made them long enough,\nand measured them accurately enough, we should be able to detect any\ndivergence that might exist. This is precisely what we must do with\nour time axes if we wish to verify Einstein's statement that they are\nnot precisely parallel; and what better evidence could we demand of\nthe truth of this statement than the evidence already presented--that\nwhen we measure our respective time components between two events,\nwe get different results?\n\n\n\n\nWHAT IT ALL LEADS TO\n\nThe preceding chapters have been compiled and written with a view to\nputting the reader in a state of mind and in a state of informedness\nwhich shall enable him to derive profit from the reading of the actual\ncompeting essays which make up the balance of the book. For this\npurpose it has been profitable to take up in detail the preliminaries\nof the Special Theory of Relativity, and to allow the General Theory\nto go by default, in spite of the fact that it is the latter which\nconstitutes Einstein's contribution of importance to science. The\nreason for this is precisely the same as that for taking up Euclidean\ngeometry and mastering it before proceeding to the study of Newtonian\nmechanics. The fundamental ideas of the two theories, while by no\nmeans identical, are in general terms the same; and the conditions\nsurrounding their application to the Special Theory are so very\nmuch simpler than those which confront us when we apply them to the\nmore general case, that this may be taken as the controlling factor\nin a popular presentation. We cannot omit the General Theory from\nconsideration, of course; but we can omit it from our preliminary\ndiscussion, and leave its development to the complete essays which\nfollow, and which in almost every case give it the larger half of their\nspace which its larger content demands. In the process of the slow\nand difficult preparation of the lay mind for the assimilation of an\naltogether new set of fundamental ideas, it is altogether desirable to\ngive the Special Theory, with its simpler applications of these ideas,\na place out of proportion to its importance in Einstein's completed\nstructure; and this we have therefore done.\n\nThe Special Theory, postulating the relativity of uniform motion and\ndeducing the consequences of that relativity, is often referred to\nas a \"special case\" of the General Theory, in which this restriction\nof uniformity is removed. This is not strictly speaking correct. The\nGeneral Theory, when we have formulated it, will call our attention to\nsomething which we really knew all the time, but to which we chose not\nto give heed--that in the regions of space to which we have access,\nuniform motion does not exist. All bodies in these regions are under\nthe gravitational influence of the other bodies therein, and this\ninfluence leads to accelerated motion. Nothing in our universe can\npossibly travel at uniform velocity; the interference of the rest of\nthe bodies in the universe prevents this.\n\nObviously, we ought not to apply the term \"special case\" to a case that\nnever occurs. Nevertheless, this case is of extreme value to us in our\nmental processes. Many of the motions with which we are concerned are\nso nearly at constant velocity that we find it convenient to treat\nthem as though they were uniform, either ignoring the resulting\nerror or correcting for it at the end of our work. In many other\ncases we are able to learn what actually occurs under accelerated\nmotion by considering what would have occurred under uniform motion\nwere such a thing possible. Science is full of complications which\nwe unravel in this fashion. The physicist deals with gas pressures\nby assuming temperatures to be constant, though he knows temperature\nnever is constant; and in turn he deals with temperatures by assuming\npressures to be constant. After this, he is able to predict what will\nhappen when, as in nature, pressures and temperatures are varying\nsimultaneously. By using as a channel of attack the artificially simple\ncase that never occurs, we get a grip on the complex case that gives\nus a true picture of the phenomenon. And because in actual nature we\ncan come as close as we please to this artificial case, by supposing\nthe variable factor to approach constancy, so when we assume it to be\nabsolutely constant we speak of the result as the limiting case. This\nsituation does not occur, but is the limiting case for those that\ndo occur.\n\nWhen, in the matter of motion, we abandon the artificial, limiting\ncase of uniform velocity and look into the general, natural one of\nunrestricted motion, we find that the structure which we have built up\nto deal with the limiting case provides us with many of the necessary\nideas and viewpoints. This is what we expect--in it lies the value\nof the limiting case. We shall see that the relativity of time and\nspace, established for the limiting case, holds good in the general\none. We shall see that the idea of the four-dimensional space-time\ncontinuum as representing the external world persists, forming the\nwhole background of the General Theory much more definitely than\nin the Special Theory. Incidentally we shall see that the greater\ngenerality of the case under consideration will demand a greater degree\nof generality in the geometry of this continuum, a non-Euclideanism of\na much more whole-hearted type than that of the Special Theory. But\nall the revisions of fundamental concepts which we have been at such\npains to make for the sake of the Special Theory will remain with us\nin the General. With this we may consider our preliminary background\nas established, and give our attention to the essayists, who will\ntry to take us more deeply into the subject than we have yet gone,\nwithout losing us in its intricacies.\n\n\n\n\n\n\n\n\n\nVII\n\nRELATIVITY\n\nThe Winning Essay in the Contest for the Eugene Higgins $5,000 Prize\n\nBY LYNDON BOLTON\nSENIOR EXAMINER, BRITISH PATENT OFFICE LONDON\n\n\nThe reader is probably acquainted with the method of specifying\npositions of points in a plane by their distances from two mutually\nperpendicular lines, or if the points are in space by their distances\nfrom three mutually perpendicular planes like adjacent sides of a\nflat-sided box. The method is in fact in common use for exhibiting\nrelations between quantities by graphs or diagrams. These sets of axes,\nas they are called, together with any scales used for measuring, must\nbe supposed rigid, otherwise the events or points which they are used\nto specify are indefinite. The lengths which locate any point with\nreference to a set of axes are called its coordinates.\n\nWhen such systems are used for physical purposes, they must be\nsupplemented by clocks to enable the times at which events occur to\nbe determined. The clocks must be synchronized, and must go at the\nsame rate, but it must suffice here to state that this is possible\nwithout indicating how these conditions can be attained. A system of\naxes with its clocks will hereinafter be called a Frame of Reference,\nand every observer will be supposed to be provided with such a frame\npartaking of his motion. All the objects which partake of an observer's\nmotion will be called his system.\n\nIt is a question whether among all possible frames of reference\nany one frame or class of frames is more suited than another for\nthe mathematical statement of physical laws. This is for experience\nto decide, and a Principle of Relativity is a statement embodying\nthe answer.\n\n\n\n\nTHE MECHANICAL PRINCIPLE OF RELATIVITY\n\nIt has been ascertained that all such frames are equally suitable for\nthe mathematical statement of general mechanical laws, provided that\ntheir motion is rectilinear and uniform and without rotation. This\nfact is comprehended in the general statement that all unaccelerated\nframes of reference are equivalent for the statement of the general\nlaws of mechanics. This is the mechanical principle of relativity.\n\nIt is well recognized however that the laws of dynamics as hitherto\nstated involve the assumptions that the lengths of rigid bodies are\nunaffected by the motion of the frame of reference, and that measured\ntimes are likewise unaffected; that is to say that any length measured\non his own system by either of two relatively moving observers appears\nthe same to both observers, or that lengths of objects and rates of\nclocks do not alter whatever the motion relative to an observer. These\nassumptions seem so obvious that it is scarcely perceived that they\nare assumptions at all. Yet this is the case, and as a matter of fact\nthey are both untrue.\n\n\n\n\nTHE SPECIAL PRINCIPLE OF RELATIVITY\n\nAlthough all unaccelerated frames of reference are equivalent for\nthe purposes of mechanical laws, this is not the case for physical\nlaws generally as long as the above suppositions are adhered\nto. Electromagnetic laws do alter their form according to the motion\nof the frame of reference; that is to say, if these suppositions are\ntrue, electromagnetic agencies act in different ways according to the\nmotion of the system in which they occur. There is nothing a priori\nimpossible in this, but it does not agree with experiment. The motion\nof each locality on the earth is continually changing from hour to\nhour but no corresponding changes occur in electromagnetic actions. It\nhas however been ascertained that on discarding these suppositions\nthe difficulty disappears, and electromagnetic laws retain their\nform under all circumstances of unaccelerated motion. According\nto the theory of relativity, the correct view which replaces these\nsuppositions is deducible from the following postulates:\n\n  (1) By no experiment conducted on his own system can an observer\n      detect the unaccelerated motion of his system.\n  (2) The measure of the velocity of light in vacuo is unaffected\n      by relative motion between the observer and the source of light.\n\nBoth these postulates are well established by experiment. The first\nmay be illustrated by the familiar difficulty of determining whether a\nslowly moving train one happens to be sitting in, or an adjacent one,\nis in motion. The passenger has either to wait for bumps (that is,\naccelerations) or else he has to look out at some adjacent object which\nhe knows to be fixed, such as a building (that is, he has to perform\nan experiment on something outside his system), before he can decide.\n\nThe second postulate is an obvious consequence of the wave theory of\nlight. Just as waves in water, once started by a ship, travel through\nthe water with a velocity independent of the ship, so waves in space\ntravel onward with a speed bearing no relation to that of the body\nwhich originated them. The statement however is based on experiment,\nand can be proved independently of any theory of light.\n\nIt is not difficult to deduce from these postulates certain remarkable\nconclusions relating to the systems of two observers, A and B, in\nrelative motion, among them the following:\n\n  (1) Objects on B's system appear to A to be shorter in the\n      direction of relative motion than they appear to B.\n  (2) This opinion is reciprocal. B thinks that A's measurements\n      on A's system are too great.\n  (3) Similarly for times: each observer thinks that the other's\n      clocks have a slower rate than his own, so that B's durations\n      of time appear shorter to B than to A, and conversely.\n  (4) Events which appear simultaneous to A do not in general appear\n      so to B, and conversely.\n  (5) Lengths at right angles to the direction of motion are\n      unaffected.\n  (6) These effects vary with the ratio of the relative velocity\n      to that of light. The greater the relative velocity, the\n      greater the effects. They vanish if there is no relative\n      velocity.\n  (7) For ordinary velocities the effects are so small as to escape\n      notice. The remarkable point however is their occurrence rather\n      than their magnitude.\n  (8) The observers similarly form different estimates of the\n      velocities of bodies on each other's systems. The velocity of\n      light however appears the same to all observers.\n\nTaking into account these revised views of lengths and times the\nmechanical principle of relativity may be extended to physical laws\ngenerally as follows: All unaccelerated frames of reference are\nequivalent for the statement of the general laws of physics. In this\nform the statement is called the Special, or Restricted, Principle\nof Relativity, because it is restricted to unaccelerated frames of\nreference. Naturally the laws of classical mechanics now require some\nmodification, since the suppositions of unalterable lengths and times\nno longer apply.\n\n\n\n\nTHE FOUR DIMENSIONAL CONTINUUM\n\nLengths and times therefore have not the absolute character formerly\nattributed to them. As they present themselves to us they are relations\nbetween the object and the observer which change as their motion\nrelative to him changes. Time can no longer be regarded as something\nindependent of position and motion, and the question is what is the\nreality? The only possible answer is that objects must be regarded as\nexisting in four dimensions, three of these being the ordinary ones of\nlength, breadth and thickness, and the fourth, time. The term \"space\"\nis applicable only by analogy to such a region; it has been called a\n\"continuum,\" and the analogue of a point in ordinary three-dimensional\nspace has been appropriately called an \"event.\" By \"dimension\" must be\nunderstood merely one of four independent quantities which locate an\nevent in this continuum. In the nature of the case any clear mental\npicture of such a continuum is impossible; mankind does not possess\nthe requisite faculties. In this respect the mathematician enjoys\na great advantage. Not that he can picture the thing mentally any\nbetter than other people, but his symbols enable him to abstract the\nrelevant properties from it and to express them in a form suitable\nfor exact treatment without the necessity of picturing anything,\nor troubling whether or not the properties are those on which others\nrely for their conceptions.\n\n\n\n\nGRAVITATION AND ACCELERATION\n\nThe limitation of statements of general law to uniformly moving systems\nis hardly satisfactory. The very concept of general law is opposed\nto the notion of limitation. But the difficulties of formulating a\nlaw so that the statement of it shall hold good for all observers,\nwhose systems may be moving with different and possibly variable\naccelerations, are very great. Accelerations imply forces which\nmight be expected to upset the formulation of any general dynamical\nprinciples, and besides, the behavior of measuring rods and clocks\nwould be so erratic as to render unmeaning such terms as rigidity\nand measured time, and therefore to preclude the use of rigid scales,\nor of a rigid frame of reference which is the basis of the foregoing\ninvestigation.\n\nThe following example taken from Einstein will make this clear,\nand also indicate a way out of the difficulty. A rotating system is\nchosen, but since rotation is only a particular case of acceleration\nit will serve as an example of the method of treating accelerated\nsystems generally. Moreover, as it will be seen, the attribution of\nacceleration to the system is simply a piece of scaffolding which\ncan be discarded when the general theory has been further developed.\n\nLet us note the experiences of an observer on a rotating disk which\nis isolated so that the observer has no direct means of perceiving\nthe rotation. He will therefore refer all the occurrences on the disk\nto a frame of reference fixed with respect to it, and partaking of\nits motion.\n\nHe will notice as he walks about on the disk that he himself and all\nthe objects on it, whatever their constitution or state, are acted upon\nby a force directed away from a certain point upon it and increasing\nwith the distance from that point. This point is actually the center\nof rotation, though the observer does not recognize it as such. The\nspace on the disk in fact presents the characteristic properties of a\ngravitational field. The force differs from gravity as we know it by\nthe fact that it is directed away from instead of toward a center,\nand it obeys a different law of distance; but this does not affect\nthe characteristic properties that it acts on all bodies alike, and\ncannot be screened from one body by the interposition of another. An\nobserver aware of the rotation of the disk would say that the force\nwas centrifugal force; that is, the force due to inertia which a body\nalways exerts when it is accelerated.\n\nNext suppose the observer to stand at the point of the disk where\nhe feels no force, and to watch someone else comparing, by repeated\napplications of a small measuring rod, the circumference of a circle\nhaving its center at that point, with its diameter. The measuring rod\nwhen laid along the circumference is moving lengthwise relatively\nto the observer, and is therefore subject to contraction by his\nreckoning. When laid radially to measure the diameter this contraction\ndoes not occur. The rod will therefore require a greater proportional\nnumber of applications to the circumference than to the diameter,\nand the number representing the ratio of the circumference of the\ncircle to the diameter thus measured will therefore be greater than\n3.14159+, which is its normal value. Moreover the relative velocity\ndecreases as the center is approached, so that the contraction of the\nmeasuring rod is less when applied to a smaller circle; and the ratio\nof the circumference to the diameter, while still greater than the\nnormal, will be nearer to it than before, and the smaller the circle\nthe less the difference from the normal. For circles whose centers\nare not at the point of zero force the confusion is still greater,\nsince the velocities relative to the observer of points on them now\nchange from point to point. The whole scheme of geometry as we know\nit is thus disorganized. Rigidity becomes an unmeaning term since\nthe standards by which alone rigidity can be tested are themselves\nsubject to alteration. These facts are expressed by the statement\nthat the observer's measured space is non-Euclidean; that is to say,\nin the region under consideration measurements do not conform to the\nsystem of Euclid.\n\nThe same confusion arises in regard to clocks. No two clocks will in\ngeneral go at the same rate, and the same clock will alter its rate\nwhen moved about.\n\n\n\n\nTHE GENERAL PRINCIPLE OF RELATIVITY\n\nThe region therefore requires a space-time geometry of its own, and\nbe it noted that with this special geometry is associated a definite\ngravitational field, and if the gravitational field ceases to exist,\nfor example if the disk were brought to rest, all the irregularities\nof measurement disappear, and the geometry of the region becomes\nEuclidean. This particular case illustrates the following propositions\nwhich form the basis of this part of the theory of relativity:\n\n  (1) Associated with every gravitational field is a system of\n      geometry, that is, a structure of measured space peculiar to\n      that field.\n  (2) Inertial mass and gravitational mass are one and the same.\n  (3) Since in such regions ordinary methods of measurement fail,\n      owing to the indefiniteness of the standards, the systems of\n      geometry must be independent of any particular measurements.\n  (4) The geometry of space in which no gravitational field exists\n      in Euclidean. [1]\n\nThe connection between a gravitational field and its appropriate\ngeometry suggested by a case in which acceleration was their common\ncause is thus assumed to exist from whatever cause the gravitational\nfield arises. This of course is pure hypothesis, to be tested by\nexperimental trial of the results derived therefrom.\n\nGravitational fields arise in the presence of matter. Matter is\ntherefore presumed to be accompanied by a special geometry, as though\nit imparted some peculiar kink or twist to space which renders the\nmethods of Euclid inapplicable, or rather we should say that the\ngeometry of Euclid is the particular form which the more general\ngeometry assumes when matter is either absent or so remote as to\nhave no influence. The dropping of the notion of acceleration is\nafter all not a very violent change in point of view, since under\nany circumstances the observer is supposed to be unaware of the\nacceleration. All that he is aware of is that a gravitational field\nand his geometry coexist.\n\nThe prospect of constructing a system of geometry which does not depend\nupon measurement may not at first sight seem hopeful. Nevertheless this\nhas been done. The system consists in defining points not by their\ndistances from lines or planes (for this would involve measurement)\nbut by assigning to them arbitrary numbers which serve as labels\nbearing no relation to measured distances, very much as a house\nis located in a town by its number and street. If this labeling\nbe done systematically, regard being had to the condition that the\nlabel-numbers of points which are close together should differ from\none another by infinitesimal amounts only, it has been found that\na system of geometry can actually be worked out. Perhaps this will\nappear less artificial when the fact is called to mind that even\nwhen standards of length are available no more can be done to render\nlengths of objects amenable to calculation than to assign numbers to\nthem, and this is precisely what is done in the present case. This\nsystem of labeling goes by the name of \"Gaussian coordinates\" after\nthe mathematician Gauss who proposed it.\n\nIt is in terms of Gaussian coordinates that physical laws must be\nformulated if they are to have their widest generality, and the general\nprinciple of relativity is that all Gaussian systems are equivalent for\nthe statement of general physical laws. For this purpose the labeling\nprocess is applied not to ordinary space but to the four dimensional\nspace-time continuum. The concept is somewhat difficult and it may\neasily be aggravated into impossibility by anyone who thinks that he\nis expected to visualize it. Fortunately this is not necessary; it is\nmerely one of these irrelevancies to which those who are unaccustomed\nto think in symbols are liable.\n\nIt will now be seen that among physical laws the law of gravitation\nstands pre-eminent, for it is gravitating matter which determines\nthe geometry, and the geometry determines the form of every other\nlaw. The connection between the geometry and gravitation is the law\nof gravitation. This law has been worked out, with the result that\nNewton's law of the inverse square is found to be approximate only,\nbut so closely approximate as to account for nearly all the motions of\nthe heavenly bodies within the limits of observation. It has already\nbeen seen that departure from the Euclidean system is intensified\nby rapidity of motion, and the movements of these bodies are usually\ntoo slow for this departure to be manifest. In the case of the planet\nMercury the motion is sufficiently rapid, and an irregularity in its\nmotion which long puzzled astronomers has been explained by the more\ngeneral law.\n\nAnother deduction is that light is subject to gravitation. This has\ngiven rise to two predictions, one of which has been verified. The\nverification of the other is as yet uncertain, though the extreme\ndifficulty of the necessary observations may account for this.\n\nSince light is subject to gravitation it follows that the constancy\nof the velocity of light assumed in the earlier part of this\npaper does not obtain in a gravitational field. There is really no\ninconsistency. The velocity of light is constant in the absence of\ngravitation, a condition which unaccelerated motion implies. The\nspecial principle of relativity is therefore a limiting case of the\ngeneral principle.\n\n\n\n\n\n\n\n\n\nVIII\n\nTHE NEW CONCEPTS OF TIME AND SPACE\n\nThe Essay in Behalf of Which the Greatest Number of Dissenting Opinions\nHave Been Recorded\n\nBY MONTGOMERY FRANCIS NEW YORK\n\n\nWe have all had experiences, on trains and boats, illustrating our\ninability to tell, without looking off to some external body, whether\nwe are at rest or moving uniformly; and when we do so look, to tell,\nwithout reference to the ground or some other point external to both\nsystems, whether ours or the other be the seat of motion. Uniform\nmotion must be relative, because we find nowhere in the universe a\nbody in the unique state of absolute rest from which alone absolute\nmotion might be measured.\n\nTrue, the wave theory of light with its homogeneous space-filling\nether seemed to provide a reference standard for the concept of\nabsolute motion, and for its measurement by experiment with light\nrays. But when Michelson and Morley looked for this absolute motion\nthey found no trace of it. To the physicist, observational student\nof the external world, nothing exists save observationally; what he\ncan never observe is not there. So: I. By no means whatever may we\nregard uniform straight-line motion as other than relative.\n\nAs a further direct consequence of the Michelson-Morley experiment\nwe have: II. Light in a vacuum presents the same velocity, $C =\n186,330$ miles per second, to all observers whatever their velocity\nof relative motion. In addition to being experimentally established,\nthis is necessary to support I, for if light will distinguish between\nour velocities, its medium is necessarily a universal standard for\nabsolute motion. But it is contrary to common sense to suppose that\nif I pass you at 100 miles per hour, the same light impulse can pass\nus both at the same speed, $C$. We feel, instinctively, that space and\ntime are not so constituted as to make this possible. But the fact has\nbeen repeatedly demonstrated. And when common sense and fundamental\nconcepts clash with facts, it is not the facts that must yield. We\nhave survived such crises, notably one where we had to change the\nfundamental concept of up-and-down; if another one is here, says\nEinstein, let us meet it.\n\nThis the Special Theory of Relativity does. It accepts Postulates\nI and II above; their consequences it deduces and interprets. For\nextensive demonstration of these I lack space, and this has been\nsatisfactorily done by others so it is not my chief duty; but clearly\nthey will be startling. For the very ray of light which refuses to\nrecognize our relative motion is the medium through which I must\nobserve your system and you mine.\n\nIt turns out that I get different values for lengths and time intervals\nin your system than you get, and vice versa. And we are both right! For\nme to accept your \"correction\" were for me to admit that you are\nat absolute rest and I in absolute motion, that your measure of\nlight velocity is right and mine wrong: admissions barred by the\npostulates. We have nothing to correct; we can only recognize the\nreason for the discrepancy; and knowing our relative velocity, each\ncan calculate from his own results what the other's will be. We find,\nof course, that at ordinary velocities the discrepancy is many times\ntoo small for detection; but at relative velocities at all comparable\nwith that of light it rises above the observational horizon.\n\nTo inquire the \"true\" length is meaningless. Chicago is east of Denver,\nwest of Pittsburgh, south of Milwaukee; we do not consider this\ncontradictory, or demand the \"true\" direction of Chicago. Einstein\nfinds that the concept of length, between points in space or events\nin time, does not as we had supposed represent an intrinsic property\nof the points or the events. Like direction, it is merely a relation\nbetween these and the observer--a relation whose value changes with\nthe observer's velocity relative to the object. If our ideas of the\npart played in the world by time and space do not permit us to believe\nthis, we must alter these ideas. Let us see how we may do this.\n\n\n\n\nA WORLD OF POINTS\n\nTo deal with points in a plane the mathematician draws two\nperpendicular lines, and locates any point, as $P$, by measuring its\ndistances, $X$ and $Y$, from these \"coordinate axes.\" The directions\nof his axes acquire for him a peculiar significance, standing out\nabove other directions; he is apt to measure the distances $X-x$ and\n$Y-y$ between the points $P$ and $Q$ in these directions, instead of\nmeasuring the single distance $PQ$. We do the same thing when we say\nthat the railroad station is five blocks north and two east.\n\nThe mathematician visualizes himself as an observer, located on his\ncoordinate framework. For another observer on another framework, the\nhorizontal and vertical distances $X'-x'$ and $Y'-y'$ between $P$ and\n$Q$ are different. But for both, the distance from $P$ direct to $Q$\nis the same. In each case the right triangle tells us that:\n\n    $$PQ = \\sqrt{(X-x)^2 + (Y-y)^2} = \\sqrt{(X'-x')^2 + (Y'-y')^2}$$\n\nImagine an observer so dominated by his coordinate system that he\nknows no way of relating $P$ with $Q$ save by their horizontal and\nvertical separation. His whole scheme of things would be shattered\nby the suggestion that other observers on other reference frames find\ndifferent horizontal and vertical components. We have to show him the\nline $PQ$. We have to convince him that this length is the absolute\nproperty enjoyed by his pair of points; that horizontals and verticals\nare merely relations between the points and the observer, result of\nthe observer's having analyzed the distance $PQ$ into two components;\nthat different observers effect this decomposition differently; that\nthis seems not to make sense to him only because of his erroneous\nconcept of a fundamental difference between verticals and horizontals.\n\n\n\n\nTHE FOUR-DIMENSIONAL WORLD OF EVENTS\n\nWe too have created a distinction in our minds corresponding to no\nsufficient reality. Our minds seize on time as inherently separable\nfrom space. We see the world made up of things in a continuum of three\nspace dimensions; to make this dead world live there runs through it\na one-dimensional time continuum, imposed from without, unrelated.\n\nBut did you ever observe anything suggesting the presence of time\nin the absence of space, or vice versa? No; these vessels of the\nuniverse always occur together. Association of the space dimensions\ninto a manifold from which time is excluded is purely a phenomenon\nof the mind. The space continuum cannot begin to exist until the time\ndimension is supplied, nor can time exist without a place to exist in.\n\nThe external world that we observe is composed, not of points,\nbut of events. If a point lacks position in time it does not exist;\ngive it this position and it becomes an event. This world of events\nis four-dimensional--which means nothing more terrifying than that\nyou must make four measures to locate an event. It does not mean,\nat all, that you must visualize four mutually perpendicular lines in\nyour accustomed three-space or in a four-space analogous to it. If\nthis world of four dimensions seems to lack reality you will be\nable to exhibit no better reality for your old ideas. Time belongs,\nwithout question; and not as an afterthought, but as part of the\nworld of events.\n\nTo locate an event we use four measures: $X$, $Y$ and $Z$ for space,\n$T$ for time. Using the same reference frame for time and space, we\nlocate a second event by the measures $x$, $y$, $z$, $t$. Minkowski\nshowed that the quantity\n\n    $$\\sqrt{(X-x)^2 + (Y-y)^2 + (Z-z)^2 - (CT-Ct)^2}$$\n\nis the same for all observers, no matter how different their $x$'s,\n$y$'s, $z$'s and $t$'s; just as in the plane the quantity\n\n    $$\\sqrt{(X-x)^2 + (Y-y)^2}$$\n\nis the same for all observers, no matter how different their $x$'s\nand $y$'s.\n\nSuch a quantity, having the same value for all observers, is\nabsolute. In the plane it represents the true, absolute distance\nbetween the points--their intrinsic property. In dealing with events it\nrepresents the true, absolute \"interval,\" in time and space together\nbetween the events. It is not space, nor time, but a combination of\nthe two. We have always broken it down into separate space and time\ncomponents. In this we are as naive as the plane observer who could\nnot visualize the distance $PQ$ until it was split into separate\nhorizontals and verticals. He understood with difficulty that another\nobserver, employing a different reference frame because in different\nposition, would make the decomposition differently. We understand with\ndifficulty that another observer, employing a different reference\nframe because in uniform motion relative to us, will decompose the\n\"interval\" between events into time and space components different\nfrom ours. Time and space are relative to the observer; only the\ninterval representing space-time is absolute. So common sense stands\nreconciled to the Special Theory of Relativity.\n\n\n\n\nSUCCESSIVE STEPS TOWARD GENERALITY\n\nIs then our laboriously acquired geometry of points in a\nthree-dimensional space to go into the discard? By no means. Jeans,\ninvestigating the equilibrium of gaseous masses, found the general\ncase too difficult for direct attack. So he considered the case where\nthe masses involved are homogeneous and incompressible. This never\noccurs; but it throws such light on the general case as to point the\nway toward attack on it.\n\nEuclidean geometry excludes motion, save that engineered by the\nobserver; and then the time is immaterial. Time does not enter at all;\nthe three space dimensions suffice. This simple case never occurs\nwhere matter exists; but its conclusions are of value in dealing with\nmore general cases.\n\nWhen we look into a world alleged to be that of Euclid and find motion,\nwe may retain the Euclidean concept of what constitutes the world and\ninvent a machinery to account for the motion; or we may abandon the\nEuclidean world, as inadequate, in favor of a more general one. We\nhave adopted the second alternative.\n\nNewton's laws tells us that a body free to move will do so, proceeding\nin a straight line at uniform velocity until interfered with. We\ndo not ask, nor does the theory tell us, whence comes the initial\nmotion. There is no machinery to produce it; it is an inherent property\nof Newton's world--assured by the superposition of the time continuum\nupon Euclid's world to make Newton's, accepted without question along\nwith that world itself.\n\nBut Newton saw that his world of uniform motion, like Euclid's,\nwas never realized. In the neighborhood of one particle a second is\ninterfered with, forced to give up its uniform motion and acquire a\nconstant acceleration. This Newton explained by employing the first\nof the alternatives mentioned above. He tells us that in connection\nwith all matter there exists a force which acts on other matter in a\ncertain way. He does not display the actual machinery through which\nthis \"force\" works, because he could not discover any machinery;\nhe had to stop with his brilliant generalization of the observed\nfacts. And all his successors have failed to detect the slightest\ntrace of a machinery of gravitation.\n\nEinstein asks whether this is not because the machinery is\nabsent--because gravitation, like position in Euclid's world and\nmotion in Newton's, is a fundamental property of the world in which it\noccurs. His point of attack here lay in precise formulation of certain\nfamiliar facts that had never been adequately appreciated. These\nfacts indicate that even accelerated motion is relative, in spite of\nits apparently real and absolute effects.\n\n\n\n\nGRAVITATION AND ACCELERATION\n\nAn observer in a closed compartment, moving with constant acceleration\nthrough empty space, finds that the \"bottom\" of his cage catches up\nwith objects that he releases; that it presses on his feet to give\nhim the sensation of weight, etc. It displays all the effects that\nhe would expect if it were at rest in a gravitational field. On the\nother hand, if it were falling freely under gravitational influence,\nits occupant would sense no weight, objects released would not leave\nhis hand, the reaction from his every motion would change his every\nposition in his cage, and he could equally well assume himself at\nrest in a region of space free from gravitational action. Accelerated\nmotion may always be interpreted, by the observer on the system,\nas ordinary force effects on his moving system, or as gravitational\neffects on his system at rest.\n\nAn alternative statement of the Special Theory is that the observed\nphenomena of uniform motion may equally be accounted for by supposing\nthe object in motion and the observer with his reference frame at\nrest, or vice versa. We may similarly state the General Theory: The\nobserved phenomena of uniformly accelerated motion may in every case be\nexplained on a basis of stationary observer and accelerated objective,\nor of stationary objective with the observer and his reference system\nin accelerated motion. Gravitation is one of these phenomena. It\nfollows that if the observer enjoy properly accelerated axes (in\ntime-space, of course), the absolute character of the world about him\nmust be such as to present to him the phenomenon of gravitation. It\nremains only to identify the sort of world, of which gravitation as\nit is observed would be a fundamental characteristic.\n\nEuclid's and Newton's systems stand as first and second approximations\nto that world. The Special Relativity Theory constitutes a correction\nof Newton, presumably because it is a third approximation. We must\nseek in it those features which we may most hopefully carry along,\ninto the still more general case.\n\nNewton's system retained the geometry of Euclid. But Minkowski's\ninvariant expression tells us that Einstein has had to abandon this;\nfor in Euclidean geometry of four dimensions the invariant takes\nthe form:\n\n    $$\\sqrt{(X-x)^2 + (Y-y)^2 + (Z-z)^2 + (T-t)^2}\\,,$$\n\nanalogous to that of two and three dimensions. It is not the presence\nof the constant $C$ in Minkowski's formula that counts; this is merely\nan adjustment so that we may measure space in miles and time in the\nunit that corresponds to a mile. It is the minus sign where Euclidean\ngeometry demands a plus that makes Minkowski's continuum non-Euclidean.\n\nThe editor has told us what this statement means. I think he has made\nit clear that when we speak of the geometry of the four-dimensional\nworld, we must not read into this term the restrictions surrounding\nthe kind of geometry we are best acquainted with--that of the\nthree-dimensional Euclidean continuum. So I need only point out that\nif we are to make a fourth (and we hope, final) approximation to the\nreality, its geometry must preserve the generality attained by that\nof the third step, if it goes no further.\n\n\n\n\nEINSTEIN'S TIME-SPACE WORLD\n\nEinstein accordingly examined the possible non-Euclidean geometries\nof four dimensions, in search of one displaying fundamental\ncharacteristics which, interpreted in terms of space-time, would\nlead to the observed facts of gravitation. The mathematics of this\ninvestigation is that part of his work which, we are told, but twelve\nmen can follow; so we may only outline his conclusions.\n\nIf we assume that in the neighborhood of matter the world of\nspace-time is non-Euclidean, and that its curvature or distortion or\nnon-Euclideanism is of a certain type already known to mathematicians;\nthat the curvature of this world in the neighborhood of matter\nincreases with the mass, and decreases as the distance from the matter\nincreases; and that every particle of matter that is not interfered\nwith travels through space-time in the most direct path possible in\nthat continuum; then the observed facts of gravitation are accounted\nfor as an inherent geometric property of this space-time world. We\nusually say that the presence of matter distorts this world, and\nthat this distortion gives the track of particles through the region\naffected its non-uniform character.\n\nGravitation then is not a force at all; it is the fundamental nature of\nthings. A body free to move through the world must follow some definite\npath. Euclid says it will stand still; Newton that it will traverse\na straight line in three-space at uniform time-rate; Einstein that it\nwill move in a \"geodesic\" through time-space--in every-day language,\nthat it will fall.\n\nThe numerical consequences of Einstein's theory are, within the\nlimits of observation, the same as those of Newton's for all bodies\nsave one--Mercury. This planet shows a small deviation from the\npath predicted by Newton's law; Einstein's theory gives its motion\nexactly. Again, when modern research showed that light must be\naffected by gravitation, Einstein's theory, because of the extreme\nvelocity of light, deviates from Newton's, where the speed is less a\ndetermining factor; and observations of starlight deflected by the\nsun during the eclipse were in much better accord with Einstein's\ntheory than Newton's. Moreover, the Special Theory predicts that mass\nis an observational variable like length and duration. Radioactive\nemanations have a velocity high enough to give appreciable results\nhere, and the prediction is verified, tending to support the general\ntheory by supporting its limiting case.\n\nWe like always to unify our science; and seldom, after effecting\na unification, are we forced to give it up. Einstein for the first\ntime brings mechanical, electromagnetic and gravitational phenomena\nwithin one structure. This is one reason why physicists are so open\nminded toward his theory--they want it to be true.\n\n\n\n\nTHE LAYMAN'S LAST DOUBT\n\nThe final answer to any series of questions is inevitably \"because the\nworld is so constructed.\" The things we are content to leave on that\nbasis are those to which we are accustomed, and which we therefore\nthink we understand; those for which this explanation leaves us\nunsatisfied are those which are new and unfamiliar. Newton told us\nthat the world of three-dimensional space with one-dimensional time\nsuperposed was so constructed that bodies left to themselves would go\non forever in a straight line at constant speed. We think we understand\nthis, but our understanding consists merely of the unspoken query,\n\"Why, of course; what is there to prevent?\" The Greeks, an intelligent\npeople, looked at this differently; they would have met Newton with\nthe unanimous demand \"Why so; what is there to keep them going?\" So\nif, in seeking an explanation of anything, we come sooner than we\nhad expected to the finality \"Because the world is so constructed,\"\nlet us not feel that we have been cheated.\n\n\n\n\n\n\n\n\n\nIX\n\nTHE PRINCIPLE OF RELATIVITY\n\nA Statement of What it is All About, in Ideas of One Syllable\n\nBY HUGH ELLIOT\nCHISLEHURST, KENT, ENGLAND\n\n\nThe invariance of the laws of nature was one of the most popular themes\nof nineteenth century philosophy. For it was not till last century that\ngeneral acceptance was accorded to the doctrine of the \"Uniformity\nof Law,\" adumbrated in ancient times by Epicurus and Lucretius. It\nis now a cardinal axiom of science that the same cause in the same\nconditions is always followed by the same effect. There exists in\nnature no indeterminate element; all things are governed by fixed laws,\nand the discovery of these laws is the main business of science.\n\n\n    It is necessary to guard against reading into this statement an\n    erroneous idea of the content of a \"law of nature.\" Such a law\n    is of course not an enactment of any sort; and it is not even to\n    be thought of as an actual explanation of the how and why of the\n    phenomena with which it has to do. It really is nothing but an\n    expression of our belief in the pronouncement of the preceding\n    paragraph, that like conditions do produce like results. It is\n    a prediction based on past experience, and is of value merely in\n    that past experience leads us to credit its accuracy. The composite\n    essay beginning on page 19 discusses this question of the reality\n    of natural laws, and should be consulted in connection with the\n    present contribution.--Editor.\n\n\nThis great philosophic principle was derived of course from the study\nof natural science; i.e., from observations and experiments conducted\nupon the earth. Their comprehensiveness is therefore limited by the\nfact that the observer is always in a state of rest, or nearly so,\nas compared with the earth. All observers upon the earth are moving\nthrough space at the same velocity; and it was possible to argue that\nthe uniformity of law might only hold good, when experiments were\nconducted at this velocity. An observer moving at very different\nvelocity might discover that the laws of nature under these new\nconditions were somewhat different.\n\nSuch a view could indeed never be very plausible, for motion is only a\nrelative conception. Imagine a universe consisting of infinite \"empty\"\nspace, in which there is poised a single material body. How shall\nwe determine whether this body is at rest, or whether it is moving\nat high or low velocity through space? It is never getting nearer\nto anything or farther from anything, since there is no other body\nfor it to get nearer to or farther from. If we say it is moving at a\nuniform velocity of a thousand miles a second, our statement really\nhas no significance. We have no more reason for affirming that it is\nin motion than we have for affirming that it is at rest. In short,\nthere is no such thing as absolute motion; the conception of motion\nonly arises when there are two or more bodies changing their position\nrelatively to one another. This is what is meant by the relativity\nof motion. It seemed therefore improbable that the laws of nature\nwould be different if the observer were moving at high velocity;\nfor the movement of the observer is not an absolute quantity, but\nmerely a statement of his relation to other bodies, and if there were\nno other bodies, the statement itself would be meaningless.\n\n\n\n\nTHE BEHAVIOR OF LIGHT\n\nNow among the established laws of nature is that which specifies\nthe velocity of light moving through a vacuum. If the laws of nature\nare invariable, this velocity will always be the same. But consider\nwhat would happen under the following circumstances: Suppose that\nwe are at rest, and that an observer on another body flies past us\nat 150,000 miles a second. Suppose that at the moment he passes, a\npiece of flint projecting from him grazes a piece of steel projecting\nfrom us, giving rise to a spark; and that we both thereupon set about\nto measure the velocity of the light so produced. After one second,\nwe should find that the light had traveled about 186,000 miles away,\nand since during this second the other observer had traveled 150,000\nmiles, we should infer that the light traveling in his direction was\nonly about 36,000 miles ahead of him. We should also infer that he\nwould find this out by his experiment, and that he would estimate the\nvelocity of light as only 36,000 miles a second in his own direction,\nand 336,000 miles a second in the opposite direction. But if this is\nso, then that law of nature which specifies the velocity of light\nis quite different for him and for us: the laws of nature must be\ndependent upon the observer's motion--a conclusion which appears\nincompatible with the idea of the relativity of motion.\n\nAnd it so happens that it is also contradictory to experimental\nconclusions. Experiments undertaken to settle the point show that\neach observer finds the same velocity for the light of the spark;\nand after one second, each observer finds that the light has traveled\n186,000 miles from himself. But how is it possible that when it has\ntraveled 186,000 miles in the same direction as the other observer\nwho himself has moved 150,000 miles meanwhile, he should still think\nit 186,000 miles ahead of him? That is the initial paradox; and since\nthere has been no room for error in the experiments, we are forced\nto conclude that there was something wrong in the assumptions and\npreconceptions with which we started.\n\n\n\n\nSPACE AND TIME\n\nThere can in fact be only one interpretation. If we each find that the\nlight has moved the same number of miles in the same number of seconds,\nthen we must be meaning something different when we speak of miles and\nseconds. We are speaking in different languages. Some subsidence has\noccurred in the foundations of our systems of measurement. We are each\nreferring to one and the same objective fact; but since we describe\nit quite differently, and at first sight incompatibly, some profound\nalteration must have occurred in our perceptions--all unsuspected by\nourselves. It has been shown precisely what this alteration is. A body\nmoving at high velocity must become flattened in the direction of its\nmotion; all its measuring apparatus, when turned in that direction,\nis shortened, so that no hint of the flattening can be obtained from\nit. Furthermore, the standards of time are lengthened out, and clocks\ngo slower. The extent of this alteration in standards of space and time\nis stated in the equations of the so-called Lorentz transformation.\n\n\nObjection might be urged to the above paragraph on the ground that the\nconnection of the observer with the variability of measured lengths\nand times is not sufficiently indicated, and that this variability\ntherefore might be taken as an intrinsic property of the observed\nbody--which of course it is not.--Editor.\n\n\nWe are accustomed to describe space as being of three dimensions, and\ntime as being of one dimension. As a matter of fact, both space and\ntime are \"ideas,\" and not immediate sense-perceptions. We perceive\nmatter; we then infer a universal continuum filled by it, which\nwe call space. If we had no knowledge of matter, we should have no\nconception of space. Similarly in the case of time: we perceive one\nevent following another, and we then invent a continuum which we call\ntime, as an abstraction based on the sequence of events. We do not see\nspace, and we do not see time. They are not real things, in the sense\nthat matter is real, and that events are real. They are products of\nimagination: useful enough in common life, but misleading when we\ntry to look on the universe as a whole, free from the artificial\ndivisions and landmarks which we introduce into it for practical\nconvenience. Hence it is perhaps not so surprising after all that\nin certain highly transcendental investigations, these artificial\ndivisions should cease to be a convenience, and become a hindrance.\n\nTake for instance our conception of time. It differs from our\nconception of space in that it has only one dimension. In space, there\nis a right and left, an up and down, a before and after. But in time\nthere is only before and after. Why should there be this limitation of\nthe time-factor? Merely because that is the verdict of all our human\nexperience. But is our human experience based on a sufficiently broad\nfoundation to enable us to say that, under all conditions and in all\nparts of the universe, there can be only one time-direction? May not\nour belief in the uniformity of time be due to the uniformity of the\nmotion of all observers on the earth? Such in fact is the postulate of\nrelativity. We now believe that, at velocities very different from our\nown, the standard of time would also be different from ours. From our\npoint of view, that different standard of time would not be confined\nto the single direction fore and aft, as we know it, but would also\nhave in it an element of what we might call right and left. True,\nit would still be of only one dimension, but its direction would\ndiffer from the direction of our time. It would still run like a\nthread through the universe, but not in the direction which we call\nstraight forward. It would have a slant in it, and the angle of the\nslant depends upon the velocity of motion. It does not follow that\nbecause we are all traveling in the same direction down the stream of\ntime, therefore that stream can only flow in the direction which we\nknow. \"Before\" and \"after\" are expressions which, like right and left,\ndepend upon our personal situation. If we were differently situated,\nif to be precise we were moving at very high velocity, we should,\nso to speak, be facing in a new direction and \"before\" and \"after\"\nwould imply a different direction of progress from that with which\nwe are now familiar.\n\n\n\n\nTHE WORLD OF REALITY\n\nBut, after all, the objective universe is the same old universe\nhowever fast we are moving about in it, and whatever way we are\nfacing. These details merely determine the way we divide it up into\nspace and time. The universe is not affected by any arbitrary lines\nwhich we draw through it for our personal convenience. For practical\npurposes, we ascribe to it four dimensions, three in space and one in\ntime. Clearly if the time direction is altered, all dimensions both\nof space and time must have different readings. If, for instance,\nthe time direction slopes away to the left, as compared with ours,\nthen space measurements to right and left must be correspondingly\naltered. An analogy will simplify the matter.\n\nSuppose we desire to reach a point ten miles off in a roughly\nnortheasterly direction. We might do so by walking six miles due east\nand then eight miles due north. We should then be precisely ten miles\nfrom where we started. But suppose our compass were out of order, so\nthat its north pole pointed somewhat to the west of north. Then in\norder to get to our destination, we might have to walk seven miles\nin the direction which we thought was east, and a little more than\nseven miles in the direction which we thought was north. We should\nthen reach the same point as before. Both observers have walked\naccording to their lights, first due east and then due north, and\nboth have reached the same point: the one observer is certain that\nthe finishing point is six miles east of the starting-point, while\nthe other is sure it is seven miles.\n\nNow we on the earth are all using a compass which points in the same\ndirection as regards time. But other observers, on bodies moving with\nvery different velocity, have a compass in which the time-direction is\ndisplaced as compared with ours. Hence our judgments of distances will\nnot be alike. In our analogy, the northerly direction corresponds to\ntime, and the easterly direction to space; and so long as we use the\nsame compass we do not differ in our measurements of distances. But\nfor any one who has a different notion of the time-direction, not\nonly time intervals but space distances will be judged differently.\n\nIn short, the universe is regarded as a space-time continuum of four\ndimensions. A \"point\" in space-time is called an \"event\"--that which\noccurs at a specified moment and at a specified place. The distance\nbetween two points in space-time is called their \"interval.\" All\nobservers will agree as to the magnitude of any interval, since it\nis a property of the objective universe; but they will disagree as\nto its composition in space and time separately. In short, space\nand time are relative conceptions; their relativity is a necessary\nconsequence of the relativity of motion. The paradox named at the\noutset is overcome; for the two observers measuring the velocity of\nthe light produced as they passed one another, were using different\nunits of space and time. And hence emerges triumphant the Special\nPrinciple of Relativity, which states that the laws of nature are\nthe same for all observers, whether they are in a state of rest or\nof uniform motion in a straight line.\n\n\n\n\nACCELERATED MOTION\n\nUniform motion in a straight line is however a very special kind of\nmotion. Our experience in ordinary life is of motions that are neither\nuniform nor in a straight line; both speed and direction of motion\nare altering. The moving body is then said to undergo \"acceleration\":\nwhich means either that its speed is increasing or diminishing, or\nthat its direction of motion is changing, or both. If we revert to our\nformer supposition of a universe in which there is only a single body\nin \"empty\" space, we clearly cannot say whether it has acceleration\nany more than whether it is moving, there being no outside standard\nof comparison; and the General Principle of Relativity asserts the\ninvariance of the laws of nature for all states of motion of the\nobserver. In this case, however, a difference might be detected\nby an observer on the moving body itself. It would be manifested\nto him as the action of a force; such for instance as we feel when\na train in which we are traveling is increasing or reducing speed,\nor when, without changing speed, it is rounding a corner. The force\ndies away as soon as the velocity becomes uniform. Thus acceleration\nreveals itself to us under the guise of action by a force. Force and\nacceleration go together, and we may either say that the acceleration\nis due to the force, or the impression of force to the acceleration.\n\nNow when we are traveling with accelerated motion, we have quite a\ndifferent idea of what constitutes a straight line from that which\nwe had when at rest or in uniform motion. If we are moving at uniform\nvelocity in an airplane and drop a stone to the earth it will appear\nto us in the airplane to fall in a straight line downward, while to\nan observer on the earth it will appear to describe a parabola. This\nis due to the fact that the stone gathers speed as it falls; it is\nsubject to the acceleration associated with gravity. Acceleration\nobliterates the fundamental difference between a straight and\ncurved line. Unless we know what is the absolute motion of the\nstone, and the two observers, we cannot say whether the line is\n\"really\" a straight or a curved line. Since absolute motion is an\nillegitimate conception, it follows that there is no such thing as\n\"really\" straight or \"really\" curved. These are only appearances\nset up as a consequence of our relative motions with respect to the\nbodies concerned. If there were no such thing as acceleration--if the\nstone fell to the earth at uniform velocity--then an observer on the\nearth or anywhere else would agree that it fell in a straight line;\nand straight lines would always be straight lines.\n\nUnder these circumstances, Euclidean geometry would be absolutely\ntrue. But if we are in a state of acceleration, then what we think are\nstraight lines are \"really\" curved lines, and Euclidean geometry, based\non the assumption that its lines are straight, must founder when tested\nby more accurate measurements. And in point of fact we are in a state\nof acceleration: for we are being acted upon by a force--namely, the\nforce of gravitation. Wherever there is matter, there is gravitation;\nwherever there is gravitation there is acceleration; wherever there\nis acceleration Euclidean geometry is inaccurate. Hence in the space\nsurrounding matter a different geometry holds the field; and bodies\nin general move through such space in curved lines.\n\nDifferent parts of space are thus characterized by different\ngeometrical properties. All bodies in the universe proceed on\ntheir established courses through space and time. But when they\ncome to distorted geometrical areas, their paths naturally seem\nto us different from when they were moving through less disturbed\nregions. They exhibit the difference by acquiring an acceleration;\nand we explain the acceleration by alleging the existence of a force,\nwhich we call the force of gravitation. But their motions can in fact\nbe perfectly predicted if we know the geometry of the space through\nwhich they are traveling. The predictions so based have in fact proved\nmore accurate than those based on the law of gravitation.\n\n\n\n\n\n\n\n\n\nX\n\nSPACE, TIME AND GRAVITATION\n\nAn Outline of Einstein's Theory of General Relativity\n\nBY W. DE SITTER\nPROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF LEYDEN\n\n\"Henceforth space by itself and time by itself shall sink to mere\nshadows, and only a union of the two shall preserve reality.\"\n\n\nThe prophecy contained in the above-quoted words, spoken by Minkowski\nat the meeting of German \"Naturforscher und Aerzte\" at Cologne in 1908,\nhas, however, only been completely fulfilled by Einstein's \"Allgemeine\nRelativit\u00e4ts-theorie\" of 1915, which incorporated gravitation into\nthe union. In the following pages an attempt is made to set forth,\nwithout using any technical language, the leading ideas of that theory:\nI will confine myself to the theory as published by Einstein in\nNovember, 1915, which forms a consistent whole, complete in itself;\nand I will not refer to later developments, which are still more\nor less tentative, and not necessary for the understanding of the\ntheory. The mathematics used by Einstein is the so-called Absolute\nDifferential Calculus. It is not more difficult or recondite than that\nused in other branches of theoretical physics, but it is somewhat\nunfamiliar to most of us, because it is not generally taught in the\nregular university courses. I will, however, in this essay abstain\nfrom using any mathematics at all, at least, I will not be using it\nopenly. It is of course unavoidable to use at least the results of\nthe mathematical reasoning, if not the reasoning itself; but so long\nas they are not put into formulas they will, it is hoped, not look\nso formidable to the reader.\n\nReferring to the quoted words of Minkowski, we may ask what is meant by\n\"reality.\" Physical science, like common sense, takes for granted that\nthere is a reality behind the phenomena, which is independent of the\nperson by whom, and the particular methods by which it is observed,\nand which is also there when it is not observed. Strictly speaking,\nall talk about what is not observed is metaphysics. Nevertheless\nthe physicist unhesitatingly believes that his laws are general, and\nthat the phenomena continue to happen according to them when nobody\nis looking. And since it would be impossible to prove that they did\nnot, he is fully entitled to his belief. The observed phenomena are\nthe effects of the action of this reality, of which we assume the\nexistence, on the observer's senses--or apparatus, which are extended\nand refined sense-organs. The laws governing the phenomena therefore\nmust convey some information regarding this reality. We shall never\nby any means be able to know anything else about it but just these\nlaws. To all intents and purposes the laws are the reality, if we\neliminate from them all that refers to the observer alone. What refers\nto the reality is called \"absolute,\" and what involves reference to\nthe observer \"relative.\" The elimination of the relative is one of\nthe things the theory of relativity has set out to do.\n\n\n\n\nTHE EXTERNAL WORLD AND ITS GEOMETRY\n\nTo describe the phenomena and derive laws from them, we locate them\nin space and time. To do this we use geometry. Here it is that the\npart contributed by the observer comes in. There are an infinite\nnumber of geometries, and a priori there seems to be no reason to\nchoose one rather than the other. Taking geometry of two dimensions\nas an example, we can draw figures on a piece of paper, and discuss\ntheir properties, and we can also do so on the shell of an egg. But\nwe cannot draw the same figures on the egg as on the paper. The\nones will be distorted as compared with the others: the two surfaces\nhave a different geometry. Similarly it is not possible to draw an\naccurate map of the earth on a sheet of paper, because the earth is\nspherical and its representation on the flat paper is always more or\nless distorted. The earth requires spherical geometry, which differs\nfrom the flat, or Euclidean, geometry of the paper.\n\nUp to a few years ago Euclidean (i.e. flat) geometry of three\ndimensions had been exclusively used in physical theories. Why? Because\nit is the true one, is the one answer generally given. Now a statement\nabout facts can be true or false, but a mathematical discipline is\nneither true nor false; it can only be correct--i.e. consistent in\nitself--or incorrect, and of course it always is correct. The assertion\nthat a certain geometry is the \"true\" one can thus only mean, that\nit is the geometry of \"true\" space, and this again, if it is to have\nany meaning at all, can only mean that it corresponds to the physical\n\"reality.\" Leaving aside the question whether this reality has any\ngeometry at all, we are confronted with the more immediately practical\nconsideration how we shall verify the asserted correspondence. There is\nno other way than by comparing the conclusions derived from the laws\nbased upon our geometry, with observations. It thus appears that the\nonly justification for the use of the Euclidean geometry is its success\nin enabling us to \"draw an accurate map\" of the world. As soon as any\nother geometry is found to be more successful, that other must be used\nin physical theories, and we may, if we like, call it the \"true\" one.\n\nAccurate observations always consist of measures, determining the\nposition of material bodies in space. But the positions change,\nand for a complete description we also require measures of time. An\nimportant remark must be made here. Nobody has ever measured a\npure space-distance, nor a pure lapse of time. The only thing that\ncan be measured is the distance from a body at a certain point of\nspace and a certain moment of time, to a body (either the same or\nanother) at another point and another time. We can even go further\nand say that time cannot be measured at all. We profess to measure\nit by clocks. But a clock really measures space, and we derive the\ntime from its space-measures by a fixed rule. This rule depends\non the laws of motion of the mechanism of the clock. Thus finally\ntime is defined by these laws. This is so, whether as a \"clock\"\nwe use an ordinary chronometer, or the rotating earth, or an atom\nemitting light-waves, or anything else that may be suggested. The\nphysical laws, of course, must be so adjusted that all these devices\ngive the same time. About the reality of time, if it has any, we\nknow nothing. All we know about time is that we want it. We cannot\nadequately describe nature with the three space-coordinates alone,\nwe require a fourth one, which we call time. We might thus say with\nsome reason that the physical world has four dimensions. But so long\nas it was found possible adequately to describe all known phenomena\nby a space of three dimensions and an independent time, the statement\ndid not convey any very important information. Only after it had been\nfound out that the space-coordinates and the time are not independent,\ndid it acquire a real meaning.\n\nAs is well known the observation by which this was found out is the\nfamous experiment of Michelson and Morley. It led to the \"special\"\ntheory of relativity, which is the one referred to by Minkowski\nin 1908. In it a geometry of four dimensions is used, not a mere\ncombination of a three-dimensional space and a one-dimensional time,\nbut a continuum of truly fourfold order. This time-space is not\nEuclidean, since the time-component and the three space-components\nare not on the same footing, but its fundamental formula has a\ngreat resemblance to that of Euclidean geometry. We may call it\n\"pseudo-Euclidean.\"\n\nThis theory, which we need not explain here, was very satisfactory so\nfar as the laws of electromagnetism, and especially the propagation\nof light, were concerned, but it did not include gravitation, and\nmechanics generally. We then had this curious state of affairs, that\nphysicists actually believed in two different \"realities.\" When they\nwere thinking of light they believed in Minkowski's time-space; when\nthey were thinking of gravitation they believed in the old Euclidean\nspace and independent time. This, of course, could not last. Attempts\nwere made so to alter Newton's law of gravitation that it would fit\ninto the four-dimensional world of the special relativity-theory,\nbut these only succeeded in making the law, which had been a model\nof simplicity, extremely complicated, and, what was worse, it became\nambiguous.\n\nIt is Einstein's great merit to have perceived that gravitation is\nof such fundamental importance, that it must not be fitted into a\nready-made theory, but must be woven into the space-time geometry from\nthe beginning. And that he not only saw the necessity of doing this,\nbut actually did it.\n\n\n\n\nGRAVITATION AND ITS PLACE IN THE UNIVERSE\n\nTo see the necessity we must go back to Newton's system of\nmechanics. Newton did two things (amongst others). He canonised\nGalileo's system of mechanics into his famous \"laws of motion,\"\nthe most important of which is the law of inertia, which says that:\n\n\n    a body, that is not interfered with, moves in a straight line\n    with constant velocity.\n\n\nThe velocity, of course, can be nil, and the body at rest. This is\na perfectly general law, the same for all material bodies, whatever\ntheir physical or chemical status. Newton took good care exactly to\ndefine what he meant by uniform motion in a straight line, and for\nthis purpose he introduced the absolute Euclidean space and absolute\ntime as an essential part of his system of laws at the very beginning\nof his great work. The other thing Newton did was to formulate the\nlaw of gravitation. Gravitation was in his system considered as\nan interference with the free, or inertial, motion of bodies, and\naccordingly required a law of its own.\n\nBut gravitation has this in common with inertia, and in this it\ndiffers from all other interferences, that it is perfectly general. All\nmaterial bodies are equally subjected to it, whatever their physical\nor chemical status may be. But there is more. Gravitation and inertia\nare actually indistinguishable from each other, and are measured\nby the same number: the \"mass\". This was already remarked by Newton\nhimself, and from his point of view it was a most wonderful accidental\ncoincidence. If an apple falls from the tree, that which makes it fall\nis its weight, which is the gravitational attraction by the earth,\ndiminished by the centrifugal force due to the earth's rotation and the\napple's inertia. In Newton's system the gravitational attraction is a\n\"real\" force, whereas the centrifugal force is only \"fictitious\". But\nthe one is as real as the other. The most refined experiments,\nalready begun by Newton himself, have not succeeded in distinguishing\nbetween them. Their identity is actually one of the best established\nfacts in experimental physics. From this identity of \"fictitious,\" or\ninertial, and \"real,\" or gravitational, forces it follows that locally\na gravitational field can be artificially created or destroyed. Thus\ninside a closed room which is falling freely, say a lift of which the\ncable has been broken, bodies have no weight: a balance could be in\nequilibrium with different weights in the two scales.\n\nHaving thus come to the conclusion that gravitation is not an\ninterference, but is identical with inertia, we are tempted to restate\nthe law of motion, so as to include both, thus:\n\n\n    Bodies which are not interfered with--do not move in straight\n    lines, but--fall.\n\n\nNow this is exactly what Einstein did. Only the \"falling\" of\ncourse requires a precise mathematical definition (like the uniform\nmotion in a straight line), and the whole gist of his theory is the\nfinding of that definition. In our earthly experience the falling\nnever lasts long, very soon something--the floor of the room, or\nthe earth itself--interferes. But in free space bodies go on falling\nforever. The motion of the planets is, in fact, adequately described as\nfalling, since it consists in nothing else but obeying Newton's law of\ngravitation together with his law of inertia. A body very far removed\nfrom all other matter is not subjected to gravitation, consequently it\nfalls with constant velocity in a straight line according to the law\nof inertia. The problem was thus to find a mathematical definition of\n\"falling,\" which would embrace the uniform straight-line motion very\nfar from all matter as well as the complex paths of the planets around\nthe sun, and of an apple or a cannon-ball on earth.\n\n\n\n\nGRAVITATION AND SPACE-TIME\n\nFor the definition of the uniform rectilinear motion of pure inertia\nNewton's Euclidean space and independent time were sufficient. For\nthe much more complicated falling under the influence of gravitation\nand inertia together, evidently a more complicated geometry\nwould be needed. Minkowski's pseudo-Euclidean time-space also was\ninsufficient. Einstein accordingly introduced a general non-Euclidean\nfour-dimensional time-space, and enunciated his law of motion thus:\n\n\n    Bodies which are not interfered with move in geodesics.\n\n\nA geodesic in curved space is exactly the same thing as a straight\nline in flat space. We only call it by its technical name, because\nthe name \"straight line\" would remind us too much of the old Euclidean\nspace. If the curvature gets very small, or zero, the geodesic becomes\nvery nearly, or exactly, a straight line.\n\nThe problem has now become to assign to time-space such curvatures\nthat the geodesics will exactly represent the tracks of falling\nbodies. Space of two dimensions can just be flat, like a sheet of\npaper, or curved, like an egg. But in geometry of four dimensions\nthere are several steps from perfect flatness, or \"pseudo-flatness,\" to\ncomplete curvature. Now the law governing the curvature of Einstein's\ntime-space, i.e., the law of gravitation, is simply that it can never,\noutside matter, be curved more than just one step beyond perfect\n(pseudo-)flatness.\n\nSince I have promised not to use any mathematics I can hardly convey\nto the reader an adequate idea of the difficulty of the problem,\nnor do justice to the elegance and beauty of the solution. It is,\nin fact, little short of miraculous that this solution, which was\nonly adopted by Einstein because it was the simplest he could find,\ndoes so exactly coincide in all its effects with Newton's law. Thus the\nremarkably accurate experimental verification of this law can at once\nbe transferred to the new law. In only one instance do the two laws\ndiffer so much that the difference can be observed, and in this case\nthe observations confirm the new law exactly. This is the well known\ncase of the motion of the perihelion of Mercury, whose disagreement\nwith Newton's law had puzzled astronomers for more than half a century.\n\nSince Einstein's time-space includes Minkowski's as a particular case,\nit can do all that the other was designed to do for electro-magnetism\nand light. But it does more. The track of a pulse of light is also a\ngeodesic, and time-space being curved in the neighborhood of matter,\nrays of light are no longer straight lines. A ray of light from a star,\npassing near the sun, will be bent round, and the star consequently\nwill be seen in a different direction from where it would be seen if\nthe sun had not been so nearly in the way. This has been verified by\nthe observations of the eclipse of the sun of 1919 of May 29.\n\nThere is one other new phenomenon predicted by the theory, which falls\nwithin the reach of observation with our present means. Gravitation\nchiefly affects the time-component of the four-dimensional\ncontinuum, in such a way that natural clocks appear to run slower\nin a strong gravitational field than in a weak one. Thus, if we\nmake the hypothesis--which, though extremely probable, is still\na hypothesis--that an atom emitting or absorbing light-waves is a\nnatural clock, and the further hypothesis--still very probable, though\nless so than the former--that there is nothing to interfere with its\nperfect running, then an atom on the sun will give off light-waves\nof smaller frequency than a similar atom in a terrestrial laboratory\nemits. Opinions as yet differ as to whether this is confirmed or\ncontradicted by observations.\n\n\n\nThe great strength and the charm of Einstein's theory do however\nnot lie in verified predictions, nor in the explanation of small\noutstanding discrepancies, but in the complete attainment of its\noriginal aim: the identification of gravitation and inertia, and in\nthe wide range of formerly apparently unconnected subjects which it\nembraces, and the broad view of nature which it affords.\n\n\n\nOutside matter, as has been explained, the law of gravitation restricts\nthe curvature of time-space. Inside continuous matter the curvature\ncan be of any arbitrary kind or amount; the law of gravitation then\nconnects this curvature with measurable properties of the matter,\nsuch as density, velocity, stress, etc. Thus these properties define\nthe curvature, or, if preferred, the curvature defines the properties\nof matter, i.e. matter itself.\n\nFrom these definitions the laws of conservation of energy, and of\nconservation of momentum, can be deduced by a purely mathematical\nprocess. Thus these laws, which at one time used to be considered\nas the most fundamental ones of mechanics, now appear as simple\ncorollaries from the law of gravitation. It must be pointed out that\nsuch things as length, velocity, energy, momentum, are not absolute,\nbut relative, i.e. they are not attributes of the physical reality,\nbut relations between this reality and the observer. Consequently\nthe laws of conservation are not laws of the real world, like the\nlaw of gravitation, but of the observed phenomena. There is, however\none law which, already before the days of relativity, had come to be\nconsidered as the most fundamental of all, viz: the principle of least\naction. Now action is absolute. Accordingly this principle retains its\ncentral position in Einstein's theory. It is even more fundamental than\nthe law of gravitation, since both this law, and the law of motion,\ncan be derived from it. The principle of least action, so far as we\ncan see at present, appears to be the law of the real world.\n\n\n\n\n\n\n\n\n\nXI\n\nTHE PRINCIPLE OF GENERAL RELATIVITY\n\nHow Einstein, to a Degree Never Before Equalled, Isolates the External\nReality from the Observer's Contribution\n\nBY E. T. BELL\nUNIVERSITY OF WASHINGTON\nSEATTLE\n\n\nEinstein's general relativity is of such vast compass, being\ncoextensive with the realm of physical events, that in any brief\naccount a strict selection from its numerous aspects is prescribed. The\nold, restricted principle being contained in the general, we shall\ntreat the latter, its close relations with gravitation, and the\nsignificance of both for our knowledge of space and time. The essence\nof Einstein's generalization is its final disentanglement of that\npart of any physical event which is contributed by the observer from\nthat which is inherent in the nature of things and independent of\nall observers.\n\nThe argument turns upon the fact that an observer must describe any\nevent with reference to some framework from which he makes measurements\nof time and distance. Thus, suppose that at nine o'clock a ball is\ntossed across the room. At one second past nine the ball occupies a\ndefinite position which we can specify by giving the three distances\nfrom the centre of the ball to the north and west walls and the\nfloor. In this way, refining our measurements, we can give a precise\ndescription of the entire motion of the ball. Our final description\nwill consist of innumerable separate statements, each of which contains\nfour numbers corresponding to four measurements, and of these one\nwill be for time and three for distances at the time indicated.\n\nImagine now that a man in an automobile looks in and observes the\nmoving ball. Suppose he records the motion. To do so, he must refer to\na timepiece and some body of reference. Say he selects his wrist-watch,\nthe floor of his auto and two sides meeting in a corner. Fancy that\njust as he begins his series of observations his auto starts bucking\nand the main-spring of his watch breaks, so that he must measure\n\"seconds\" by the crazy running-down of his watch, and distances with\nreference to the sides of his erratic auto. Despite these handicaps\nhe completes a set of observations, each of which consists of a\ntime measured by his mad watch and three distances reckoned from\nthe sides of his bucking machine. Let us assume him to have been so\nabsorbed in his experiment that he noticed neither the disorders\nof his watch nor the motion of his auto. He gives us his sets of\nmeasurements. We remark that his seconds are only small fractions\nof ours, also his norths and wests are badly mixed. If we interpret\nhis sets in terms of our stationary walls and sober clock we find\nthe curious paradox that the ball zigzagged across the room like an\nintoxicated bee. He obstinately argues that we know no more than he\nabout how the ball actually moved. For we got a smooth description,\nhe asserts, by choosing an artificially simple reference framework,\nhaving no necessary relations whatever to the ball. The crooked\npath plotted from his observations proves, he declares, that the\nball was subject to varying forces of which we in the room suspected\nnothing. He contends that our room was being jarred by a system of\nforces which exactly compensated and smoothed out the real jaggedness\nof path observed by himself. But if we know all about his watch and\nauto we can easily apply necessary corrections to his measurements,\nand, fitting the corrected set to our reference-framework of walls\nand clock, recover our own smooth description.\n\nFor consistency we must carry our readjustments farther. The path\nmapped from our measurements is a curve. Perhaps the curvature was\nintroduced by some peculiarity of our reference framework? Possibly\nour own room is being accelerated upward, so that it makes the ball's\ntrue path--whatever that may be--appear curved downward, just as the\nautoist's zigzags made the path he mapped appear jagged. Tradition\nattributes the downward curving to the tug of gravity. This force we\nsay accelerates the ball downward, producing the curved path. Is this\nthe only possible explanation? Let us see.\n\n\n\n\nGRAVITATION AND ACCELERATION\n\nImagine a man in a room out of which he cannot see. He notices that\nwhen he releases anything it falls to the floor with a constant\nacceleration. Further he observes that all his objects, independently\nof their chemical and physical properties, are affected in precisely\nthe same way. Now, he previously has experimented with magnets, and\nhas remarked that they attract certain bodies in essentially the same\nway that the things which he drops are \"attracted\" to whatever is\nbeneath the floor. Having explained magnetic attraction in terms of\n\"forces,\" he makes his first hypothesis: (A) He and his room are in\na strong \"field of force,\" which he designates gravitational. This\nforce pulls all things downward with a constant acceleration. Here\nhe notes a singular distinction between magnetic and gravitational\n\"forces\": magnets attract only a few kinds of matter, notably iron;\nthe novel \"force,\" if indeed a force at all, acts similarly upon all\nkinds of matter. He makes another hypothesis: (B) His room and he\nare being accelerated upward.\n\n\n\nEither (A) or (B) describes the facts perfectly. By no experiment\ncan he discriminate between them. So he takes the great step, and\nformulates the Equivalence Hypothesis:\n\n\nA gravitational field of force is precisely equivalent in its effects\nto an artificial field of force introduced by accelerating the\nframework of reference, so that in any small region it is impossible\nto distinguish between them by any experiment whatever.\n\n\nNext reconsidering his magnetic \"forces,\" he extends the equivalence\nhypothesis to cover all manifestations of force: The effects\nattributed to forces of any kind whatever can be described equally\nwell by saying that our reference frameworks are accelerated; and\nmoreover there is possible no experiment which will discriminate\nbetween the descriptions.\n\nIf the accelerations are null, the frameworks are at rest or in\nuniform motion relatively to one another. This special case is the\n\"restricted\" principle of relativity, which asserts that it is\nimpossible experimentally to detect a uniform motion through the\nether. Being thus superfluous for descriptions of natural phenomena,\nthe ether may be abandoned, at least temporarily. The older physics\nsought this absolute ether framework to which all motions could be\nunambiguously referred, and failed to find it. The most exacting\nexperiments, notably that of Michelson-Morley, revealed no trace of\nthe earth's supposed motion through the ether. Fitzgerald accounted\nfor the failure by assuming that such motion would remain undetected if\nevery moving body contracted by an amount depending upon its velocity\nin the direction of motion. The contraction for ordinary velocities\nis imperceptible. Only when as in the case of the beta particles,\nthe velocity is an appreciable fraction of the velocity of light,\nis the contraction revealed. This contraction follows immediately\nfrom Einstein's generalization constructed upon the equivalence\nhypothesis and the restricted relativity principle. We shall see that\nthe contraction inevitably follows from the actual geometry of the\nuniverse. [2]\n\nLet us return for a moment to the moving ball. Four measures,\nthree of distances and one of time, are required in specifying its\nposition with reference to some framework at each point and at each\ninstant. All of these measures can be summed up in one compendious\nstatement--the equations of motion showed how in changing from our\nroom to his accelerated auto we found a new summary, \"transformed\nequations,\" which seemed to indicate that the ball had traversed\na strong, variable field of force. Is there then in the chaos of\nobservational disagreements anything which is independent of all\nobservers? There is, but it is hidden at the very heart of nature.\n\n\n\n\nPATHS THROUGH THE WORLD OF FOUR DIMENSIONS\n\nTo exhibit this, we must recall a familiar proposition of geometry:\nthe square on the longest side of a right-angled triangle is equal\nto the sum of the squares on the other two sides. It has long been\nknown that from this alone all the metrical properties of Euclidean\nspace--the space in which for 2,000 years we have imagined we were\nliving--can be deduced. Metrical properties are those depending upon\nmeasurement. Now, in the geometry of any space, Euclidean or not,\nthere is a single proposition of a similar sort which tells us how\nto find the most direct distance between any two points that are\nvery close together. This small distance is expressed in terms of\nthe two sets of distance measurements by which the end-points are\nlocated, just as two neighboring positions of our ball were located\nby two sets of four measurements each. We say by analogy that two\nconsecutive positions of the ball are separated by a small interval of\ntime-space. From the formula for the very small interval of time-space\nwe can calculate mathematically all the metrical properties of the time\nand space in which measurements for the ball's motion must be made. So\nin any geometry mathematical analysis predicts infallibly the truth\nabout all facts depending upon measurements from the simple formula\nof the interval between neighboring points. Thus, on a sphere the sum\nof the angles of any triangle formed by arcs of great circles exceeds\n180\u00b0, and this follows from the formula for the shortest (\"geodesic\")\ndistance between neighboring points on the spherical surface.\n\nWe saw that it takes four measurements, one for time and three for\ndistances, to fix an elementary event, viz., the position of the\ncentre of our ball at any instant. A system of all possible such sets\nof four measurements each, constitutes what mathematicians call a\nfour-dimensional space. The study of the four-dimensional time-space\ngeometry, once its shortest-distance proposition is known, reveals all\nthose relations in nature which can be ascertained by measurements,\nthat is, experimentally. We have then to find this indispensable\nproposition.\n\nImagine the path taken by a particle moving solely under the influence\nof gravitation. This being the simplest possible motion of an actual\nparticle in the real world, it is natural to guess that its path\nwill be such that the particle moves from one point of time-space to\nanother by the most direct route. This in fact is verified by forming\nthe equations of the free particle's motion, which turn out to be\nprecisely those that specify a geodesic (most direct line) joining\nthe two points. On the (two-dimensional) surface of a sphere such a\nline is the position taken by a string stretched between two points\non the surface, and this is the shortest distance on the surface\nbetween them. But in the time-space geometry we find a remarkable\ndistinction: the interval between any two points of the path taken is\nthe longest possible, and between any two points there is only one\nlongest path. Translated into ordinary space and time this merely\nasserts that the time taken between any two points on the natural\npath is the longest possible.\n\nRecall now that when the line-formula for any kind of space is known\nall the metrical properties of that space are completely determined,\nand combine with this what we have just found, namely, the equations\nof motion of a particle subject only to gravitation are the same\nequations as those which fix the line-formula for the four-dimensional\ntime-space. Since gravitation alone determines the motion of the\nparticle, and since this motion is completely described by the very\nequations which fix all the metrical properties of time-space, it\nfollows that the metrical (experimentally determinable) properties\nof time-space are equivalent to those of gravitation, in the sense\nthat each set of properties implies the other.\n\n\n\n\nTHE UNIVERSE OF SPACE-TIME\n\nWe have found the thing in nature which is independent of all\nobservers, and it turns out to be the very structure of time-space\nitself. The motion of the free particle obviously is a thing\nunconditioned by accidents of observation; the particle under the\ninfluence of gravitation alone must go a way of its own. And if some\nobserver in an artificial field of force produced by the acceleration\nof his reference framework describes the path as knotted, he merely is\nfoisting eccentricities of his own motion upon the direct path of the\nparticle. The conclusion is rational, for we believe that time-space\nexists independently of any man's way of perceiving it.\n\nIncidentally note that this space is that of the physical world. For\nonly by measurements of distances and times can we become aware of\nour extension in time and space. If beyond this time-space geometry\nof measurements there is some \"absolute geometry,\" science can have\nno concern with it, for never can it be revealed by the one exploring\ndevice we possess--measurement.\n\nWe have followed a single particle. Let us now form a picture of\nseveral. Any event can be analyzed into a multitude of coincidences\nin time-space. For consider two moving particles--say electrons. If\nthey collide they both are in very approximately one place at the\nsame time. We imagine the path of an electron through time-space\nplotted by a line (in four-dimensional space), which will deviate\nfrom a \"most direct\" (geodesic) path if the electron is subjected\nto forces. This is the \"world-line\" of the electron. If the world\nlines of several electrons intersect at one point in time-space,\nthe intersection pictures the fact of their coincidence somewhere\nand somewhen; for all their world-lines having a time-space point in\ncommon, at some instant they must have been in collision. Each point\nof a world-line pictures the position at a certain place at a certain\ntime; and it is the intersections of world-lines which correspond to\nphysical events. Of what lies between the intersections we have no\nexperimental knowledge.\n\nImagine the world-lines of all the electrons in the universe threading\ntime-space like threads in a jelly. The intersections of the tangle\nare a complete history of all physical events. Now distort the\njelly. Clearly the mutual order of the intersections will be unchanged,\nbut the distances between them will be shortened or lengthened. To a\ndistortion of the jelly corresponds a special choice (by some observer)\nof a reference framework for describing the order of events. He cannot\nchange the natural sequence of events. Again we have found something\nwhich is independent of all observers.\n\nWe can now recapitulate our conclusions and state the principle of\nrelativity in its most general form.\n\n(1) Observers describe events by measures of times and distances made\nwith regard to their frameworks of reference.\n\n(2) The complete history of any event is summarized in a set of\nequations giving the positions of all the particles involved at\nevery instant.\n\n(3) Two possibilities arise. (A) Either these equations are the\nsame in form for all space-time reference frameworks, persisting\nformally unchanged for all shifts of the reference scheme; or (B),\nthey subsist only when some special framework is used, altering their\nform as they are referred to different frameworks. If (B) holds,\nwe naturally assume that the equations, and the phenomena which they\nprofess to represent, owe their existence to some peculiarity of the\nreference framework. They do not, therefore, describe anything which is\ninherent in the nature of things, but merely some idiosyncrasy of the\nobserver's way of regarding nature. If (A) holds, then obviously the\nequations describe some real relation in nature which is independent\nof all possible ways of observing and recording it.\n\n(4) In its most general form the principle of relativity states that\nthose relations, and those alone, which persist unchanged in form\nfor all possible space-time reference frameworks are the inherent\nlaws of nature.\n\nTo find such relations Einstein has applied a mathematical method\nof great power--the calculus of tensors--with extraordinary\nsuccess. This calculus threshes out the laws of nature, separating\nthe observer's eccentricities from what is independent of him, with\nthe superb efficiency of a modern harvester. The residue is a physical\ngeometry--or geometrical physics--of time-space, in which it appears\nthat the times and spaces contributed by the several observers'\nreference frameworks are shadows of their own contrivings; while\nthe real, enduring universe is a fourfold order of time and space\nindissolubly bound together. One observer separates this time-space\ninto his own \"time\" and \"space\" in one way, determined by his path\nthrough the world of events; another, moving relatively to the first,\nseparates it differently, and what for one is time shades into space\nfor another.\n\nThis time-space geometry is non-Euclidean. It is \"warped\" (curved),\nthe amount of warping at any place being determined by the intensity\nof the gravitational field there. Thus again gravitation is rooted in\nthe nature of things. In this sense it is not a force, but a property\nof space. Wherever there is matter there is a gravitational field,\nand hence a warping of space. Conversely, as long ago imagined by\nClifford, wherever there is a warping of space, there is matter;\nand matter is resolved ultimately into wrinkles in time-space.\n\nTo visualize a warped space, consider a simple analogy. A man walks\naway from a polished globe; his image recedes into the mirror-space,\nshortening and thinning as it goes, and thinning (in the direction of\nmotion) faster than it shortens. Everything around him experiences\na like effect. If he tries to discover this by a footrule it\nautomatically shortens faster as he turns it into the horizontal\nposition, so his purpose eludes him. The mirror-space is warped in\nthe direction of the image's motion. So is our own. For all bodies,\nas evidenced by the Fitzgerald contraction, shorten in the direction\nof motion. And just as the image can never penetrate the mirror-space\na greater distance than half its radius, so probably time-space is\ncurved in such a way that our universe, like the surface of a sphere,\nis finite in extent, but unbounded.\n\n\n\n\n\n\n\n\n\nXII\n\nFORCE VS. GEOMETRY\n\nHow Einstein Has Substituted the Second for the First in Connection\nWith the Cause of Gravitation\n\nBY SAUL DUSHMAN\nGENERAL ELECTRIC LABORATORIES SCHENECTADY, N. Y.\n\n\nThe theory of relativity represents a most strikingly original\nconception of time and space, which was suggested by Einstein in order\nto correlate with all our past experience certain observations made\nin recent years. It is therefore extremely comprehensive in its scope;\nit demands from us a radical revision in our notions of time and space;\nit throws new light on the nature of mass and energy, and finally, it\nfurnishes a totally new conception of the old problem of gravitation.\n\nThe starting point of the theory is the familiar observation that\nmotion is always relative: that is, to define the motion of any object\nwe must always use some point of reference. Thus we speak of the\nvelocity of a train as 40 miles per hour with respect to the earth's\nsurface, but would find it impossible to determine its absolute speed,\nor motion in space, since we know of no star whose position can be\nspoken of as absolutely fixed. These and similar considerations have\nled to the conclusion, pointed out by Newton and others, that it is\nimpossible by any mechanical experiments on the earth to measure its\nvelocity in space.\n\nHowever, the results of observations on the phenomena of light and\nelectricity led to the revival of the same problem under another\nform. As well known, there was evolved from these discoveries,\nthe theory that light and electrical energy are of the same nature,\nand are in each case manifestations of wave-disturbances propagated\nthrough a hypothetical medium, the ether, with a velocity of 186,000\nmiles per second.\n\nThe problem therefore arose as to whether the earth and all stellar\nbodies move through this ether. In that case it ought to be possible\nto measure the velocity of the earth with respect to this medium, and\nunder these conditions we could speak, in a sense, of absolute motion.\n\nA large number of experiments has been tried with this end in view. The\nmost famous of these, and the one which stimulated the subsequent\ndevelopment of the theory of relativity, was that carried out by\nMichelson and Morley in 1887. To understand the significance of this\nexperiment we shall refer briefly to an analogous observation which\nis quite familiar.\n\nDoes it take longer to swim to a point 1 mile up a stream and back\nor to a point 1 mile across stream and back? The experienced swimmer\nwill answer that the up-and-down journey takes longer. If we assume\nthat the swimmer has a speed of 5 miles an hour in still water and\nthat the current is 3 miles an hour, we find that, while it requires\nfive-eighths hour to make the up-and-down journey, it takes only\none-half hour for the trip across stream and back. The ratio between\nthe times required for the two journeys is thus five-fourths, and if\nthis is written in the form\n\n    $$\\frac{1}{\\sqrt{1 - (\\frac{3}{5})^2}}$$\n\nit shows how the result depends upon the square of the ratio of the\nspeeds of the swimmer and the current.\n\nNow the earth is moving in its orbit about the sun with a velocity\nof 18 miles per second. If the earth moves through the ether and a\nlight-beam passes from one mirror to another and back again, the time\ntaken for this journey ought to be longer when the light-path is in\nthe direction of the earth's motion than when it is at right angles\nto this direction. For we can consider the light as a swimmer having\na speed of 186,000 miles per second and travelling in a stream whose\ncurrent is 18 miles per second.\n\nWhen Michelson and Morley tried the experiment they could not observe\nany difference in the velocity of light in the two directions. The\nexperiment has since been repeated under various conditions, but\nalways with negative results.\n\nEinstein's contribution to science consists in interpreting this\nresult as being in accord with Newton's ideas on mechanical relativity\nin that it demonstrates the impossibility of measuring absolute\nmotion, not only by mechanical, but also by optical or electrical\nexperiments. Consequently the velocity of light must be regarded as\nconstant and independent of the motion of either source or observer.\n\n\n\n\nTHE RELATIVITY OF UNIFORM MOTION\n\nLet us consider some of the consequences which follow from this\nprinciple. An observer travelling with say one-half the velocity\nof light in the same direction as a ray of light would find that the\nlatter has the usual velocity of 186,000 miles per second. Similarly an\nobserver travelling in the opposite direction to that of the light-ray,\nwith one-half the velocity of light, would obtain the same result.\n\nEinstein has shown that these conclusions can be valid only if the\nunits of time and space used by the two observers depend upon their\nrelative motions. A careful calculation shows that the unit of length\nused by either observer appears to the other observer contracted\nwhen placed in the direction of their relative motion (but not,\nwhen placed at right angles to this direction), and the unit of time\nused by either observer appears to the other too great. Moreover, the\nratio of the units of length or of time varies with the square of the\nrelative speed of the two observers, according to a relation which is\nsimilar to that mentioned above for the swimmer in the current. This\nrelation shows that as the relative speed approaches that of light\nthe discrepancy between the units increases.\n\nThus, for an observer moving past our earth with a velocity which is\nnine-tenths that of light, a meter stick on the earth would be 44\ncentimeters as measured by him, while a second on our clocks would\nbe about two and a half seconds as marked by his clock. Similarly,\nwhat he calls a meter length would, for us, be only 44 centimeters\nand he would appear to us to be living about two and a half times\nslower than we are. Each observer is perfectly consistent in his\nmeasurements of time and space as long as he confines his observations\nto his own system, but when he tries to make observations on another\nsystem moving past his, he finds that the results which he obtains\ndo not agree with those obtained by the other observer.\n\nIt is not surprising that in accordance with this conclusion it also\nfollows that the mass of a body must increase with its velocity. For\nlow velocities the increase is so small that we cannot ever hope to\nmeasure it, but as the velocity of light is approached the difference\nbecomes more and more appreciable and a body having the velocity of\nlight would possess infinite mass, which simply means that such a\nvelocity cannot be attained by any material object. This conclusion\nhas been experimentally confirmed by observations on the mass of\nthe extremely small negatively charged particles which are emitted\nby radioactive elements. Some of these particles are ejected with\nvelocities which are over nine-tenths that of light, and measurements\nshow that the increase in mass is in accord with this theory.\n\nThe relativity theory also throws new light on the nature of mass\nitself. According to this view, mass and energy are equivalent. The\nabsolute destruction of 1 gram of any substance, if possible, would\nyield an amount of energy which is one hundred million times as much\nas that obtained by burning the same mass of coal. Conversely, energy\nchanges are accompanied by changes in mass. The latter are ordinarily\nso inappreciably small as to escape our most refined methods of\nmeasurements, but in the case of the radioactive elements we actually\nobserve this phenomenon. From this standpoint, also, the laws of\nconservation of energy and of mass are shown to be intimately related.\n\n\n\n\nUNIVERSAL RELATIVITY\n\nSo far we have dealt with what has been designated as the special\ntheory of relativity. This, as we have seen, applies to uniform motion\nonly. In extending the theory to include non-uniform or accelerated\nmotion, Einstein has at the same time deduced a law of gravitation\nwhich is much more general than that of Newton.\n\nA body falling towards the earth increases in velocity as it falls. The\nmotion is said to be accelerated. We ascribe this increase in velocity\nto a gravitational force exerted by the earth on all objects. As\nshown by Newton, this force acts between all particles of matter in\nthe universe, and varies inversely as the square of the distance,\nand directly as the product of the masses.\n\nOf course, we have had a number of theories of gravitation, and\nnone of them have proven successful. Einstein, however, was the\nfirst one to suggest a conception of gravitation which has proven\nextremely significant. He points out that a gravitational force\nis non-existent for a person falling freely with the acceleration\ndue to gravity. For this person there is no sensation of weight,\nand if he were in a closed box which is also falling with the same\nacceleration, he would be unable to decide as to whether his system\nwere falling or situated in interplanetary space where there is\nno gravitational field. Furthermore, if he were to carry out any\noptical or electrical experiments in this box he would observe the\nsame results as an experimenter on the earth. A ray of light would\ntravel in a straight line so far as this observer can perceive,\nwhile an external observer would, of course, judge differently.\n\nEinstein shows that this is equally true for all kinds of acceleration\nincluding that due to rotation. In the case of a rotating body there\nexists a centrifugal force which tends to make objects on the surface\nfly outwards, but for an external observer this force does not exist\nany more than gravity exists for the observer falling freely.\n\nThus we can draw the general conclusion that a gravitational field or\nany other field of force may be eliminated by choosing an observer\nmoving with the proper acceleration. For this observer, however,\nthe laws of optics and electricity must be just as valid as for an\nobserver on the earth.\n\nIn postulating this equivalence hypothesis Einstein merely makes use\nof the very familiar observation that, independently of the nature\nof the material, all bodies possess the same acceleration in a given\nfield of force.\n\nThe problem which Einstein now sets out to solve is that of determining\nthe law which shall describe the motion of any system in a field of\nforce in such a general manner as to leave unaltered the fundamental\nrelations of electricity and optics.\n\nIn connection with the solution of this problem he finds it necessary\nto discard the limitations placed on us by ordinary or Euclidean\ngeometry. In this manner geometrical concepts as well as those of\nforce are completely robbed of all notions of absoluteness, and the\ngoal of a general theory of relativity is attained.\n\n\n\n\nTHE GEOMETRY OF GRAVITATION\n\nLet us consider a circular disc rotating with a uniform peripheral\nspeed. According to the deductions from the \"special theory\" of\nrelativity, an observer situated near the edge of this disc, but not\nrotating with it, will observe that units of length measured along\nthe circumference of the disc are contracted. On the other hand,\nmeasurements along the diameter, which is at right angles to the\ndirection of motion of the circumference, will show no contraction\nwhatever, and, consequently the observer will find that the ratio\nof circumference to diameter has not the well known value 3.14159\n... but exceeds this value, the difference being greater and greater\nas the peripheral speed approaches that of light. That is, the laws\nof ordinary geometry no longer hold true.\n\nHowever, we know other cases in which the ordinary or Euclidean\ngeometry is not applicable. Thus suppose that on the surface of a\nsphere we describe a series of concentric circles. Since the surface\nis curved, we are not surprised at finding that the circumference of\nany one of these circles is less than 3.14159 ... times the distance\nacross the circle as measured on the surface of the sphere. What\nthis means, therefore, is that we cannot use Euclidean geometry to\ndescribe measurements on the surface of a sphere, and every schoolboy\nknows this from comparing Mercator's projection of the earth's surface\nwith the actual representation on a globe.\n\nWhen we come to think of it, the reason we realize all this is\nbecause our sense of three dimensions enables us to differentiate\nflat surfaces from those that are curved. Let us, however, imagine\na two-dimensional being living on the surface of a large sphere. So\nlong as his measurements are confined to relatively small areas he\nwill find it possible to describe all his measurements in terms of\nEuclidean geometry. As, however, his area of operation increases\nhe will begin to observe greater and greater discrepancies. Being\nunfamiliar with the existence of such a three-dimensional object as\na sphere, and therefore not realizing that he is on the surface of\none, our intelligent two-dimensional being will conclude that the\ndisturbance in his geometry is due to the action of a force, and by\nmeans of plausible assumptions on the \"law\" of this force he will\nreconcile his observations with the laws of plane geometry.\n\nNow since an acceleration in a gravitational field is identical with\nthat due to centrifugal force produced by rotation, we concluded that\nthe geometry in a gravitational field must also be non-Euclidean. That\nis, space in the neighborhood of matter is distorted or curved. The\ncurvature of space bears the same relation to three dimensions that\nthe curvature of a spherical surface bears to two dimensions, and\nthat is why we do not perceive it, any more than the intelligent\ntwo-dimensional being would be aware of the distortion of his space\n(or surface). Furthermore, like this being, we have assumed the\nexistence of a gravitational force to account for discrepancies in\nour geometrical measurements.\n\nThe identification in this manner of gravitational effects with\ngeometrical curvature of space enables Einstein to derive a general\nlaw for the path of any particle in a gravitational field, with respect\nboth to space and to time. Furthermore, the law expresses this motion\nin terms which are independent of the relative motion and position\nof the observer, and satisfies the condition that the fundamental\nlaws of physics be equally valid for all observers. The solution\nof the problem involved the use of a new kind of higher calculus,\nelaborated by two Italian mathematicians, Ricci and Levi-Civita. The\nresult is a law of motion which is extremely general in its validity.\n\nFor low velocities it approximates to Newton's solution, and in the\nabsence of a gravitational field it leads to the same conclusions as\nthe special theory of relativity. There are three deductions from this\nlaw which have aroused a great deal of interest, and the confirmation\nof two of these by actual observation must be regarded as striking\nproof of Einstein's theory.\n\n\n\n\n\n\n\n\n\nXIII\n\nAN INTRODUCTION TO RELATIVITY\n\nA Treatment in Which the Mathematical Connections of Einstein's Work\nare Brought Out More Strongly and More Successfully Than Usual in a\nPopular Explanation\n\nBY HAROLD T. DAVIS, UNIVERSITY OF WISCONSIN, Madison, Wis.\n\n\nOne of the first questions which appears in philosophy is this: What\nis the great reality that underlies space and time and the phenomena\nof the physical universe? Kant, the philosopher, dismissed it as\na subjective problem, affirming that space and time are \"a priori\"\nconcepts beyond which we can say no more.\n\nThen the world came upon some startling facts. In 1905 a paper appeared\nby Professor Albert Einstein which asserted that the explanation of\ncertain remarkable discoveries in physics gave us a new conception\nof this strange four-dimensional manifold in which we live. Thus,\nthe great difference between the space and time of philosophy and the\nnew knowledge is the objective reality of the latter. It rests upon\nan amazing sequence of physical facts, and the generalized theory,\nwhich appeared several years later, founded as it is upon the abstruse\ndifferential calculus of Riemann, Christoffel, Ricci and Levi-Civita,\nemerges from its maze of formulas with the prediction of real phenomena\nto be sought for the in the world of facts.\n\nWe shall, therefore, approach the subject from this objective point\nof view. Let us go to the realm of actual physical events and see how\nthe ideas of relativity gradually unfolded themselves from the first\ncrude wonderings of science to the stately researches that first\ndiscovered the great ocean of ether and then penetrated in such a\nmarvelous manner into some of its most mysterious properties.\n\n\n\n\nTHE ELECTROMAGNETIC THEORY OF LIGHT\n\nSuppose that we go out on a summer night and look into the dark depths\nof the sky. A thousand bright specks are flashing there, blue, red,\nyellow against the dark velvet of space. And as we look we must all\nbe impressed by the fact that such remote objects as the stars can be\nknown to us at all. How is it that light, that curious thing which\nfalls upon the optic nerve and transmits its pictures to the brain,\ncan ever reach us through the black regions of interstellar space? That\nis the question which has for its answer the electromagnetic theory\nof light.\n\nThe first theory to be advanced was Newton's \"corpuscular\" theory\nwhich supposed that the stars are sending off into space little pellets\nof matter so infinitesimally small that they can move at the rate of\n186,000 miles a second without injuring even so delicate a thing as\nthe eye when they strike against it.\n\nBut in 1801, when Thomas Young made the very important discovery of\ninterference, this had to give way to the wave theory, first proposed\nby Huyghens in the 17th century. The first great deduction from this,\nof course, was the \"luminiferous ether,\" because a wave without some\nmedium for its propagation was quite unthinkable. Certain peculiar\nproperties of the ether were at once evident, since we deduce that\nit must fill all space and at the same time be so extremely tenuous\nthat it will not retard to any noticeable degree the motion through\nit of material bodies like the planets.\n\nBut how light was propagated through the ether still remained a\nperplexing problem and various theories were proposed, most prominent\namong them being the \"elastic solid\" theory which tried to ascribe to\nether the properties of an elastic body. This theory, however, laid\nitself open to serious objection on the ground that no longitudinal\nwaves had been detected in the ether, so that it began to appear\nthat further insight into the nature of light had to be sought for\nin another direction.\n\nThis was soon forthcoming for in 1864 a new theory was\nproposed by James Clerk Maxwell which seemed to solve all of the\ndifficulties. Maxwell had been working with the facts derived from\na study of electrical and magnetic phenomena and had shown that\nelectromagnetic disturbances were propagated through the ether at a\nvelocity identical with that of light. This, of course, might have been\nmerely a strange coincidence, but Maxwell went further and demonstrated\nthe interesting fact that an oscillating electric charge should give\nrise to a wave that would behave in a manner identical with all of the\nknown properties of a light wave. One particularly impressive assertion\nwas that these waves, consisting of an alternating electric field\naccompanied by an alternating magnetic field at right angles to it,\nand hence called electromagnetic waves, would advance in a direction\nperpendicular to the alternating fields. This satisfied the first\nessential property of light rays, i.e., that they must be transverse\nwaves, and the ease with which it explained all of the fundamental\nphenomena of optics and predicted a most striking interrelation\nbetween the electrical and optical properties of material bodies,\ngave it at once a prominent place among the various theories.\n\nThe electromagnetic theory, however, had to wait until 1888 for\nverification when Heinrich Hertz, in a series of brilliant experiments,\nsucceeded in producing electromagnetic waves in the laboratory and\nin showing that they possessed all of the properties predicted by\nMaxwell. These waves moved with the velocity of light: they could be\nreflected, refracted, and polarized: they exhibited the phenomenon\nof interference and, in short, could not be distinguished from light\nwaves except for their difference in wave length.\n\n\n\n\nTHE MICHELSON-MORLEY EXPERIMENT\n\nWith the final establishment of the electromagnetic theory of light as\na fact of physics, we have at last endowed the ether with an actual\nsubstantiality. The \"empty void\" is no longer empty, but a great\nocean of ether through which the planets and the suns turn without\never being aware that it is there.\n\nIn 1881 A. A. Michelson undertook an experiment, originally suggested\nby Maxwell, to determine the relative motion of our earth to the\nether ocean and six years later he repeated it with the assistance\nof E. W. Morley. The experiment is now known as the Michelson-Morley\nexperiment and since it is the great physical fact upon which the\ntheory of relativity rests, it will be well for us to examine it\nin detail.\n\nSince we can scarcely think that our earth is privileged in the\nuniverse and that it is at rest with respect to this great ether ocean\nthat fills space, we propose to discover how fast we are actually\nmoving. But the startling fact is that the experiment devised for this\npurpose failed to detect any motion whatever of the earth relative\nto the ether. [3]\n\nThe explanation of this very curious fact was given by both\nH. A. Lorentz and G. F. Fitzgerald in what is now widely known under\nthe name of the \"contraction hypothesis.\" It is nothing more nor less\nthan this:\n\nEvery solid body undergoes a slight change in dimensions, of the order\nof ($v^2/c^2$), when it moves with a velocity $v$ through the ether.\n\nThe reason why the experiment failed, then, was not because the earth\nwas not moving through the ether, but because the instruments with\nwhich the experiment was being conducted had shrunk just enough to\nnegative the effect that was being looked for. [4]\n\n\n\n\nTHE LORENTZ TRANSFORMATION\n\nWe can not at this point forebear introducing a little mathematics\nto further emphasize the theory and the very logical nature of this\ncontraction hypothesis.\n\nLet us suppose that we were on a world that was absolutely motionless\nwith respect to the ether and were looking at a ray of light. The\nmagnetic and electric fields which form the ray can be described by\nmeans of four mathematical expressions which have come to bear the\nname of \"Maxwell's field equations.\" Now suppose that we ask ourselves\nthe question: How must these equations be changed so that they will\napply to a ray of light which is being observed by people on a world\nthat is moving with a velocity v through the ether?\n\nThe answer is immediate. From the Michelson-Morley experiment we know\nthat we can not tell how fast or how slowly we are moving with respect\nto the ether. This means that no matter what world we may be upon,\nthe form of the Maxwell field equations will always be the same,\neven though the second set of axes (or frame of reference) may be\nmoving with high velocity with respect to the first.\n\nStarting from this hypothesis (called in technical language the\ncovariance of the equations with respect to a transformation of\ncoordinates), Lorentz found that the transformation which leaves the\nfield equations unchanged in form was the following:\n\n    $$x' = k(x - vt), y' = y, z' = z, t' = k(t - vx/c)$$\n\nwhere $k$ is as on page 92.\n\nAnd what, now, can be deduced from these very simple looking\nequations? In the first place we see that the space of $x'$, $y'$,\n$z'$, $t'$ is not our ordinary concept of space at all, but a space in\nwhich time is all tangled up with length. To put it more concretely,\nwe may deduce from them the interesting fact that whenever an aviator\nmoves with respect to our earth, his shape changes, and if he were to\ncompare his watch with one on the earth, he would find that his time\nhad changed also. A sphere would flatten into an ellipse, a meter\nstick would shorten up, a watch would slow down and all because, as\nH. Minkowski has shown us from these very equations, we are really\nliving in a physical world quite different from the world of Euclid's\ngeometry in which we are accustomed to think we live.\n\nA variety of objections has very naturally been made to this rather\nradical hypothesis in an attempt to discredit the entire theory, but\nit is easily seen that any result obtained through the field equations\nmust necessarily be in conformity with the theory of contraction, since\nthis theory is only the physical interpretation of that transformation\nwhich leaves the field equations unaltered. Indeed, it is even possible\nto postulate the Lorentz transformation together with the assumption\nthat each element of charge is a center of uniformly diverging tubes\nof strain and derive the Maxwell field equations from this, which\nshows from another point of view the truly fundamental nature of\nthe transformation.\n\n\n\n\nTHE FIRST THEORY OF RELATIVITY\n\nThe whole question of the ether had arrived at this very\ninteresting point when Professor Einstein in 1905 stated the theory\nof relativity. He had noticed that the equations of dynamics as\nformulated by Newton did not admit the Lorentz transformation, but\nonly the simple Galilean transformation:\n\n    $$x' = x - vt, y' = y, z' = z, t' = t\\,.$$\n\nHere, indeed, was a curious situation. Two physical principles,\nthat of dynamics and that of electromagnetism, were coexistent and\nyet each one admitted a different transformation when the system of\nreference was transferred to axes moving with constant velocity with\nrespect to the ether.\n\nNow the electromagnetic equations and their transformation had been\nshown to be in accord with experimental fact, whereas it had long\nbeen felt that Newton's equations were only a first approximation\nto the truth. For example, the elliptic orbit of a planet had been\nobserved by Leverrier to exhibit a disquieting tendency to rotate\nin the direction of motion. This precession, which in the case of\nMercury was as large as 43'' per century, could not be accounted\nfor in any way by the ordinary Newtonian laws and was, consequently,\na very celebrated case of discordance in gravitational astronomy.\n\nWith this example clearly before him, Einstein took the great step\nand said that the laws of dynamics and all other physical laws had to\nbe remade so that they, also, admit the Lorentz transformation. That\nis to say,\n\nThe laws of physical phenomena, or rather the mathematical expressions\nfor these laws, are covariant (unchanged in form) when we apply the\nLorentz transformation to them.\n\nThe deductions from the Michelson-Morley experiment now seem to have\nreached their ultimate conclusion.\n\nOne discordant fact in this new theory remained, however. That same\nprecession of the perihelion of Mercury which had first lead Einstein\nto his theory remained unsettled. When the new approximations were\napplied to the formula of orbital motion, a precession was, indeed,\nobtained, but the computed value fell considerably below that of the\nobserved 43'' per century.\n\n\n\n\nTHE INCLUSION OF GRAVITATION\n\nWith the idea of investigating the problem from the very bottom,\nEinstein now undertook a broader and more daring point of view. In\nthe first place he said that there is no apparent reason in the great\nscheme of world events why any one special system of coordinates should\nbe fundamental to the description of phenomena, just as in the special\ntheory a ray of light would appear the same whether viewed from a\nfixed system or a system moving with constant velocity with respect\nto the ether. This makes the very broad assumption that no matter\nwhat system of coordinates we may use, the mathematical expressions\nfor the laws of nature must be the same. In Einstein's own words,\nthen, the first principle of this more general theory of relativity\nmust be the following:\n\n\"The general laws of nature are expressed through equations which\nhold for all systems of coordinates, that is, they are covariant with\nrespect to arbitrary substitutions.\" [5]\n\nBut this was not enough to include gravitation so Einstein\nnext formulated what he was pleased to call his \"equivalence\nhypothesis.\" This is best illustrated by an example. Suppose that\nwe are mounting in an elevator and wish to investigate the world\nof events from our moving platform. We mount more and more rapidly,\nthat is with constant acceleration, and we appear to be in a strong\ngravitational field due to our own inertia. Suppose, on the other\nhand, that the elevator descends with an acceleration equal to that\nof gravity. We would now feel certain that we were in empty space\nbecause our own relative acceleration has entirely destroyed that of\nthe earth's gravitational field and all objects placed upon scales\nin an elevator would apparently be without weight.\n\nApplying this idea, then, Einstein decided to do away with gravitation\nentirely by referring all events in a gravitational field to a new set\nof axes which should move with constant acceleration with respect to\nthe first. In other words we are going to deal with a system moving\nwith uniform acceleration with respect to the ether, just as we\nconsidered a system moving with uniform velocity in the special theory.\n\nThe next step in the construction of this complicated theory is to\nreduce these two hypotheses to the language of mathematics and this\nwas accomplished by Einstein with the help of M. Grossmann by means\nof the theory of tensors.\n\nOn account of the very great intricacy of the details, we must\ncontent ourselves with the mere statement that this really involved\nthe generalization of the famous expressions known as Laplace's\nand Poisson's equations, on the explicit assumption that these\ntwo equations would still describe the gravitational field when we\nare content to use a first approximation to the truth. The set of\nten differential equations which Einstein got as a result of his\ngeneralization he called his field equations of gravitation. [6]\n\n\n\n\n\n\n\n\n\nXIV\n\nNEW CONCEPTS FOR OLD\n\nWhat the World Looks Like After Einstein Has Had His Way With It\n\nBY JOHN G. McHARDY, COMMANDER R.N.,\nLONDON\n\n                   \"The new-created world, which fame in heaven\n                    Long had foretold, a fabric wonderful,\n                    Of absolute perfection.\"\n\n\nEinstein's Theory of Relativity has led to determining a key law\nof nature--the law of gravitation--which is also the basic law of\nmechanics. Thus it embraces a whole realm of physics, and promises,\nthrough the researches of Professor Weyl, to embrace another\nrealm--electro-dynamics. Its limitations are not yet reached, for\nEinstein has already postulated therefrom a theory of a finite, yet\nunbounded, universe. This essay, however, is mainly concerned with\nmechanics, and electrical forces are not considered.\n\nTo have synthesised Newton's two great principles--his law of motion\nand law of gravitation--interpreting in the process the empirical law\nof equality of gravitational and inertial mass, is alone an immense\nachievement; but Einstein's researches have opened up a new world\nto the physicist and philosopher which is of greater importance. He\nhas given us a vision of the immaterial world, a geometrical or\nmathematical vision, which is more satisfying than the \"ether\"\nconceptions hitherto presented. The fabric of his vision is not\nbaseless. It is this fabric we shall consider, touching on certain\naspects of the Einstein theory in the endeavor to present an image in\nminiature of his edifice of thought and to show the firmness of its\nfoundations. That they are well and truly laid was demonstrated by\nthe verification, from observations made during the solar eclipse in\n1919, of Einstein's prediction of the displacement of a wave of light\nin a gravitational field, showing light to have the property of weight.\n\nThe physical world is shown by Einstein to be a world of\n\"relations.\" Underlying it there is an absolute world of which physical\nphenomena are the manifestation. \"Give me matter and motion,\" says\nDescartes, \"and I will construct the world.\" \"Give me a world in\nwhich there are ordered relations,\" says the Relativist, \"and I will\nshow you the behavior of matter therein\" (mechanics). We first view\nthis underlying world as an abstraction, abstracting energy (\"bound\"\nas in matter and electrons, \"free\" as in light), and its attribute\nforce. This abstraction we will call the \"World-Frame.\" Later, we\nwill study the underlying world in connection with energy, and will\ncall this absolute world the \"World-Fabric.\" The connection between\nthe geometrical character of the World-Frame and the geometrical\ncharacters of the World-Fabric is the key to the law of gravitation.\n\n\n\n\nTHE WORLD-FRAME\n\nThis is our conception of a world, if such were possible, entirely free\nfrom the influence of energy. We may conceive of it as an amorphous\nimmaterial something containing \"point-events\" (a point-event\nbeing an instant of time at a point in space--a conception, not a\ndefinition). These point-events have a fourfold order and definite\nrelation in this Frame, i.e. they can be specified by four variables\nor coordinates in reference to some base called a reference system,\nwith respect to which they are forward or backward, right or left,\nabove or below, sooner or later. This shows the World-Frame to be\nfour-dimensional. Thus an aggregate of point-events (or an \"event,\"\nwhich implies limited extension in space and limited duration\nin time) [7] would have what we familiarly describe as length,\nbreadth, height and time. To express these metrical properties most\nsimply we must choose a four-dimensional reference system having\na particular form--rectilinear axes (Cartesian coordinates), and a\nparticular motion--uniform and rectilinear, i.e. unaccelerated, and\nnon-rotating with respect to the path of a light ray. We call this\nan inertial system because Newton's Law of Inertia holds for such\na system alone. This system indicates how observers partition the\nWorld-Frame into space and time. It restricts observers to uniform\nrectilinear motion, and observations to bodies and light-pulses in\nsuch motion. Thus gravitational and other forces are discounted,\nand we obtain World-Frame conditions notwithstanding the fact that\nobservers are in the presence of energy.\n\nNow the separation between point-events which have a definite relation\nto each other must be absolute. The separation between two points in\na plane is defined by the unique distance between them (the straight\nline joining them). Between point-events the analogue of this unique\ndistance, which we call the \"separation-interval\" (to indicate its\ntime-like and space-like nature), is also unique. Its unique and\nabsolute character give it great importance as thereby it is the same\nfor all observers regardless of their reference system.\n\nIf, in place of the rather cumbersome expression $X-x$ to indicate\nthe difference between the $x$-coordinates of two points, we employ\nthe more compact expression $dx$; if for the benefit of readers who\nhave a little algebra but no analysis we state explicitly that this\nexpression is a single symbol for a single quantity, and has nothing\nto do with any product of two quantities $d$ and $x$; and if we\nextend this notation to all our coordinates: then it is clear from\nprevious essays that the distance $S$ between two points in a plane\nreferred to a rectilinear system $OX$, $OY$, is given by the simple\nequation $S^2 = (dx)^2 + (dy)^2$. Einstein and Minkowski show that\nthe value for the separation interval $\\Omega$, the analogue of $S$,\nreferred to an inertial system is given by the equation\n\n    $$\\Omega^2 = (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2\\,,$$\n\nwhich is seen to be a modified extension to four dimensions of the\nequation for $S$. We must measure $t$ in the same units as $x$, $y$,\n$z$. By taking the constant velocity of light (300,000 kilometres\nper second) as unit velocity, we can measure in length or time\nindiscriminately. [8]\n\nWe will analyse briefly this equation as it epitomizes the Special\nTheory of Relativity. If the World-Frame had been Euclidean the\nequation would have been\n\n    $$\\Omega^2 = (dx)^2 + (dy)^2 + (dz)^2 + (dt)^2$$\n\nbut this would not satisfy the \"transformation equations\" which\nresulted from the Special Theory. These transformation equations\narose directly from a reconciliation between two observed facts; (a)\nthe observed agreement of all natural phenomena with the \"Restricted\nPrinciple of Relativity\"--a principle which shows that absolute\nrectilinear motion cannot be established--(as regards mechanics this\nwas recognized by Newton; the Michelson-Morley and other experiments\nshowed this principle also applied to optical and electro-dynamical\nphenomena); and (b) the observed disagreement of optical and\nelectro-dynamical phenomena (notably the constancy of light velocity)\nwith the laws of dynamics as given by classical mechanics, e.g., in\nregard to the compounding of relative velocities. Einstein effected\nthis reconciliation by detecting a flaw in classical mechanics. He\nshowed that by regarding space and time measurements as relative\nto the observer--not absolute as Newton defined them--there was\nnothing incompatible between the Principle of Relativity and the\nlaws of dynamics so modified. Newton's definitions were founded on\nconception. Einstein's recognition of the relativity of space and\ntime is based on observation.\n\nEquation (1) shows that the geometry of the World-Frame referred to\nan inertial system is semi-Euclidean (hyperbolic), and that space and\ntime measurements are relative to the observer's inertial reference\nsystem. The equation shows that the World-Frame has a certain\ngeometrical character which we distinguish as four-dimensional\n\"flatness.\" It is everywhere alike (homaloidal). Its flat character\nis shown by the straight line nature of the separation-interval and\nof the system to which it is most simply referred.\n\nThus we have found two absolute features in the World-Frame--(1) Its\ngeometrical character--\"flatness\"; (2) The separation-interval--which\ncan be expressed in terms of measurable variables called space and time\npartitions, this partitioning being dependent on the observer's motion.\n\nWe are now in a position to explore the World-Fabric. Already we see\nthat, studied under inertial conditions (free of force), it agrees\nwith the World-Frame.\n\n\n\n\nTHE WORLD-FABRIC\n\nThe General Theory of relativity is largely concerned with the\ninvestigation of the World-Fabric. Consider the World-Frame to be\ndisturbed. We may regard this disturbance, which manifests itself in\nphysical phenomena, as energy, or more correctly \"action.\"\n\nWhen energy is thwarted in its natural flow, force is manifested,\nwith which are associated non-uniform motions such as accelerations\nand rotations. This disturbed World-Frame we distinguish as the\nWorld-Fabric. It is found to have various non-Euclidean characters\ndiffering from the simple \"flat\" character of the World-Frame according\nto the degree of disturbance (action) in the region. Disturbance\ngives the fabric a geometrical character of \"curvature\"; the more\nconsiderable the disturbance, the greater the curvature. Thus an\nempty region (not containing energy, but under its influence) has\nless curvature than a region in which free energy abounds.\n\nOur problem, after showing the relativity of force (especially\ngravitational force), is to determine the law underlying the fabric's\ngeometrical character; to ascertain how the degree of curvature is\nrelated to the energy influencing a region, and how the curvature of\none region is linked by differential equations to that of neighboring\nregions. Such a law will be seen to be the law of gravitation.\n\nWe study the World-Fabric by considering tracks on which material\nparticles and light-pulses progress; we find such tracks regulated\nand defined by the Fabric's curvature, and not, as hitherto supposed,\nby attractive force inherent in matter. As a track is measurable by\nsumming the separation-intervals between near-by point-events on it,\nall observers will agree which is the unique track between two distant\npoint-events. Einstein postulates that freely progressing bodies\nwill follow unique tracks, which are therefore called natural tracks\n(geodesics).\n\nIf material bodies are prevented from following natural tracks by\ncontact with matter or other causes, the phenomenon of gravitational\nforce is manifested relative to them. Whenever the natural flow\nof energy is interrupted force is born. For example, when the\npiston interrupts the flow of steam, or golf ball flow of club,\nforce results--the interruption is mutual, and the force relative\nto both. Likewise when the earth interrupts the natural track of a\nparticle (or observer) gravitational force is manifested relative\nto both.\n\nSo long as a body moves freely no force is appreciated by it. A\nfalling aviator (neglecting air resistance) will not appreciate any\ngravitational force. He follows a natural track, thereby freeing\nhimself from the force experienced in contact with matter. He\nacquires an accelerating motion with respect to an inertial system. By\nacquiring a particular accelerating motion an observer can annul any\nforce experienced in any small region where the field of force can\nbe considered constant.\n\nThus Einstein, interpreting the equality of gravitational and inertial\nmass, showed that the same quality manifests itself according to\ncircumstances as \"weight\" or as inertia, and that all force is purely\nrelative and may be treated as one phenomenon (an interruption in\nenergy flow). This \"Principle of Equivalence\" shows that small portions\nof the World-Fabric, observed from a freely moving particle (free of\nforce), could be treated as small portions of the World-Frame. [9]\n\nIf such observations were practicable, we could determine the Fabric\ncurvature by referring point-event measurements to equation (1). We\ncannot observe from unique tracks but we can observe them from our\nrestrained situation. Their importance is now apparent, because, by\ntracing them over a region, we are tracing something absolute in the\nFabric--its geometrical character. We study this curvature by exploring\nseparation-intervals on the tracks of freely moving bodies, relating\nthese separation-intervals to actual measurements in terms of space\nand time components depending on the observer's reference system. The\nlaw of curvature must be the law of gravitation. To illustrate the\nlines on which Einstein proceeded to survey the World-Fabric from\nthe earth we will consider a similar but more simple problem--the\nsurvey of the sea-surface curvature from an airship. We study this\ncurvature by exploring small distances on the tracks of ships (which we\nmust suppose can only move uniformly on unique tracks--arcs of great\ncircles), relating such distances to actual measurements in terms of\nlength and breadth components depending on the observer's reference\nsystem. This two-dimensional surface problem can be extended to the\nfour-dimensional Fabric one.\n\nWe consider the surface to be covered by two arbitrarily drawn\nintersecting series of curves: curves in one series not intersecting\neach other, vide figure. This Gaussian system of coordinates is\nappropriate only when the smaller the surface considered, the\nmore nearly it approximates to Euclidean conditions. It admits of\ndefining any point on the surface by two numbers indicating the curves\nintersecting at that point. $P$ is defined by $x_1$, $x_2$. $P_1$\n(very near $P$) is defined by $x_1 + dx_1$, $x_2 + dx_2$. The equation\nfor the minute distance $s$ between two adjacent points in such a\nsystem is given by the general formula\n\n    $$s^2 = g_{11}d{x_1}^2 + g_{12}dx_1dx_2 + g_{22}d{x_2}^2\\,.$$\n\nThe $g$'s may be constants or functions of $x_1$, $x_2$. Their\nvalue is dependent on the observer's reference system and on the\ngeometrical character of the surface observed. The curves being\narbitrary, the formula is appropriate for any reference system,\nor even if the observer does not know exactly what his reference\nsystem is. (The Fabric observer does not know what his space and time\npartitioning actually is because he is in a gravitational field). It\nis the $g$'s which disclose the geometry of an observer's partitions,\nand their values also contain a reflection of the character of the\nregion observed.\n\nWe find $s$ by direct exploration with a moving ship ($\\Omega$ is found\nby direct exploration with a freely moving particle); $dx_1$, $dx_2$\nare the observed length and breadth measurement differences which we\nhave to relate to $s$. By making sufficient observations in a small\narea and referring them to the general formula we can find the values\nof the $g$'s for the observer's particular reference system. Different\nvalues for $g$'s will be found if the observer changes his reference\nsystem, but there is a limitation to the values so obtainable owing to\nthe part played by the surface itself, which is diffidently expressing\nits intrinsic geometrical character in the $g$'s in each observation.\n\n\n\n\nEINSTEIN'S RESULTS\n\nThus we approach the absolute character of the surface through\nthe relative nature of the observer's reference system. There is a\nrelationship common to all values of the $g$'s that belong to the\nsame curvature. This relationship is expressed by a differential\nequation. It is this equation of curvature that the airship's observer\nmust find. Einstein's problem was similar, but he was concerned with\nfour dimensions, which entailed a general formula with ten $g$'s,\nand he had to find a set of differential equations of the second\norder to determine the law of Fabric curvature. He divided the Fabric\ninto regions: I. World-Frame--beyond influence of energy. II. Empty\nregion--free of energy, but under its influence. III. Region containing\nfree energy only. Each region has a characteristic curvature. By\nmeans of an absolute differential calculus--a wonderful mathematical\nscaffolding erected by Riemann, Christoffel and others--involving the\ntheory of tensors, he succeeded in finding such a set of equations. He\nkept the following points in view: (1) The equations must not only\ngive the character of region II, but must satisfy the special case\nof region I; (2) They must be independent of any partitioning system,\nbecause the General Theory of Relativity demands that a law of nature\nbe in a form appropriate for all observers whatever their position\nand motion; (3) They must be concerned with energy which is conserved,\nnot mass which the Special Theory showed dependent on velocity. This\nset of differential equations which shows how the curvature of the\nFabric at any point links to the curvature at neighboring points is\nthe law of gravitation, a law which has been severely tested by the\npractical observation of the solar eclipse already referred to. At a\nfirst approximation these equations degenerate into Newton's Law. At\na second approximation they account for the motion of the perihelion\nof Mercury, which had hitherto baffled astronomers. All the laws\nof mechanics are deducible from this law of World-Fabric curvature,\ni.e. conservation of energy (which includes conservation of mass since\nwe re-define mass as energy) and conservation of momentum (re-defined\nby a relativist). It must be noted that this law and the General\nTheory show that the velocity of light is not absolutely constant,\nbut, like everything else, a light-pulse is affected by the Fabric\ncurvature in a gravitational field. In conclusion we will contrast\nsome conspicuous differences in the old world view of classical\nmechanics and the new view presented by Einstein.\n\n1. A three-dimensional ether medium with variously conceived properties\nwhich communicated the supposed inherent attractive force in matter\nin some unexplained way, and transmitted electromagnetic waves,\nhas been replaced by a four-dimensional external World-Fabric, the\ngeometrical character of which controls the motion of matter (energy)\nand accounts for all mechanical laws.\n\n2. After separating the observer's subjective share in definitions\nfrom nature's share in the things defined, space, time, and force,\nhitherto regarded as absolute, have been shown to be purely relative\nand dependent on the observer's track. Mass has also proved to be\nrelative to velocity unless re-defined as energy. As classical\nmechanics bases all definitions on space, time, and mass units,\nthe relativity of such defined quantities is now apparent.\n\n3. Newton's laws of motion, his law of gravitation, and the laws of\nconservation, hitherto regarded as unrelated, are now synthesised in\na basic law of mechanics.\n\nEinstein has not disturbed the electric theory of matter, and both the\nold and new physics have in common the \"Principle of Least Action.\" We\nobtain a glimpse of this principle in the unique tracks pursued by\nfreely moving bodies, which may be regarded as tracks of least effort,\nforce only being manifested as an expression of the Fabric's resentment\nwhen bodies depart from these natural tracks. Einstein has approached\nnearer to the truth in regard to the laws underlying nature, and,\nas always, this means a simplification. His theory, which entails\na readjustment of such fundamental conceptions as space and time,\nopens up fresh fields to scientific investigation and to philosophic\nthought. It reveals a bridge uniting the domains of physics and\nphilosophy, and it heralds a new era in the history of science.\n\n\n\n\n\n\n\n\n\nXV\n\nTHE NEW WORLD\n\nA Universe in Which Geometry Takes the Place of Physics, and Curvature\nThat of Force\n\nBY GEORGE FREDERICK HEMENS, M.C., B.SC., LONDON\n\n\nIt is familiar knowledge that the line, the surface and ordinary\nEuclidean space are to be regarded as spaces of one, two and three\ndimensions respectively and readers of this journal are aware\nthat a hypothetical space of four dimensions has been closely\ninvestigated. The most convenient space to study is the surface or\ntwo-space, since we can regard it as embedded in a three-space. If a\nsurface is curved it is generally impossible to draw a straight line\non it, for as we see clearly, the \"straightest\" line is changing its\ndirection at every point. To describe this property accurately it\nis necessary to ascribe to each point a magnitude which expresses\nwhat happens to the direction of a short line in the region when\ndisplaced a short distance parallel to itself. This is called the\ndirection-defining magnitude. Different sets of values of this\nmagnitude relate to surfaces of different curvatures.\n\nA second fundamental property has recently been pointed out. There\nis inherent in every part of a space a measure of length peculiar\nto that particular region and which in general varies from region\nto region. To describe this variation accurately it is necessary to\nascribe to each point another magnitude called the length-defining\nmagnitude, which expresses the change from each point to the next of\nthe unit of length. These two magnitudes define the surface completely.\n\nSimilarly, a space of any number of dimensions is defined completely\nby a similar pair of magnitudes. A space is the \"field\" of such\na magnitude-pair and the nature of these magnitudes defines the\ndimensions of the space. The four-space usually described is the\nEuclidean member of an infinity of four-spaces.\n\nWhen we look into a mirror we see a space differing from ordinary\nspace in that right and left are interchanged and this is described\nmathematically by saying that if we locate points as usual by\nspecifying three distances $X_1$, $X_2$, $X_3$ of the point from three\nmutually perpendicular planes, then a point $X_1$, $X_2$, $X_3$, in\nactual space corresponds with a point $X_1$, $X_2$, $ -X_3$ in the\nmirrored space: in other words the mirrored space is derived from\nthe real space by multiplying the $X_3$ coordinates by $-1$. If we\nwere to multiply by $\\sqrt{-1}$ instead of $-1$ we should derive a\ndifferent space; in this case, however, we have no mirror to show us\nwhat it looks like. Such a space is said to have one negative dimension\nand it has the peculiar property that in the figure derived from the\nright triangle of ordinary space the square of the \"hypotenuse\" equals\nthe difference and not the sum of the squares of the other two sides,\nso that the length of a line may sometimes have to be represented by\nthe square-root of a negative number, a \"complex\" number.\n\nIn considering what at first sight may appear to be fantastic\nstatements made by this theory, it must be borne in mind that all our\nknowledge of the external universe comes through our sense-impressions,\nand our most confident statements about external things are really\nof the nature of inferences from these sense-impressions and, being\ninferences, liable to be wrong. So that if the theory says that a\nstone lying on the ground is not a simple three-dimensional object, and\nthat its substance is not the same as its substance a moment before,\nthe matter is one for due consideration and not immediate disbelief.\n\nThe idea that the universe extends in time as well as in space is not\nnew, and fiction-writers have familiarized us with wonderful machines\nin which travellers journey in time and are present at various stages\nof the world's history. This conception of the universe, to which the\nname \"space-time\" is usually applied, is adopted by the new theory\nand assigned the status of a physical reality.\n\n\n\n\nTHE WORLD GEOMETRY\n\nThe fundamental creed of the new theory is that the space-time universe\nconstitutes a true four-dimensional space of one negative dimension,\nthis dimension being time. The variations from point to point of\nthe direction-defining and length-defining magnitudes generate the\ngeometrical properties of curvature, etc., and these are cognised by\nthe human mind as physical phenomena: our sense-impressions are nothing\nmore nor less than perceptions of the geometry of a fourspace. So\ninstead of inferring from our sense-impressions the existence of\nmatter, motion and the like as we are accustomed to do, we should with\nequal justice infer the existence of a geometrical fourspace. Thus it\nbecomes necessary to prepare a dictionary in which the familiar things\nof our world are identified with those geometrical properties of the\nfour-space which really constitute them, and in so doing parts of\nour geometrical knowledge assume the guise of new physical knowledge.\n\nThrough the fourspace our consciousness travels, cognising a changing\nthree-dimensional section of it as it goes and thus giving rise\nto time. It becomes aware that the fourspace is pleated or folded\nalong lines all running roughly in the same direction, and possibly\nbecause this is the easiest direction to follow, it travels along the\nlines. The direction of this motion is the negative dimension. Thus\nconsciousness is always aware of the nearly constant forms of the\ncross-sections of the pleats along which it travels. These unvarying\nforms constitute matter: matter is the form of a section through\na uniform pleat of the fourspace--a three-dimensional aspect of a\nfour-dimensional curvature; so that in strict accuracy we should\nsay that a stone is the shape or form of a changing section of a\nfour-dimensional object, the complete object being a long fold in the\nfourspace. The physical interpretation of this conservation of form\nof the cross-section is that matter is conserved. It is thus seen that\nthe conscious mind, by following these pleats, has so determined time\nthat the law of the conservation of matter must hold. The mathematical\ntreatment of the subject makes it clear that practically all other\nphysical laws similarly follow as a direct result of this choice of\ntime. The type of order prevailing in the physical universe, the laws\nof gravitation, heat, motion and the rest are not directly imposed\nby some external power, but are apparently chosen by mind itself.\n\nIn the neighborhood of these pleats the fourspace is still curved,\nbut to a smaller degree. This we cognise as energy or as a field\nof force. Thus energy is seen to be the same kind of thing as\nmatter and would therefore be expected to have weight. This was\nexperimentally demonstrated in 1919 when light was in effect actually\nweighed. Conversely, matter consists of energy; and it is calculated\nthat one liter of water contains sufficient energy to develop a million\nhorsepower for about four years. It is now believed that the sun's\nenergy is derived from the disintegration of the matter of which it\nis made.\n\nThe method of establishing these identifications will be clear from the\nfollowing: We already knew that matter is made up of electrons and that\nradiant energy is electromagnetic and before the advent of this theory\nit was regarded as certain that practically all observed physical\nphenomena except gravitation were manifestations of the electromagnetic\nfield. The new theory has confirmed this belief. It is found that\nthe gravitational and electromagnetic conditions of the universe are\ncompletely defined if to each point of space-time a gravitational and\nan electric potential are ascribed. These are magnitudes of the same\nnature as the direction-defining and length-defining magnitudes which\nmust necessarily be associated with every point of space-time if it\nis a true \"space,\" and they are therefore identified with these. By\nperforming ordinary mathematical operations on these magnitudes\nstatements of fact clothed in mathematical form are obtained, which are\nto be interpreted on the one hand as physical laws and on the other\nas geometrical properties of the fourspace. Nearly all our physical\nlaws are derivable mathematically in this way, so that an extensive\nidentification is effected which has been fruitful of results.\n\nIt has been mentioned that a slight curvature is sometimes cognised\nas force and as this identification appeared originally as a postulate\nits history is interesting.\n\n\n\n\nTHE GENESIS OF THE THEORY\n\nAn experiment by Michelson and Morley (1887), on which the whole theory\nis based, made it appear that if a man measures the velocity at which\nlight passes him he will get the same result whether he is stationary,\nrushing to meet the light, or moving in the same direction as the\nlight. The solution was provided by Einstein in 1905. He suggested that\nsince we know the results of these determinations ought not to agree,\nsomething must have happened to the clocks and measuring-rods used in\nmeasuring the velocity so that the standards of length and time were\nnot the same in the three cases, the alterations being exactly such as\nto make the velocity of light constant. This solution is universally\naccepted as true and is the fundamental postulate. Thus the length of\na stick and the rate at which time passes will change as the velocity\nof the person observing these things changes. If a man measured the\nlength of an aeroplane going past him at 161,000 miles per second it\nwould measure only half the length observed when stationary. If the\naeroplane were going with the velocity of light, its length would\nvanish though its breadth and height would be unaltered. Similarly,\nif of two twin brothers one were continually moving with reference\nto the other their ages would gradually diverge, for time would go at\ndifferent rates for the two. If one moved with the velocity of light,\ntime would stand still for him while for the other it would go on as\nusual. To get actually younger it would be necessary to move quicker\nthan light which is believed to be impossible. The velocity of light\nis assumed to be the greatest velocity occurring in nature.\n\nEvidently then if the distance in space and the interval in time\nseparating two given events, such as the firing of a gun and the\nbursting of the shell, are measured by two observers in uniform\nrelative motion, their estimates will not agree. Consider now the\nsimple problem of measuring the distance between two points on an\nordinary drawing-board. If we draw two perpendicular axes, we can\ndefine this distance by specifying the lengths of the projections\non the two axes of the line joining the points. If we choose two\ndifferent axes the projections will not be the same but will define\nthe same length. Similarly, in a Euclidean four-space the distance\nbetween two points will be defined by the projections on the four\naxes, but if these axes be rotated slightly, the projections will be\ndifferent, but will define the same length. Now, returning to the two\nobservers just mentioned, it was noticed by Minkowski in 1908 that\nif the space measurements between the two events are split into the\nusual three components, and if the time measurements are multiplied\nby $\\sqrt{-1}$, the difference between the two sets of measurements\nis exactly the same as would have occurred had these two events\nbeen points in a Euclidean fourspace, and two different observations\nmade of their distance apart using two sets of axes inclined to each\nother. The velocity of light is made equal to 1 in this calculation by\na suitable choice of units. This discovery threw a vivid light on the\nproblem of space-time, showing that it is probably a true four-space\nof one negative dimension, a simple derivative of the much-discussed\nand now familiar Euclidean four-space.\n\nAlthough this discovery gave a tremendous impetus to the progress\nof the theory, it is probable that it holds a deeper significance\nnot yet revealed. It is probably a statement of the \"stuff\" of which\nthe four-space is made, and perhaps also of how it is made; but the\nproblem remains unsolved.\n\nIt thus becomes plain that our two observers are merely looking at the\nsame thing from different viewpoints. Each has just as much right as\nthe other to regard himself as being at rest in ordinary space (this is\nthe postulate of the relativity of uniform motion) and to regard his\ntime direction as a straight line in the four-space. The difference\nis merely that the two time axes are inclined to each other. If,\nhowever, one were moving with an acceleration with reference to the\nother his path in the four-space will appear curved to the other,\nthough he himself, since he regards it as his time axis, will still\nassume it to be straight. If there is a body moving in what one\nobserver sees to be a straight line, the other will, of course,\nin general see it as curved, and following the usual custom, since\nthis body, without apparent reason, deviates from the straight path,\nwill say there must be some force acting on it. Thus the curvature\nof his time axis, due to his accelerated motion, makes it appear\nthat there is round him a field of force, which causes freely moving\nbodies to deviate from the straight path. Now if space-time is itself\ninherently curved it is not generally possible for any line in it to\nbe straight any more than it is possible for any line on the surface\nof a sphere to be straight. Hence, all axes must be curved, and all\nobservers, whatever their states of motion, must experience fields of\nforce which are of the same nature as those due to motion only. The\nextra force experienced when a lift begins to rise is an example of\nforce due to pure motion: gravitation is the similar force due to an\ninherent curvature of the four-space, and it was the postulate that\nthese forces were similar that made possible Einstein's solution of\nthe general problem of gravitation.\n\n\n\n\nTHE TIME DIAGRAM\n\nThe correlation of time with its geometrical analogue is of absorbing\ninterest. Representing velocity by the common method of plotting\na curve showing positions at various times and marking distances\nhorizontally and times vertically, the velocity of light being 1,\n$MM'$ and $NN'$ will both represent this velocity. Since this is\nassumed to be the greatest velocity occurring in nature, all other\npossible velocities are represented by lines falling within the upper\nand lower V's. Now this diagram correctly represents two dimensions\nof Minkowski's Euclidean four-space so, transmuting to real but flat\nfour-space by multiplying times by $\\sqrt{-1}$, it is seen that there\nis a region outside which no effect can be propagated from $O$ since\nthat would involve the existence of a velocity greater than that of\nlight. This region represents the future of $O$. Similarly, $O$ can\nonly be affected by events within the region derived from the\ndownward-opening V, which therefore represents the past of $O$.\nThe region between the two represents events which may be either\nsimultaneous with $O$ or not, according to the velocity of the\nobserver at $O$. Thus in this theory an event dictated by free-will,\ncould affect points in its \"future\" region, but not in any other,\nwhich agrees with experience and shows that the theory is not\nessentially \"determinist.\" If \"free-will\" is really free, the future\nis not yet determined, and the fourspace must be in some way formed\nby the will as time progresses.\n\nThe trains of thought inspired by Einstein's postulates have already\ncarried us to a pinnacle of knowledge unprecedented in the history of\nman. On every hand, as we look out upon the universe from our new and\nlofty standpoint, unexpected and enthralling vistas open up before us,\nand we find ourselves confronting nature with an insight such as no\nman has ever before dared aspire to.\n\nIt is completely unthinkable that this theory can ever be swept\naside. Apart from experimental verifications which, in point of fact,\nlend it the strongest support, no one could work through the theory\nwithout feeling that here, in truth, the inner workings of the universe\nwere laid bare before him. The harmony with nature is far too complete\nfor any doubt to arise of its truth.\n\n\n\n\n\n\n\n\n\nXVI\n\nTHE QUEST OF THE ABSOLUTE\n\nModern Developments in Theoretical Physics, and the Climax Supplied\nby Einstein\n\nBY DR. FRANCIS D. MURNAGHAN,\nJOHNS HOPKINS UNIVERSITY\nBALTIMORE\n\n\nWe shall discuss the more important aspects of the theory popularly\nknown as the \"Einstein Theory of Gravitation\" and shall try to show\nclearly that this theory is a natural outcome of ideas long held by\nphysicists in general. These ideas are:\n\n(a) The impossibility of \"action at a distance;\" in other words we\nfind an instinctive repugnance to admit that one body can affect\nanother, remote from it, instantaneously and without the existence\nof an intervening medium.\n\n(b) The independence of natural, i.e., physical, laws of their\nmathematical mode of expression. Thus, when an equation is written down\nas the expression of a physical law it must be satisfied, no matter\nwhat units we choose in order to measure the quantities occurring in\nthe equation. As our physics teacher used to say \"the expression of the\nlaw must have in every term the same dimensions.\" More than this the\nchoice of the quantities used to express the law--if there be a choice\nopen--must have no effect on its correctness. As we were told--\"all\nphysical laws are capable of expression as relations between vectors or\nelse as relations between magnitudes of the same dimensions.\" We shall\nhope to make this clearer in its proper place in the essay, as its\nobvious generalization is Einstein's cardinal principle of relativity.\n\nThe measurements which an experimental physicist makes are always the\nexpression of a coincidence of two points in space at the same time. If\nwe ask such an experimenter what he means by a point in space he tells\nus that, for him, the term has no meaning until he has a material\nbody with reference to which he can locate the point by measurements;\nin general it requires three measurements and he expresses this by\nsaying that space has three dimensions. He measures his distance,\nas a rule, parallel to three mutually perpendicular lines fixed in\nthe material body--a Cartesian reference-frame so-called. So that a\n\"point in space\" is equivalent to a given material reference-frame\nand three numbers or coordinates. If, for any reason, we prefer to\nuse a new material reference-frame the coordinates or measurements\nwill change and, if we know the relative positions of the two material\nreference-frames, there is a definite relation between the two sets of\nthree coordinates which is termed a transformation of coordinates. But\nwhich particular material reference-frame shall we use? The first\nchoice would, we think, be that attached to the earth. But, even yet,\nwe are in doubt as there are numberless Cartesian frameworks attached\nto the earth (as to any material body) and it is here that our idea\n(b) begins to function. We say it must be immaterial which of these\nCartesian frames we use. In each frame a vector has three components\nand when we change from one frame to another the components change\nin such a way that if two vectors have their three components equal\nin one framework they will be equal in any other attached to the\nsame material system. So our idea (b), which says that our physical\nequations must be vector equations, is equivalent to saying that the\nchoice of the framework attached to any given material body can have\nno effect on the mode of expression of a natural law.\n\nShall we carry over our idea (b) to answer the next question: \"To\nwhich material body shall we attach our framework?\" To this question\nNewton gave one answer and Einstein another. We shall first consider\nNewton's position and then we may hope to see clearly where the new\ntheory diverges from the classical or Newtonian mechanics. Newton's\nanswer was that there is a particular material frame with reference\nto which the laws of mechanics have a remarkably simple form commonly\nknown as \"Newton's laws of motion\" and so it is preferable to use\nthis framework which is called an absolute frame.\n\nWhat is the essential peculiarity of an absolute frame? Newton was\nessentially an empiricist of Bacon's school and he observed the\nfollowing facts. Let us suppose we have a framework of reference\nattached to the earth. Then a small particle of matter under the\ngravitational influence of surrounding bodies, including the earth,\ntakes on a certain acceleration $A_1$. Now suppose the surrounding\nbodies removed (since we cannot remove the earth we shall have to\nview the experiment as an abstraction), and another set introduced;\nthe particle, being again at its original position, will begin to\nmove with an acceleration $A_2$. If both sets of surrounding bodies\nare present simultaneously the particle begins to move with an\nacceleration which is approximately but not quite the sum of $A_1$\nand $A_2$. Newton postulated there there is a certain absolute\nreference frame in which the approximation would be an equality;\nand so the acceleration, relative to the material frame, furnishes\na convenient measure of the effect of the surrounding bodies--which\neffect we call their gravitational force. Notice that if the effect\nof the surrounding bodies is small the acceleration is small and\nso we obtain as a limiting case, Newton's law of inertia which says\nthat a body subject to no forces has no acceleration; a law which,\nas Poincar\u00e9 justly observed, can never be subjected to experimental\njustification. The natural questions then arise: which is the\nabsolute and privileged reference-frame and how must the simple laws\nbe modified when we use a frame more convenient for us--one attached\nto the earth let us say? The absolute frame is one attached to the\nfixed stars; and to the absolute or real force defined as above,\nwe must add certain terms, usually called centrifugal forces. These\nare referred to as fictitious forces because, as it is explained,\nthey are due to the motion of the reference-frame with respect to\nthe absolute frame and in no way depend on the distribution of the\nsurrounding bodies. Gravitational force and centrifugal forces have\nin common the remarkable property that they depend in no way on the\nmaterial of the attracted body nor on its chemical state; they act\non all matter and are in this way different from other forces met\nwith in nature, such as magnetic or electric forces. Further Newton\nfound that he could predict the facts of observation accurately on the\nhypothesis that two small particles of matter attracted each other,\nin the direction of the line joining them, with a force varying\ninversely as the square of the distance between them. This law is an\n\"action at a distance\" law and so is opposed to the idea (a).\n\nWe have tacitly supposed that the space in which we make our\nmeasurements is that made familiar to us by the study of Euclid's\nelements. The characteristic property of this space is that\nstated by the theorem of Pythagoras that the distance between two\npoints is found by extracting the square root of the sum of the\nsquares of the differences of the Cartesian coordinates of the two\npoints. Mathematicians have long recognized the possibility of other\ntypes of space and Einstein has followed their lead. He abandons\nthe empiricist method and when asked what he means by a point in\nspace replies that to him a point in space is equivalent to four\nnumbers how obtained it is unnecessary to know a priori; in certain\nspecial cases they may be the three Cartesian coordinates of the\nexperimenter (measured with reference to a definite material framework)\ntogether with the time. Accordingly he says his space is of four\ndimensions. Between any two \"points\" we may insert a sequence of sets\nof four numbers, varying continuously from the first set to the second,\nthus forming what we call a curve joining the two points. Now we define\nthe \"length\" of this curve in a manner which involves all the points\non it and stipulate that this length has a physical reality, i.e.,\naccording to our idea (b) its value is independent of the particular\nchoice of coordinates we make in describing the space. Among all\nthe joining curves there will be one with the property of having\nthe smallest length; this is called a geodesic and corresponds to\nthe straight line in Euclidean space. We must now, for lack of an\na priori description of the actual significance of our coordinates,\nextend the idea of vector so that we may speak of the components of\na vector no matter what our coordinates may actually signify. In\nthis way are introduced what are known as tensors; if two tensors\nare equal, i.e., have all their components equal, in any one set of\ncoordinates they are equal in any other and the fundamental demand of\nthe new physics is that all physical equations which are not merely\nthe expression of equality of magnitudes must state the equality of\ntensors. In this way no one system of coordinates is privileged above\nany other and the laws of physics are expressed in a form independent\nof the actual coordinates chosen; they are written, as we may say,\nin an absolute form.\n\n\n\n\nTHE GRAVITATIONAL HYPOTHESIS\n\nEinstein flatly denies Newton's hypothesis that there is an absolute\nsystem (and, indeed, many others before him had found it difficult\nto admit that so insignificant a part of the universe as our fixed\nstar system should have such a privileged position as that accorded\nto it in the Newtonian Mechanics). In any system, he says, we have no\nreason to distinguish between the so-called real gravitational force\nand the so-called fictitious centrifugal forces--if we wish so to\nexpress it gravitational force is fictitious force. [10] A particle\nmoving in the neighborhood of material bodies moves according to a law\nof inertia--a physical law expressible, therefore, in a manner quite\nindependent of the choice of coordinates. The law of inertia is that\na particle left to itself moves along the geodesics or shortest lines\nin the space. If the particle is remote from other bodies the space\nhas the Euclidean character and we have Newton's law of inertia;\notherwise the particle is in a space of a non-Euclidean character\n(the space being always the four-dimensional space) and the path of\nthe particle is along a geodesic in that space. Einstein, in order\nto make the theory more concrete, makes a certain stipulation as to\nthe nature of the gravitational space which stipulation is expressed,\nas are all physical laws, by means of a tensor equation--and this is\nsometimes called his law of gravitation.\n\nPerhaps it will be well, in exemplification, to explain why light\nrays, which pass close to the sun, should be bent according to the new\ntheory. It is assumed that light rays travel along certain geodesics\nknown as minimal geodesics. The sun has an intense gravitational field\nnear it--or, as we now say, the departure of the four-dimensional\nspace from the Euclidean is very marked for points near the sun--but\nfor points so remote as the earth this departure is so small as\nto be negligible. Hence the form of the geodesics near the sun is\ndifferent from that near the earth. If the space surrounding the\nsun were Euclidean the actual paths of the light rays would appear\ndifferent from geodesics or straight-lines. Hence Einstein speaks\nof the curvature of the light rays due to the gravitational field of\nthe sun; but we must not be misled by a phrase. Light always travels\nalong geodesics (or straight lines--the only definition we have of a\nstraight line is that it is a geodesic); but, owing to the \"distortion\"\nof the space they traverse, due to the sun, these geodesics reach us\nwith a direction different from that they would have if they did not\npass through the markedly non-Euclidean space near the sun.\n\nThe consideration of the fundamental four-dimensional space as\nbeing non-Euclidean where matter is present gives a possibility of\nan answer to the world old question: Is space finite or infinite? Is\ntime eternal or finite? The fascinating possibility arises that the\nspace may be like the two-dimensional surface of a sphere which to a\nlimited experience seems infinite in extent and flat or Euclidean in\ncharacter. A new Columbus now asks us to consider other possibilities\nin which we should have a finite universe--finite not only as to space\nmeasurement but as to time (for the space may be such that all of the\nfour coordinates of its points are bounded in magnitude). However,\nalthough Einstein speaks of the possibility of a finite universe, we\ndo not, personally, think his argument convincing. Points on a sphere\nmay be located by the Cartesian coordinates of their stereographic\nprojections on the equatorial plane and these coordinates, which\nmight well be those actually measured, are not bounded.\n\n\n\n\nTHE SPECIAL RELATIVITY THEORY\n\nIn our account of the Einstein theory we have not followed its\nhistorical order of development for two reasons. Firstly, the\nearlier Special Relativity Theory properly belongs to a school\nof thought diametrically opposed to that furnishing the \"General\nTheory of Relativity\" and, secondly, the latter cannot be obtained\nfrom the former by the process of generalization as commonly\nunderstood. Einstein, when proposing the earlier theory, adopted\nthe position of the empiricist so that to him the phrase, a point in\nspace, had no meaning without a material framework of reference in\nwhich to measure space distances. When he came to investigate what\nis meant by time and when he asked the question \"what is meant by\nthe statement that two remote events are simultaneous?\" it became\nevident that some mode of communication between the two places is\nnecessary; the mode adopted was that by means of light-signals. The\nfundamental hypothesis was then made that the velocity of such signals\nis independent of the velocity of their source (some hypothesis is\nnecessary if we wish to compare the time associated with events, when\none material reference-system is used, and the corresponding time when\nanother in motion relative to the first is adopted). It develops that\ntime and space measurements are inextricably interwoven; there is no\nsuch thing as the length of a body or the duration of an event but\nrather these are relative to the reference-system. [11] Minkowski\nintroduced the idea of the space of events--of four dimensions--but\nthis space was supposed Euclidean like the three-dimensional space\nof his predecessors. To Einstein belongs the credit of taking from\nthis representation a purely formal mathematical character and of\ninsisting that the \"real\" space--whose distances have a physical\nsignificance--is the four-dimensional space. But we cannot insist\ntoo strongly on the fact that in the gravitational space of the\ngeneral theory there is no postulate of the constancy of velocity of a\nlight-signal and accordingly no method of assigning a time to events\ncorresponding to that adopted in the special theory. In this latter\ntheory attention was confined to material systems moving with uniform\nvelocity with respect to each other and it developed that the velocity\nof light was the ultimate velocity faster than which no system could\nmove--a result surprising and a priori rather repugnant. It is merely\na consequence of our mode of comparing times of events; if some other\nmethod--thought transference, let us say--were possible the velocity\nof this would be the \"limiting velocity.\"\n\nIn conclusion we should remark that the postulated equivalence of\n\"gravitational\" and \"centrifugal\" forces demands that anything\npossessed of inertia will be acted upon by a gravitational field and\nthis leads to a possible identification of matter and energy. Further\nour guiding idea (a) will prompt us to say, following the example\nof Faraday in his electrical researches, that the geodesics of a\ngravitational space have a physical existence as distinct from a mere\nmathematical one. The four-dimensional space we may call the ether, and\nso restore this bearer of physical forces to the position it lost when,\nas a three-dimensional idea in the Special Relativity Theory, it had to\nbear an identical relation to a multitude of relatively moving material\nsystems. The reason for our seemingly paradoxical title for an essay on\nRelativity will be clear when it is remembered that in the new theory\nwe consider those space-time properties which are absolute or devoid\nof reference to any particular material reference-frame. Nevertheless,\nalthough the general characteristics of the theory are thus described,\nwithout reference to experiment, when the theory is to be tested it\nis necessary to state what the four coordinates discussed actually\nare--how they are determined by measurement. It is our opinion that\nmuch remains to be done to place this portion of the subject on a\nsatisfactory basis. For example, in the derivation of the nature\nof the gravitational space, surrounding a single attracting body,\nmost of the accounts use Cartesian coordinates as if the space were\nEuclidean and step from these to polar coordinates by the formul\u00e6\nfamiliar in Euclidean geometry. But these details are, perhaps,\nlike matters of elegance, if we shall be allowed to give Einstein's\nquotation from Boltzmann, to be left to the \"tailor and the cobbler.\"\n\n\n\n\n\n\n\n\n\nXVII\n\nTHE PHYSICAL SIDE OF RELATIVITY\n\nThe Immediate Contacts Between Einstein's Theories and Current Physics\nand Astronomy\n\nBY PROFESSOR WILLIAM H. PICKERING\nHARVARD COLLEGE OBSERVATORY,\nMANDEVILLE, JAMAICA\n\n\nThe Theory of Relativity will be treated first from the physical\nside, leaving the three astronomical tests to which it has been put\nto be discussed later. There is one astronomical fact however that\nmust be mentioned in this connection, and this is the discovery of\nthe aberration of light by Bradley in 1726. It is found that every\nstar in the heavens apparently describes a small annual ellipse,\nwhose major axis is 41'' in length. This Bradley showed to be due to a\ncombination of the velocity of the earth in its orbit, and the velocity\nof light; and it is so explained in all the elementary text-books on\nastronomy. It implies a stationary ether through which the earth is\nmoving. The importance of this statement will appear presently.\n\nThe subject is usually illustrated by supposing a man to go out in a\nrainstorm carrying a vertical tube. If the rain is falling vertically,\nand the man stands still, the sides of the tube will not be wet,\nsave by an occasional drop, but if the tube is moved, it must then be\ninclined forward in order to keep it dry. The angle of inclination,\nwhich corresponds to aberration, will depend on the relative velocity\nof the tube, corresponding to the earth, and the rain drops which\ncorrespond to the waves of light.\n\nIf three lines are dropped upon a point in space, each line being\nperpendicular to the plane containing the other two, we have what\nis known as a system of coordinates. Einstein's original theory\nof relativity, which he now designates as the \"special theory,\"\ndepends on two principles. The first is that \"Every law of nature\nwhich holds good with respect to a coordinate system $K$ must also\nhold good for any other system $K'$, provided that $K$ and $K'$ are\nin uniform movement of translation.\" The second principle is that\n\"Light in a vacuum has a definite and constant velocity, independent\nof the velocity of its source.\"\n\nThese two sentences may be considered as authoritative, being quoted\nin Einstein's own words.1 The first of these principles need not\ngreatly surprise us. The second is not well expressed, because it\nis ambiguous. He does not say how the first \"velocity\" is measured,\nwhether relatively to the ether or relatively to the observer. In\nfact this is the very gist of the whole matter, as we shall presently\nsee. In the case of sound the velocity is constant with regard to the\nmedium, the air, in the case of light it is supposed to be constant\nwith regard to the observer. It reaches him with a constant velocity,\nno matter how he moves.\n\nIn order to understand this statement clearly let us consider the\nappended tabular diagram. On a calm day imagine a source of sound\nat $S$ in line a. This may be either a gun or a bell. Imagine an\nobserver 1,100 feet distant, located at $O$. The velocity of sound in\nair is 1,100 feet per second. This velocity we will take as unity,\nas indicated in the third column, and the velocity with which the\nsound reaches the observer is also 1, as shown in the fourth. It will\nreach him in a unit interval of 1 second, as shown in the fifth. If\nthe bell is struck, it will give its normal pitch or frequency,\nwhich we will also call unity, in the sixth column.\n\nNow imagine case b where the observer is on a train advancing toward\n$S$. When he is 1,100 feet distant, the gun is fired, but as he is\nadvancing toward it, he hears it at $O$ in rather less than a second,\nas shown in the fifth column. The velocity of the sound with regard\nto him is rather more than unity, as shown in the fourth column. If\nthe bell is sounded, the pitch, that is the frequency, is raised,\nbecause he receives more sound waves per second than before.\n\nIn case c the observer is stationary, but the source of sound is\nreceding. At a distance of 1,100 feet the gun is fired, and the\nobserver hears it after an interval of just one second, as in case\na. The velocities with regard to the observer and through the medium\nare also unity. If the bell is struck the pitch is lowered, since he\nreceives fewer sound waves per second, the reverse of case b.\n\n\n                    Velocity\n    Source     in       to         Interval   Frequency   Observer\n               Medium   Observer\n\n                                  Air\n\na       S     1         1          1          1             O\nb       S     1         1+         1-         1+        O\nc   S         1         1          1          1-            O\nd   S         1         1+         1-         1         O\n\n                                 Ether\n\nA       S     1         1           1         1             O\nB       S     1-        1           1         1+        O\nC   S         1         1           1         1-            O\nD   S         1 -       1           1         1         O\n\n\nIn case d imagine the source and the observer 1,100 feet apart, and\nadvancing on the same train. When the gun is fired, the velocity of\nthe sound waves will be greater with regard to the observer, and he\nwill hear the sound in less than a second, as in case b. When the\nbell is struck it will have the normal pitch, the same as in case a.\n\nWe find therefore that for sound the velocity with regard to\nthe medium is always unity, while the velocity with regard to the\nobserver, and the interval elapsed, depend only on the motion of the\nobserver himself, and are independent of the motion of the source. The\nfrequency of the vibrations, on the other hand, depends only on the\nrelative motion of the observer and the source, but is independent of\ntheir common motion in any direction. Further, it makes no difference\nwhether the source and the observer are moving on a train, or whether\nthey are stationary, and a uniform wind is blowing past them.\n\nIn the case of light waves we shall find a very different state of\naffairs, although the rules for frequency are the same as they are for\nsound. In case A we have the normal conditions, where both the source\nand observers are stationary. In case B we have a representation of\nthe Michelson-Morley experiment as supplemented by that of Majorana,2\nwhere the source is stationary and the observer advances. Unlike the\ncase of sound, the interval elapsed, as shown by the experiment, is now\nthe same as in case A, and since the distance to the observer is less,\nthe velocity of light with respect to the ether must also be less than\nunity. Since the observer is advancing against the light, this will\npermit the velocity of light with regard to the observer to remain\nunity, in conformity with the second principle of relativity. Compare\nwith case b for sound. As Jeans expresses it, \"The velocity of light\nin all directions is the same, whatever the motion of the observer.\"3\nThat is to say it appears to be the same to him, however he moves.\n\nCase C represents Einstein's statement, as confirmed by Majorana's\nexperiment. It does not differ from case c for sound. Case D is\nmore complex, but accepting the statement above that the velocity\nis constant with regard to the observer, we see that the velocity\nthrough the medium must be less, and that the interval elapsed will\nbe constant, as in case B. Could we use the brighter stars and planets\nas sources of light, several of these cases could be further tested.\n\nThis brings us at once to statements that contradict our common\nsense. For instance, Jeans says \"no matter what the velocity of the\nobserver is, the light surface, as observed by that observer, is\ninvariably a sphere having that observer as center.\"3 That is to say\nthe light surface, or wave front, is a contracting, not an expanding,\nsphere. This, if confirmed, would go a long way toward making our\nuniverse a subjective rather than an objective phenomenon. Again\nimagine a flash of light, such as an explosion, to occur when an\nobserver is in a given position. It makes no difference how the\nobserver may move while the light is approaching him, whether several\nmiles forward or backward, the light will reach him in exactly the\nsame time, as is shown by Michelson's experiment. Or if two observers\nare at the same spot when the explosion occurs, and one moves forward,\nand the other backward, they will both see the explosion at exactly\nthe same instant.\n\nThis sounds ridiculous, but not only is it what Jeans says, but it is\nthe logical interpretation of Einstein's second principle, if Einstein\nmeans by velocity, velocity with regard to the observer. If he means\nvelocity with regard to the medium, then the case is exactly the same\nas that of sound in air, and Michelson's experiment as well as the\nMaxwell-Lorentz theory of light are contradicted. This theory is now\nuniversally accepted, and Michelson's experiment has been carefully\nrepeated by other observers, and fully confirmed. This is the very\nheart of the relativity question.\n\nIf we state the matter objectively it comes to this. The velocity\nof light with regard to the ether is a variable quantity, depending\nmerely on where the observer chooses to go. As Eddington well says,\n\"these relations to the ether have no effect on the phenomena and can\nbe disregarded--a step which appears to divest the ether of the last\nremnants of substantiality.\"4\n\nThe only way of avoiding this apparent absurdity seems to be to\nconsider that the ether moves with the earth. Michelson's result would\nthen be fully explained. Of course this can only be true for a few\nmiles above the earth's surface. Beyond that the ether must either\nbe stationary or move with the sun. The velocity of light with regard\nto the ether would then be a constant, just as the velocity of sound\nis constant with regard to the air. This would contradict Einstein's\nsecond principle as it is generally understood. The trouble with\nthis suggestion is that it fails to account for aberration, which,\nas already explained, appears to require that the earth should be\nmoving through the ether. To meet this emergency would involve some\nmodification of the undulatory theory of light, which apparently\nwould not be impossible, but has not yet been made.\n\nIn 1915 Einstein brought out an extension of his first principle. This\nhe calls the \"general theory of relativity.\" It states that in our\nchoice of coordinate systems we \"should not be limited in any way\nso far as their state of motion is concerned.\"1 This leads to the\nthree astronomical consequences mentioned later in this paper, two\nof which have been more or less confirmed, and the third practically\ncontradicted as far as quantitative measures are concerned.5\n\nAs is well known the kinetic energy of a moving body may be\nexpressed as $e = 1/2mv^2$, but if the body is charged electrically,\nthe fraction becomes $1/2(m + m')v^2$, where $m'$ is a quantity\ndependent on the square of the electrical charge. That is to say,\nwe have the normal mass of the body, and also what we may call its\nelectrical mass. If when in this condition a portion of the mass is\nelectrical, the question at once occurs to us, why may not the whole\nmass be electrical, in other words, a form of energy? Although this\nhas not been satisfactorily proved hitherto, yet such is the general\nbelief among physicists. As Einstein puts it \"inert mass is nothing\nelse than latent energy.\"1 The same idea is sometimes expressed as\n\"the mass of ordinary matter is due to the electromagnetic energy of\nits ultimate particles, and electromagnetic energy wherever found\nmust possess mass, i.e., inertia.\"6 If that is so, since a ray of\nlight on the undulatory theory is a form of electromagnetic energy,\nit too must possess mass. Since all mass with which we are familiar is\nsubject to the attraction of gravitation, it seemed likely that a ray\nof light would be bent out of its course in passing near the sun, and\nthis as we have seen was proved to be true at the recent solar eclipse.\n\nThat portion of the mass of a body due to its electrical charge can\nbe readily shown experimentally to vary with the velocity of the\nbody. Einstein has shown the same to be true of the normal mass,\nas is illustrated in the advance of the perihelion of the orbit\nof Mercury. He has also pointed out that gravitation, inertia and\ncentrifugal force are all closely related, and obey similar laws. Thus\nif we rise from the earth with accelerated velocity, we apparently\nincrease our weight. Again if the velocity of rotation of the earth on\nits axis should be increased, our weight would be diminished. These\nfacts are suggestive when we come to consider the ultimate cause\nof gravitation.\n\nAnother fact which must be rather startling to the older school\nof scientists is that momentum is no longer simply $mv$, mass times\nvelocity, but that the velocity of light $c$, comes into the question,\nand the formula for momentum now assumes the form of\n\n    $$\\frac{m v}{\\sqrt{1 - \\frac{v^2}{c^2}}}$$\n\nFor ordinary velocities this correction is extremely small, but it\nhas been shown to be necessary, both theoretically and experimentally,\nwhen dealing with the high velocities with which we are now familiar.\n\nThe theory of relativity is so widespread in its application that\nseveral other theories have become more or less intimately combined\nwith it, for which Einstein is in no way responsible. One of these is\nknown as the Fitzgerald-Lorentz theory, that all bodies are subject to\na contraction in the direction of their motions through space. This was\nfirst suggested in order to explain the Michelson-Morley experiment,\nbut has proved inadequate to do so, particularly when the observer is\nreceding from the source. This contraction is expressed by the same\nfactor used in the denominator of the revised expression for momentum,\ngiven above. Again the quantity $c$ is so enormous, that even for\nlarge bodies at planetary velocities the contraction amounts to very\nlittle. Thus the earth moving at a speed of eighteen miles per second\nin its orbit, is flattened only 1/200,000,000, or 2.5 inches. On the\nother hand for high velocities of many thousand miles per second,\nsuch as we have become familiar with in the case of the radioactive\nsubstances, the flattening is a very considerable fraction of the\ndiameter of the moving body, one-half or more, and in the case of\nthe corpuscles of light, if that theory were adopted, this flattening\nbecomes equal to the diameter, and their thickness is reduced to zero.\n\nWhen we view Einstein's theories from the astronomical standpoint,\nthe earliest fact bearing on relativity that we need consider was\nthe discovery of aberration, by Bradley, in 1726, as seen above. In\n1872 Airy observed the star g Draconis through a telescope filled\nwith water. Since the velocity of light is less in water than in\nair, we should naturally expect to find the aberration appreciably\nincreased. It was found, on the other hand, however, to be unaffected.\n\nIn 1887 the results of the famous Michelson-Morley experiment were\npublished.7 In this experiment the velocity of light was measured\nin various directions with regard to the motion of the earth in its\norbit. If the ether were stationary, and the earth moving through it,\ndifferent velocities should be obtained in different directions. Such\nwas not the case however, and the experiment indicated that the ether\nmoved with the earth. It thus flatly contradicted the conclusions\nfounded on aberration.\n\nEinstein's Special Theory of Relativity, of 1905, as we have seen,\nresolves this contradiction. But as we shall presently see, it is the\nGeneral Theory, of 1915, that leads to astronomical applications of\nbroad scope. It indicates, for instance, that there is no essential\ndifference between gravitation and inertia. This idea may be crudely\nillustrated by our feelings of increased weight when an elevator\nstarts rapidly upwards. A man while falling freely in space ceases\nto feel the pull of gravitation.\n\nBut we must not as yet conceive of the theory of relativity as a\nuniversally accepted and unquestioned truth of science. Eddington is\nits leading English exponent, and he is supported by such men as Jeans,\nLarmor, and Jeffreys. On the other hand, the theory has been severely\ncriticised by Lodge, Fowler, Silberstein, and Sampson. Few American\nscientists have expressed any opinions in print on the subject,\nand the recent eclipse observations, to which we shall refer later,\nare to be repeated with more suitable instruments for verification\nin 1922, in the hope of obtaining more accurate and accordant results.8\n\nAn appurtenance of the Einstein theories which bears much the\nsame relation to them as does the Lorentz-Fitzgerald contraction,\nmentioned above, is the idea, first clearly stated by Minkowski,\nthat time is a kind of space--a fourth dimension. This the reader\nwill doubtless find to be the most difficult portion of the theory\nto picture in his own mind. It is entirely unsupported by experiment\nor observation, necessarily so, and is based wholly on mathematical\nand philosophical conceptions. Our distinction between space and time\nseems to be that the direction in which we progress without effort\nis time; the other directions, in which we have to make an exertion\nto move ourselves, or in which we are carried, are space. How many\ndimensions empty space may have, we really have no means of knowing,\nbecause we can neither see nor feel it. Matter we know has three,\nlength, breadth, and thickness, also that it lies remote from us in\nthree corresponding directions. These facts may have given us the\nerroneous impression that space too has only three dimensions. Now\nit is claimed that time is a fourth, and that there are also others.\n\nIn order to illustrate this, Eddington asks us to imagine a movie film\ntaken of a man or of any moving object. Let the separate pictures be\ncut apart and piled on one another. This would form a sort of pictorial\nhistory of the individual for a brief interval in his life, in the form\nof a cube. If we attempt to pick it up, it falls apart, thus clearly\nshowing the difference between time and space. But suppose it now\nall glued together in one solid cube, so that it is no easier to cut\na section in one direction than in another. That is Minkowski's idea\nof space and time, and further, that the direction in which we should\ncut it depends merely on the velocity with which we are moving through\nspace. I should cut it parallel to the films, but a man on a rapidly\nmoving star, in order to separate it into space and time, would cut\nit in an inclined direction. That is a thing which may be true, but it\nis one which we believe no mortal man can clearly picture to himself.\n\nOn the other hand Turner has recently made a very interesting\npoint,9 namely, that the fourth dimension as actually treated\nby the mathematicians is not time itself, but time multiplied\nby a constant--the velocity of light.10 Without affecting the\nastronomical proofs of relativity at all, this simplifies our\nconceptions enormously. In ordinary everyday life time and space\ncannot be identical, any more than a yard can be identical with a\nquart. On what is known to physicists as the centimeter-gram-second\nsystem, distance is represented by $l$, mass by $m$, and time by\n$t$. Velocity is then distance divided by time, $l/t$, or as we say\nin English units, so many feet per second, and the fourth dimension\nmay be expressed as time multiplied by velocity, $t \\times l/t =\nl$. That is to say, it is simply distance, just like the other three\ndimensions. To say that time is the fourth dimension from this point\nof view, appears to us just as ridiculous as it would be to attempt\nto measure the velocity of a train in quarts. It is quite correct,\nhowever, although unusual, to speak of a given train as moving at a\nspeed of 10 quarts per square inch per second, $l^3/l^2 t = l/t$. This\nwould be equivalent to a velocity of 33 miles per hour.\n\nIf I wish to give a complete dimensional description of myself in my\nfour dimensions, I must give my length, my breadth, and my thickness,\never since I came into being, and also the course I have traversed\nthrough space since that time. This latter distance will be expressed\nin terms of a unit whose length is 186,000 miles, the distance\ntraversed by light in one second. The distance which I travel through\nspace annually is enormous, and very complex as to direction. It\ninvolves not merely my own motions as I cross the room, or take a\ntrain or steamer, but also those due to the rotation of the earth on\nits axis, its revolution round the sun, and the motion of the latter\nthrough the heavens. In general I travel, or in other words increase\nmy length in the fourth dimension, by over 4,000 units a year. The\nfourth dimension accordingly, if this view is accepted, is simply a\ndistance like the other three, and perfectly easy to understand.\n\nWe now come to the three actual tests by which the theory has been\ntried. The planets as is well known revolve about the sun in ellipses,\nwith the sun in one of the foci. That is to say, the sun is not in\nthe center, but a little on one side of it. The end of the ellipse\nwhere the planet comes nearest to the sun is called the perihelion,\nand here the planet is moving most rapidly. The other end is called\nthe aphelion, and here the motion is slowest. According to Newton's\ntheory of gravitation, if a spherical sun possesses a single planet\nor companion, its orbit will be permanently fixed in space unless\nperturbed by some other body. If a second planet exist, it will\ncause the perihelion of the first slowly to advance. According to\nEinstein the mass of a planet depends in part on its velocity. It\nwill therefore be less at aphelion where it is moving slowly than at\nperihelion where it is moving rapidly, consequently in addition to\nthe Newtonian attraction we have another one which increases as we\napproach the sun. The effect of this will be to cause the perihelion\nof the orbit to advance, whether there is a second planet or not.\n\nAmong the larger planets Mercury has the most eccentric orbit, and it\nalso moves most rapidly, so that it is particularly well adapted to\ntest the relativity theory. The observed advance of its perihelion is\n574'' per century, instead of the theoretical figure 532'', due to the\nother planets--a difference of 42''.11 This has long been a puzzling\ndiscrepancy between observation and the law of gravitation. Prior to\nEinstein, attempts were made to eliminate it by assuming a certain\noblateness of the solar disk. If the equatorial diameter exceeded the\npolar by only 0''.5 the whole advance would be accounted for, but not\nonly has this ellipticity failed of detection, but if it existed,\nit should produce a very noticeable and inadmissible change in the\ninclination of Mercury's orbit, amounting to about 3'' per century,\nas has been demonstrated by both Herzer and Newcomb.12\n\nEinstein from computations alone, without introducing any new\nconstants or hypotheses whatever, showed, if the theory of relativity\nbe accepted, that the sun should produce an acceleration of 43''\nper century, thus entirely accounting for the observed discrepancy,\nfar within the limits of accuracy of the observations. The only other\nplanet whose orbit has a large eccentricity, and that is suitable\nfor investigation, is the planet Mars. Here the discrepancy between\nobservation and theory is very slight, only 4'', and a portion of\nthat may be due to the attraction of the asteroids. This deviation\nis so slight that it may well be due entirely to accidental errors\nof observation, but however that may be, Einstein's theory reduces\nit to 2''.7.\n\nThis all seems very satisfactory and complete, but the trouble\nwith it is that the coincidence for Mercury is rather too good. It\nis based on the assumption that the sun is a perfect sphere, and\nthat the density of its surface is uniform from the equator to the\npoles. This would doubtless be true if the sun did not revolve on\nits axis. In point of fact it does revolve, in a period in general of\nabout 26 days. Consequently an object on its equator must experience\na certain amount of centrifugal force. Therefore if its surface were\nof uniform density the shape of the sun would be an oblate spheroid.\n\nIt can be readily shown that the theoretical excess of the equatorial\nover the polar diameter, due to the centrifugal force, should amount to\nonly 0''.04, an amount which could hardly be detected by observation,\nand might readily be concealed by a slight excess of equatorial over\npolar density. Any reasonable excess of density at the center would\ndiminish this result but slightly. The molecular weight of the central\nmaterial13 is probably about 2. This computed equatorial excess is\none-twelfth of the amount necessary to cause the observed advance,\nand should therefore cause an advance of the perihelion of about 3''.5\nper century, reducing the difference between the observed advance\nand that caused by gravitation to 38''.5. According to Einstein the\nadvance due to relativity should be, as we saw, 43'', a discrepancy\nof 4''.5 per century, or 10 per cent. Jeffreys has remarked that\nany discrepancy such as 10'' \"would be fatal to a theory such as\nEinstein's, which contains no arbitrary constituent capable of\nadjustment to suit empirical facts.\"14 It must be pointed out here\nhowever, that so far as known, this small correction to the motion\nof Mercury's perihelion has not previously been suggested, so that\nthere has been no opportunity hitherto for its criticism by others.\n\nIt was due largely to the success with Mercury that it was decided to\nput the relativity theory to another test. According to the Newtonian\ntheory, as stated by Newton himself, corpuscles as well as planets\nhave mass, and must therefore be attracted by the sun. According to\nEinstein, owing to their high velocity, this attraction must be twice\nas great as it would be according to the theory of gravitation. If the\nray of light proceeding from a star were to pass nearly tangent to the\nsun's limb it should be deflected 0''.87 according to Newton. According\nto the theory of relativity it should be deflected 1''.75. Stars\nof course cannot usually be observed near the sun. It is therefore\nnecessary to take advantage of a total solar eclipse, when the sun is\ncompletely hidden by the moon, in order to secure these observations.\n\nTwo expeditions, one to Africa, and one to South America, observed\nsuccessfully the total eclipse of May 29, 1919. The former was\nlocated on the Island of Principe in the Gulf of Guinea. The latter\nwas located at Sobral, Brazil. Their equipment and results are shown\nin the following table, where the successive columns give the location,\nthe aperture in inches of the telescopes employed, their focus in feet,\nthe number of plates secured, the number of stars measured, their\nmean deduced deflection from their true positions by the attraction\nof the sun, and the deviations from the theoretical results.15 In\nthe first and last line of the table shown herewith, this\n\n\n    Location   Aperture   Focus   Plates   Stars   Defl.    Dev.\n\n    Principe         13      11        2       5   1''.60   -0''.15\n    Sobral           13      11       19      12   0 .93    (+0 .06)\n    \u00a0\u00a0,,\u00a0\u00a0            4      19        8       7   1 .98    +0 .23\n\n\ndeviation is taken from Einstein's computed value of 1''.75. In the\nsecond line the difference shown is from the value required by the\nNewtonian theory, 0''.87. The results obtained with this telescope\nwere rejected however, although they were much the most numerous,\nbecause it was found that for some reason, supposed to be the heating\nof the mirror by the sun before the eclipse, the star images were\nslightly out of focus, and were therefore considered unreliable. The\nresults with the two other telescopes were not very accordant, but the\n4-inch had the longer focus, secured the greater number of plates,\nand showed the greater number of stars. The results obtained with\nit therefore appear to have been the more reliable. They differ from\nEinstein's prediction by 13 per cent. In future expeditions to test\nthis question, the mirror in front of the telescope will be eliminated.\n\nWe now come to the final test which has been applied to Einstein's\ntheory. Einstein showed that in the intense gravitational field of the\nsun, the theory of relativity required that all of the spectrum lines\nshould be shifted slightly toward the red end. The shift however is\nexceedingly small, and can only be detected and measured with the\nmost powerful modern instruments. Moreover only certain lines can\nbe used, because owing to varying pressure in the solar atmosphere,\nwhich affects many lines, as well as to rapid motion in the line of\nsight, which may affect all of them, still larger displacements are\nliable to occur.\n\nAccording to the theory of relativity the displacement of the lines\nshould be $+0.0080 A$. St. John at Mt. Wilson found a displacement\nfor the cyanogen lines of only $+0.0018 A$.16 Evershed at Kodaikanal\nfound +0.0060 at the north pole of the sun, and +0.0080 at the south\npole. These latter values however were only for the stronger lines. The\nweaker lines give much smaller shifts, as do those of calcium and\nmagnesium.17 According to Einstein all lines should give nearly the\nsame shift, an amount proportional to the wave length. It therefore\nappears that we must conclude by saying that Einstein's theory of\nrelativity has been partially, but not completely, verified.\n\n\n\nThe reference numbers in the above text have nothing to do with the\nnumbers used in other parts of this volume to acknowledge the work\nof the various contestants; they refer to Dr. Pickering's sources,\nas follows:\n\n1 Journ. Brit. Astron. Assoc., 1919, 30, 76.\n\n2 Comptes Rendus, 165, 424, and 167, 71.\n\n3 Monthly Notices R. A. S., 1919, 80, 104.\n\n4 Monthly Notices R. A. S., 1917, 77, 379.\n\n5 Astro-Physical Journal, 1917, 46, 249. Journ. Brit. Astron. Assoc.,\n1920, 30, 276.\n\n6 Monthly Notices, R. A. S., 1917, 77, 377.\n\n7 Amer. Journ. Sci., 34, 333.\n\n8 Monthly Notices, R. A. S., 1920, 80, 628.\n\n9 The Observatory 1920, April. From an Oxford Note Book.\n\n10 Monthly Notices, R. A. S., 1917. 78, 3 De Sitter, 1919, 80, 121,\nJeans, 80, 146 Jeffreys.\n\n11 \"Gravitation and the Principle of Relativity,\" Eddington. Royal\nInstitution of Great Britain, 1918.\n\n12 Journ. Brit. Astron. Assoc., 1920, 30, 125.\n\n13 \"The Interior of a Star,\" Eddington. Scientia, 1918, 23, 15.\n\n14 Monthly Notices, R. A. S., 1919, 80, 138.\n\n15 Monthly Notices. R. A. S., 1920, 80,\n415. Journ. Brit. Astron. Assoc., 1919, 30, 46.\n\n16 Astro-Physical Journ., 1917, 46, 249.\n\n17 Journ. Brit. Astron. Assoc., 1920, 30, 276.\n\n\n\n\n\n\n\n\n\nXVIII\n\nTHE PRACTICAL SIGNIFICANCE OF RELATIVITY\n\nThe Best Discussion of the Special Theory Among All the Competing\nEssays\n\nBY PROFESSOR HENRY NORRIS RUSSELL, PRINCETON UNIVERSITY\n\n\nCan a small child catch a baseball moving sixty miles an hour without\ngetting hurt? We should probably answer \"No\"--but suppose that the boy\nand his father were sitting side by side in an express train, and the\nball was tossed lightly from one to the other. Then there would be\nno trouble about it, whether the train was standing still, or going\nat full speed. Only the relative motion of ball and boy would count.\n\nThis every-day experience is a good illustration of the much discussed\nPrinciple of Relativity, in its simplest form. If there were no\njolting, the motion of the train, straight ahead at a uniform speed,\nwould have no effect at all upon the relative motions of objects\ninside it, nor on the forces required to produce or change these\nmotions. Indeed, the motion of the earth in its orbit, which is free\nfrom all jar, but a thousand times faster, does not influence even\nthe most delicate apparatus. We are quite unconscious of it, and\nwould not know that the earth was moving, if we could not see other\nbodies outside it. This sort of relativity has been recognized for\nmore than two centuries and lies at the bottom of all our ordinary\ndynamical reasoning, upon which both science and engineering are based.\n\nBut there are other things in nature besides moving bodies,--above\nall, light, which is intimately related to electricity and magnetism,\nand can travel through empty space, between the stars. It moves at\nthe enormous speed of 186,000 miles per second, and behaves exactly\nlike a series of vibrations or \"waves.\" We naturally think of it as\ntravelling through some medium, and call this thing, which carries\nthe light, the \"ether.\"\n\nCan we tell whether we are moving through this ether, even though all\nparts of our apparatus move together, and at the same rate? Suppose\nthat we have two mirrors, $M$ and $N$, at equal distances, $d$, from a\npoint $O$, but in directions at right angles to one another, and send\nout a flash of light from $O$. If everything is at rest, the reflected\nflashes will evidently come back to $O$ at the same instant, and the\nelapsed time will be $2d/c$ seconds if $c$ is the velocity of light.\n\nBut suppose that $O$, $M$, and $N$ are fastened to a rigid frame work,\nand all moving in the direction $O M$, with velocity $V$. The light\nwhich goes from $O$ toward $M$, at the speed $c$, will overtake it\nwith the difference of their speeds, $c-v$, taking $d/(c-v)$ seconds to\nreach $M$. On the way back, $O$ will be advancing to meet it, and the\nreturn trip will occupy $d/(c+v)$ seconds. The elapsed time for the\nround trip comes out $2cd/(c^2-v^2)$ seconds, which is longer than\nwhen the system was at rest--the loss of time in the \"stern chase\"\nexceeding the saving on the return.\n\nThe light which is reflected from N has a different history. When\nit starts, $O$ and $N$ have certain positions in the ether, $O_1$\nand $N_1$. By the time it reaches the mirror, this is at $N_2$, and\n$O$ is at $O_2$, and when it returns, it finds $O$ at $O_3$. The\ndistances for the outward and inward journeys are now equal, but\n(as is obvious from the figure), each of them is greater than $d$,\nor $O_2N_2$, and the time for the round trip will be correspondingly\nincreased. A simple calculation shows that it is $2d/\\sqrt{c^2-v^2}$.\n\nThe increase above the time required when the system was at rest\nis less in this case than the preceding. Hence, if the apparatus\nis moving through the ether, the flashes reflected from $M$ and $N$\nwill not return at the same instant.\n\nFor such velocities as are attainable--even the 18 miles per second of\nthe earth in its orbit--the difference is less than a hundred-millionth\nof the elapsed time. Nevertheless, Michelson and Morley tried to\ndetect it in their famous experiment.\n\nA beam of light was allowed to fall obliquely upon a clear glass\nmirror (placed at $O$ in the diagram) which reflected part of it\ntoward the mirror, $M$, and let the rest pass through to the mirror\n$N$. By reuniting the beams after their round trips, it was possible\nto tell whether one had gained upon the other by even a small fraction\nof the time of vibration of a single light wave. The apparatus was\nso sensitive that the predicted difference, though amounting to less\nthan a millionth part of a billionth of a second, could easily have\nbeen measured; but they actually found no difference at all--though\nthe earth is certainly in motion.\n\nOther optical experiments, more intricate, and even more delicate,\nwere attempted, with the same object of detecting the motion of the\nearth through the ether; and they all failed.\n\n\n\n\nTHE SPECIAL THEORY AND ITS SURPRISING CONSEQUENCES\n\nIt was upon these facts that Einstein based his original, or \"special\"\ntheory of Relativity. He assumed boldly that the universe is so\nconstituted that uniform straight-ahead motion of an observer and all\nhis apparatus will not produce any difference whatever in the result\nof any physical process or experiment of any kind. Granting this,\nit follows that if all objects in the visible universe were moving\nuniformly together in any direction, no matter how fast, we could\nnot find this out at all. We cannot determine whether the universe,\nas a whole, is at rest or in motion, and may as well make one guess\nas another. Only the relative motions of its parts can be detected\nor studied.\n\nThis seems simple and easy enough to understand. But the consequences\nwhich follow from it are extraordinary, and at first acquaintance\nseem almost absurd.\n\nIn the first place, if an observer measures the velocity of light,\nhe must always get the same result, no matter how fast he and his\napparatus are moving, or in what direction (so long as the motion is\nuniform and rectilinear). This sounds harmless; but let us go back to\nthe Michelson-Morley experiment where the light came back in exactly\nthe same time from the two mirrors. If the observer supposes himself\nto be at rest, he will say that the distances $O M$ and $O N$ were\nequal. But if he fancies that the whole universe is moving in the\ndirection $O M$, he will conclude that $M$ is nearer to $O$ than $N$\nis--for if they were equidistant, the round-trip would take longer in\nthe first case, as we have proved. If once more he fancies that the\nuniverse is moving in the direction $O N$, he will conclude that N is\nnearer to $O$ than $M$ is. His answer to the question which of the\ntwo distances, $O M$ or $O N$, is the greater will therefore depend\non his assumption about the motion of the universe as a whole.\n\nSimilar complications arise in the measurement of time. Suppose that\nwe have two observers, $A$ and $B$, provided with clocks which run\nwith perfect uniformity, and mirrors to reflect light signals to one\nanother. At noon exactly by his clock, $A$ sends a flash of light\ntowards $B$. $B$ sees it come in at 12:01 by his clock. The flash\nreflected from $B$'s mirror reaches $A$ at 12:02 by $A$'s clock. They\ncommunicate these observations to one another.\n\nIf $A$ and $B$ regard themselves as being at rest, they will agree that\nthe light took as long to go out as it did to come back, and therefore\nthat it reached $B$ at just 12:01 by $A$'s clock, and that the two\nclocks are synchronized. But they may, if they please, suppose that\nthey (and the whole universe) are moving in the direction from $A$\ntowards $B$, with half the speed of light. They will then say that\nthe light had a \"stern chase\" to reach $B$, and took three times as\nlong to go out as to come back. This means that it got to $B$ at 1\n1/2 minutes past noon by $A$'s clock, and that $B$'s clock is slow\ncompared with $A$'s. If they should assume that they were moving with\nthe same speed in the opposite direction, they would conclude that\n$B$'s clock is half a minute fast.\n\nHence their answer to the question whether two events at different\nplaces happen at the same time, or at different times, will depend\non their assumption about the motion of the universe as a whole.\n\nOnce more, let us suppose that $A$ and $B$, with their clocks and\nmirrors, are in relative motion, with half the speed of light, and pass\none another at noon by both clocks. At 12:02 by $A$'s clock, he sends a\nflash of light, which reaches $B$ at 12:04 by his clock, is reflected,\nand gets back to $A$'s clock at 12:06. They signal these results to\neach other, and sit down to work them out. $A$ thinks that he is at\nrest, and $B$ moving. He therefore concludes that the light had the\nsame distance to go out as to return to him and took two seconds each\nway, reaching $B$ at 12:04 by $A$'s clock, and that the two clocks,\nwhich agreed then, as well as at noon, are running at the same rate.\n\n$B$, on the contrary, thinks that he is at rest and $A$ in motion. He\nthen concludes that $A$ was much nearer when he sent out the flash\nthan when he got it back, and that the light had three times as far to\ntravel on the return journey. This means that it was 12:03 by $A$'s\nclock at the instant when the light reached $B$ and $B$'s clock read\n12:04. Hence $A$'s clock is running slow, compared with $B$'s.\n\nHence the answer to the question whether two intervals of time,\nmeasured by observers who are in motion relative to one another, are\nof the same or of different durations, depends upon their assumptions\nabout the motion of the universe as a whole.\n\nNow we must remember that one assumption about the motion of the\nuniverse as a whole is exactly as good--or bad--as another. No\npossible experiment can distinguish between them. Hence on the\nPrinciple of Relativity, we have left no absolute measurement of\ntime or space. Whether two distances in different directions are to\nbe called equal or not--whether two events in different places are\nto be called simultaneous or not--depends on our arbitrary choice of\nsuch an assumption, or \"frame of reference.\" All the various schemes\nof measurement corresponding to these assumptions will, when applied\nto any imaginable experiment, predict exactly the same phenomena. But,\nin certain important cases, these predictions differ from those of the\nold familiar theory, and, every time that such experiments have been\ntried, the result has agreed with the new theory, and not with the old.\n\nWe are therefore driven to accept the theory of relativity, strange\nas it is, as being more nearly \"true to nature\" than our older\nideas. Fortunately, the difference between the results of the two\nbecome important only when we assume that the whole visible universe is\nmoving together much faster than any of its parts are moving relatively\nto one another. Unless we make such an unwarranted assumption, the\ndifferences are so small that it takes the most ingenious and precise\nexperiments to reveal them.\n\n\n\n\nTHE GENERALIZATION\n\nNot content with all this, Einstein proceeded, a few years ago, to\ndevelop a \"general\" theory of relativity, which includes the effects\nof gravitation.\n\nTo make this idea clear, let us imagine two observers, each, with\nhis measuring instruments, in a large and perfectly impervious box,\nwhich forms his \"closed system.\"\n\nThe first observer, with his box and its contents, alone in space,\nis entirely at rest.\n\nThe second observer, with his box and its contents, is, it may be\nimagined, near the earth or the sun or some star, and falling freely\nunder the influence of its gravitation.\n\nThis second box and its contents, including the observer, will then\nfall under the gravitational force, that is, get up an ever-increasing\nspeed, but at exactly the same rate, so that there will be no tendency\nfor their relative positions to be altered.\n\nAccording to Newton's principles, this will make not the slightest\ndifference in the motions of the physical objects comprising the system\nor their attractions on one another, so that no dynamical experiment\ncan distinguish between the condition of the freely falling observer\nin the second box and the observer at rest in the first.\n\nBut once more the question arises: What could be done by an optical\nexperiment?\n\nEinstein assumed that the principle of relativity still applied in\nthis case, so that it would be impossible to distinguish between the\nconditions of the observers in the two boxes by any optical experiment.\n\nIt can easily be seen that it follows from this new generalized\nrelativity that light cannot travel in a straight line in a\ngravitational field.\n\nImagine that the first observer sets up three slits, all in a straight\nline. A ray of light which passes through the first and second will\nobviously pass exactly through the third.\n\nSuppose the observer in the freely falling system attempts the same\nexperiment, having his slits $P$, $Q$, $R$, equally spaced, and placing\nthem at right angles to the direction in which he is falling. When\nthe light passes through $P$, the slits will be in certain position\n$P_1 Q_1 R_1$ (Figure). By the time it reaches $Q$, they will have\nfallen to a lower level, $P_2 Q_2 R_2$, and when it reaches $R$,\nthey will be still lower, $P_3 Q_3 R_3$. The times which the light\ntakes to move from $P$ to $Q$ and $Q$ to $R$ will be the same: but,\nsince the system is falling ever faster and faster the distance $R_2\nR_3$ will be greater than $Q_1 Q_2$. Hence, if the light which has\npassed through $P$ and $Q$ moves in a straight line, it will strike\nabove $R$, as is illustrated by the straight line in the figure. But,\non Einstein's assumption, the light must go through the third slit,\nas it would do in the system at rest, and must therefore move in a\ncurved line, like the curved line in the figure, and bend downward\nin the direction of the gravitational force.\n\n\n\n\nTHE TESTS\n\nCalculation shows that the deviation of light by the moon or planets\nwould be too small to detect. But for a ray which had passed near\nthe sun, the deflection comes out 1.7'', which the modern astronomer\nregards as a large quantity, easy to measure. Observations to test\nthis can be made only at a total eclipse, when we can photograph stars\nnear the sun, on a nearly dark sky. A very fine chance came in May,\n1919, and two English expeditions were sent to Brazil and the African\ncoast. These photographs were measured with extreme care, and they\nshow that the stars actually appear to be shifted, in almost exactly\nthe way predicted by Einstein's theory.\n\nAnother consequence of \"general relativity\" is that Newton's law of\ngravitation needs a minute correction. This is so small that there is\nbut a single case in which it can be tested. On Newton's theory, the\nline joining the sun to the nearest point upon a planet's orbit (its\nperihelion) should remain fixed in direction, (barring certain effects\nof the attraction of the other planets, which can be allowed for). On\nEinstein's theory it should move slowly forward. It has been known for\nyears that the perihelion of Mercury was actually moving forward, and\nall explanations had failed. But Einstein's theory not only predicts\nthe direction of the motion, but exactly the observed amount.\n\nEinstein also predicts that the lines of any element in the solar\nspectrum should be slightly shifted towards the red, as compared\nwith those produced in our laboratories. Different observers have\ninvestigated this, and so far they disagree. The trouble is that\nthere are several other influences which may shift the lines, such\nas pressure in the sun's atmosphere, motion of currents on the sun's\nsurface, etc., and it is very hard to disentangle this Gordian knot. At\npresent, the results of these observations can neither be counted\nfor or against the theory, while those in the other two cases are\ndecisively favorable.\n\nThe mathematical expression of this general relativity is intricate\nand difficult. Mathematicians--who are used to conceptions which\nare unfamiliar, if not incomprehensible, to most of us--find that\nthese expressions may be described (to the trained student) in terms\nof space of four dimensions and of the non-Euclidean geometry. We\ntherefore hear such phrases as \"time as a sort of fourth dimension,\"\n\"curvature of space\" and others. But these are simply attempts--not\naltogether successful--to put mathematical relationships into ordinary\nlanguage, instead of algebraic equations.\n\nMore important to the general reader are the physical bearings of\nthe new theory, and these are far easier to understand.\n\nVarious assumptions which we may make about the motion of the universe\nas a whole, though they do not influence the observed facts of nature,\nwill lead us to different ways of interpreting our observations as\nmeasurements of space and time.\n\nTheoretically, one of these assumptions is as good as any other. Hence\nwe no longer believe in absolute space and time. This is of great\ninterest philosophically. Practically, it is unimportant, for,\nunless our choice of an assumption is very wild, our conclusions and\nmeasurements will agree substantially with those which are already\nfamiliar.\n\nFinally, the \"general\" relativity shows that gravitation and\nelectro-magnetic phenomena--(including light) do not form two\nindependent sides of nature, as we once supposed, but influence one\nanother (though slightly) and are parts of one greater whole.\n\n\n\n\n\n\n\n\n\nXIX\n\nEINSTEIN'S THEORY OF RELATIVITY\n\nA Simple Explanation of His Postulates and Their Consequences\n\nBY T. ROYDS, KODAIKANAL OBSERVATORY, INDIA\n\n\nEinstein's theory of relativity seeks to represent to us the world as\nit really is instead of the world of appearances which may be deceiving\nus. When I was in town last week to buy 5 yards of calico I watched\nthe draper very carefully as he measured the cloth to make sure I was\nnot cheated. Yet experiment can demonstrate, and Einstein's theory\ncan explain, that the draper's yardstick became longer or shorter\naccording to the direction in which it was held. The length of the\nyardstick did not appear to me to change simply because everything else\nin the same direction, the store, the draper, the cloth, the retina\nof my eye, changed length in the same ratio. Einstein's theory points\nout not only this, but every case where appearances are deceptive,\nand tries to show us the world of reality.\n\nEinstein's theory is based on the principle of relativity and\nbefore we try to follow his reasoning we must spend a little time\nin understanding what he means by \"relativity\" and in grasping how\nthe idea arises. Suppose I wish to define my motion as I travel\nalong in an automobile. I may be moving at the rate of 25 miles an\nhour relative to objects fixed on the roadside, but relative to a\nfellow-passenger I am not moving at all; relative to the sun I am\nmoving with a speed of 18 1/2 miles per second in an elliptical orbit,\nand again relative to the stars I am moving in the direction of the\nstar Vega at a speed of 12 miles per second. Thus motion can only be\ndefined relative to some object or point of reference. Now this is\nnot satisfactory to the exact scientist. Scientists are not content\nwith knowing, for example, that the temperature of boiling water is\n+100\u00b0 C. relative to the temperature of freezing; they have set out\nto determine absolute temperatures and have found that water boils at\n373\u00b0 C. above absolute zero. Why should I not, therefore, determine\nthe absolute motion of the automobile, not its motion relative to\nthe road, earth, sun or stars, but relative to absolute rest?\n\nMichelson and Morley set out in their famous experiment to measure the\nabsolute velocity of their laboratory, which was, of course, fixed\non the earth. The experiment consisted of timing two rays of light\nover two equal tracks at right angles to each other. When one track\nwas situated in the direction of the earth's motion they expected to\nget the same result as when two scullers of equal prowess are racing\nin a river, one up and down the stream and the other across and back;\nthe winner will be the sculler rowing across the stream, as working\nout an example will convince. Even if the earth had been stationary\nat the time of one experiment, the earth's motion round the sun would\nhave been reversed 6 months later and would then have given double the\neffect. They found, however, that the two rays of light arrived always\nan exact dead heat. All experimenters who have tried since have arrived\nat the same result and found it impossible to detect absolute motion.\n\nThe principle of relativity has its foundation in fact on these\nfailures to detect absolute motion. This principle states that the\nonly motion we can ever know about is relative motion. If we devise\nan experiment which ought to reveal absolute motion, nature will\nenter into a conspiracy to defeat us. In the Michelson and Morley\nexperiment the conspiracy was that the track in the direction of the\nearth's absolute motion should contract its length by just so much\nas would allow the ray of light along it to arrive up to time.\n\nWe see, therefore, that according to the principle of relativity\nmotion must always remain a relative term, in much the same way as\nvertical and horizontal, right and left, are relative terms having\nonly meaning when referred to some observer. We do not expect to\nfind an absolute vertical and are wise enough not to attempt it;\nin seeking to find absolute motion physicists were not so wise and\nonly found themselves baffled.\n\nThe principle that all motion is relative now requires to be worked\nout to all its consequences, as has been done by Einstein, and we\nhave his theory of relativity. Einstein conceives a world of four\ndimensions built up of the three dimensions of space, namely up and\ndown, backwards and forwards, right and left, with time as the fourth\ndimension. This is an unusual conception to most of us, so let us\nsimplify it into something which we can more easily picture but which\nwill still allow us to grasp Einstein's ideas. Let us confine ourselves\nfor the present to events which happen on this sheet of paper, i.e.,\nto space of two dimensions only and take time as our third dimension\nat right angles to the plane of the paper. We have thus built up a\nthree dimensional world of space-time which is every bit as useful\nto us as a four dimensional representation so long as we only need\nstudy objects moving over the sheet of paper.\n\nSuppose a fly is crawling over this sheet of paper and let us make\na movie record of it. If we cut up the strip of movie film into the\nindividual pictures and cement them together one above another in\ntheir proper order, we shall build up a solid block of film which\nwill be a model of our simplified world of space-time and in which\nthere will be a series of dots representing the motion of the fly\nover the paper. Just as I can state the exact position of an object\nin my room by defining its height above the floor, its distance\nfrom the north wall and its distance from the east wall, so we can\nreduce the positions of the dots to figures for use in calculations\nby measuring their distances from the three faces intersecting in the\nlines $OX$, $OY$, and $OT$, where $OXAYTBCD$ represents the block of\nfilm. The mathematician would call the three lines $OX$, $OY$, $OT$\nthe coordinate axes. Measuring all the dots in this way we shall\nobtain the motion of the fly relative to the coordinate axes $OX$,\n$OY$, $OT$. If we add a block $OTDYEFGH$ of plain film we can use\n$EX$, $EH$, $EF$ as coordinate axes and again obtain the motion of\nthe fly relative to these new axes; or we can add block after block\nso as to keep the axes moving. We can conceive of other changes of\naxes. The operator making the movie record might have taken the fly\nfor the hero of the piece and moved the camera about so as to keep the\nfly more or less central in the picture; or he might, by turning the\nhandle first fast and then slow and by moving the camera, have made\nthe fly appear to be doing stunts. Moving the camera would change the\naxes of $x$ and $y$, and turning the handle at different speeds would\nchange the axis of time. Again, we might change the axes by pushing\nthe block out of shape or by distorting it into a state of strain.\nWhatever change of axes we make, any dot in the block of film will\nsignify a coincidence of the fly with a certain point of the paper\nat a certain time, and the series of dots will, in every case, be a\nrepresentation of the motion of the fly. Maybe the representation\nwill be a distorted one, but who is to say which is the absolutely\nundistorted representation? The principle of relativity which we laid\ndown before says that no one set of coordinates will give the absolute\nmotion of the fly, so that one set is as good as another. The principle\nthat all motion is relative means, therefore, that no matter how we\nchange our coordinates of space-time, the laws of motion which we\ndeduce must be the same for all changes.\n\nTo use an analogy, the sculptured head of Shakespeare on my table may\nappear to have hollow cheeks when I admit light from the east window\nonly, or to have sunken eyes with light from the skylight in the roof,\nbut the true shape of the head remains the same in all lights.\n\nHence, if with reference to two consecutive dots in our block of\nfilm a mathematical quantity can be found which will not change no\nmatter how we changes our axes of coordinates, that quantity must\nbe an expression of the true law of motion of the fly between the\ntwo points of the paper and the two times represented by these two\ndots. Einstein has worked out such a quantity remaining constant for\nall changes of coordinates of the four dimensional world of space-time.\n\nIn passing we may notice a feature of Einstein's world of space-time\nwhich we shall doubtless find it difficult to conceive, namely,\nthat there is no essential difference between a time and a distance\nin space. Since one set of coordinates is as good as another, we\ncan transform time into space and space into time according as we\nchoose our axes. For example if we change $OX$, $OT$, the axes of $x$\nand time in Fig. 2, into $OX'$, $OT'$ by a simple rotation, the new\ntime represented by $OT'$ consists partly of $OA$ in the old time\nand partly of $OB in the old $x$ direction. Referring to our block\nof movie film again, it means that although I might separate the\nblock into space and time by slicing it into the original pictures,\nI can just as readily slice it in any direction I choose and still\nget individual pictures representing the motion of the fly but with,\nof course, new time and space. So whilst I may be believe that a liner\nhas travelled 3,000 miles in 4 days, an observer on a star who knows\nnothing of my particular axes in space-time may say, with equal truth,\nthat it went 2,000 miles in 7 days. Thus, time and space are not two\nseparate identities in Einstein's view; there only exists a world of\nfour dimensions which we can split up into time and space as we choose.\n\nLet us see now how Einstein explains gravitation. When a body is not\nacted on by any forces (except gravitation) the quantity which remains\nconstant for all changes of coordinates implies that the body will\nfollow that path in the space of an outside observer which takes the\nleast time. It is an observed fact that one body attracts another by\ngravitation; that is, the path of one body is bent from its course by\nthe presence of another. Now we can bend the path of the fly in our\nblock of film by straining the block in some way. Suppose, therefore,\nthat I strain the world so as to bend the path of a body exactly as\nthe gravitation due to some other body bends it; i.e., by a change\nof coordinates I have obtained the same effect as that produced\nby gravitation. Einstein's theory, therefore, explains gravitation\nas a distortion of the world of space-time due to the presence of\nmatter. Suppose first that a body is moving with no other bodies near;\naccording to Einstein it will take the path in space which requires the\nleast time, i.e., a straight line as agrees with our experience. If now\nthe world be strained by the presence of another body or by a change\nof coordinates it will still pursue the path of least time, but this\npath is now distorted from the straight line, just as in a similar way\nthe path on a globe requiring the least time to travel follows a great\ncircle. So, on Einstein's view of gravitation, the earth moves in an\nelliptical path around the sun not because a force is acting on it,\nbut because the world of space-time is so distorted by the presence of\nthe sun that the path of least time through space is the elliptical\npath observed. There is, therefore, no need to introduce any idea of\n\"force\" of gravitation. Einstein's theory explains gravitation only\nin the sense that he has explained it away as a force of nature and\nmakes it a property of space-time, namely, a distortion not different\nfrom an appropriate change of coordinates. He does not, however,\nexplain how or why a body can distort space-time. It is noteworthy\nthat whilst the law of gravitation and the law of uniform motion in\na straight line when no force is acting were separate and independent\nlaws under Newton, Einstein finds one explanation for both under the\nprinciple of relativity. [12]\n\n\n\n\n\n\n\n\n\nXX\n\nEINSTEIN'S THEORY OF GRAVITATION\n\nThe Discussion of the General Theory and Its Most Important\nApplication, from the Essay by\n\nPROF. W. F. G. SWANN, UNIVERSITY OF MINNESOTA, MINNEAPOLIS\n\n\nNewton's great discovery regarding the motion of the planets consisted\nin his showing that these could all be summed up in the following\nstatement: consider any planet in its relation to all particles in the\nuniverse. Write down, for the planet, in the line joining it to any\nparticle, an acceleration proportional to the mass of the particle and\nto the inverse square of its distance from the planet. Then calculate\nthe planet's resultant acceleration by combining all the accelerations\nthus obtained.\n\nWe have here purposely avoided the use of the word \"force,\" for\nNewton's law is complete as a practical statement of fact without it;\nand this word adds nothing to the law by way of enhancing its power\nin actual use. Nevertheless, the fact that the acceleration is made\nup as it were of non-interfering contributions from each particle\nin the line joining it to the planet strongly suggests to the mind\nsomething of the nature of an elastic pull for which the particle is\nresponsible, and to which the planet's departure from a straight-line\nmotion is due. The mind likes to think of the elastic; ever since the\ntime of Newton people have sought to devise some mechanism by which\nthese pulls might be visualized as responsible for the phenomena in\nthe same way as one pictures an elastic thread as controlling the\nmotion of a stone which swings around at its end.\n\nThis search has been always without success; and now Einstein has\nfound a rather different law which fits the facts better than Newton's\nlaw. It is of such a type that it does not lend itself conveniently\nto expression in terms of force; the mind would gain nothing by\ntrying to picture such forces as are necessary. It compensates for\nthis, however, in being capable of visualization in terms of what is\nultimately a much simpler concept.\n\nIn order to appreciate the fundamental ideas involved, suppose for\na moment that gravitation could be annihilated, completely, and\nsuppose I find myself upon this earth in empty space. You shall be\nseated at some point in space and shall watch my doings. If I am in\nthe condition of mind of the people of the reign of King Henry VIII,\nI shall believe that the earth does not rotate. If I let go a stone,\nthere being no gravity, I shall find that it flies away from me with\nan acceleration. You will know, however, that the stone really moves\nin a straight line with constant velocity, and that the apparent\nacceleration which I perceive is due to the earth's rotation. If I\nhave argued that acceleration is due to force, I shall say that the\nearth repels the stone, and shall try to find the law governing the\nvariation of this force with distance. I may go farther, and try to\nimagine some reason for the force, some pushing action transmitted from\nthe earth to the stone through a surrounding medium; and, you will\npity me for all this wasted labor, and particularly for my attempt\nto find a mechanism to account for the force, since you know that if\nI would only accept your measurements all would appear so simple.\n\nLet us probe this matter a little farther, however, from the\nstand-point of myself. I must believe in the reality of the force,\nsince I have to be tied to my chair to prevent my departure from\nthe earth. I might wonder how this field of force would affect the\npropagation of light, chemical action and so forth. For, even though I\nhad discovered that, by using your measures, I could transform away the\napparent effects of my field of force as far as concerned its power\nto hurl stones about, I could still regard this as a mathematical\naccident, and believe that the force was really there. Although I\nmight suspect that the same transformation of view-point that would\nannul the field's effect as regards the stones would also annul its\neffect as regards light, etc., I should not be sure of this, as you\nwould be; and my conscience would hardly allow me to do more than\nlook upon the assumption of complete equivalence between the apparent\nfield and a change in the system of measurement as a hypothesis. I\nshould be strongly tempted to make the hypothesis, however.\n\nNow the question raised by Einstein is whether the force of gravity,\nwhich we experience as a very real thing, may be put upon a footing\nwhich is in some way analogous to that of the obviously fictitious\ncentrifugal force cited above: whether gravitation may be regarded as\na figment of our imagination engendered by the way in which we measure\nthings. He found that it could be so regarded. He went still farther,\nand in his Principle of Equivalence, he postulated that the apparent\neffects of gravitation in all phenomena could be attributed to the same\nchange in the system of our measurements that would account for the\nordinary phenomena of gravitation. On the basis of this hypothesis\nhe was able to deduce for subsequent experimental verification,\nthe effects of gravitation on light. He did not limit himself to\nsuch simple changes in our measurements as were sufficient to serve\nthe purpose of the problem of centrifugal force cited above; but,\nemboldened by the assumptions, in the older theory of relativity,\nof change in standards of length and time on account of motion,\nhe went even farther than this, and considered the possibility of\nchange of our measures due to mere proximity to matter.\n\nHis problem amounted to an attempt to find some way in which it was\npossible to conceive our scales and clocks as altered, relatively to\nsome more fundamental set, so as to allow of the planetary motions\nbeing uniform and rectilinear with respect to these fundamental\nmeasures, although they appear as they do to us. If we allow our\nimaginations perfect freedom as to how the scales may be altered, we\nshall not balk at assuming alterations varying in any way we please,\nEinstein does, however, introduce restrictions for reasons which we\nwill now discern.\n\nIf we imagine our whole universe, with its observers, planetary\norbits, instruments, and everything else, embedded in a jelly, and\nthen distort the jelly and contents in any way, the numbers at which\nour planetary orbits (or rather their telescopic images) intersect\nour scales will be unaltered. Moreover, we could vary, in any manner,\nthe times at which all objects (including the clock hands) occupied\ntheir distorted positions, and the hand of some clock near the point\nwhere the planetary image crossed the scale would record for this\noccurrence the same dial reading as before. An inhabitant of this\ndistorted universe would be absolutely unconscious of the change. Now\nthe General Theory of Relativity which expresses itself in slightly\nvaried forms, amounts to satisfying a certain philosophical craving\nof the mind, by asserting that the laws of nature which control our\nuniverse ought to be such that another universe like the above, whose\ninhabitants would be unconscious of their change, would also satisfy\nthese laws, not merely from the standpoint of its own inhabitants,\nbut also from the standpoint of our measurements. In other words,\nthis second universe ought to appear possible to us as well as to\nits inhabitants.\n\nEinstein decides to make his theory conform to this philosophical\ndesire, and this greatly limits the modifications of clocks and\nscales which he permits himself for the purpose of representing\ngravitation. Further, if we express the alterations of the measures\nas functions of proximity to matter, velocity and so forth, our\nexpressions for these alterations will include, as a particular case,\nthat where matter is absent, although the scales and observer may still\nremain. Our alteration of the scales and clocks with velocity must thus\nrevert, for this case, to that corresponding to the older theory of\nrelativity, in order to avoid predicting that two observers, in uniform\nmotion relative to each other in empty space, will measure different\nvalues for the velocity of light. In this way, the velocity of light\ncomes to play a part in expressing the alterations of the measures.\n\nEven with these restrictions, Einstein was able to do the equivalent\nof finding an alteration of scales and clocks in the presence of\nmatter which would account for our finding that the planetary motions\ntake place very nearly in accordance with Newton's law. The new law\nhas accounted with surprising accuracy for certain astronomical\nirregularities for which Newton's law failed to account, and has\npredicted at least one previously unknown phenomenon which was\nimmediately verified.\n\nIn conclusion, it may be of interest to state how the new law describes\nthe motion of a particle in the vicinity of a body like the earth. The\nlaw amounts to stating that, if we measure a short distance, radially\nas regards the earth's center, we must allow for the peculiarity of\nour units by dividing by\n\n    $$\\sqrt{1-\\frac{2mG}{c^2r}}$$\n\nwhere $r$ is the distance from the earth's center, $m$ the mass\nof the earth, $c$ the velocity of light, and $G$ the Newtonian\ngravitational constant. Tangential measurements require no correction,\nbut intervals of time as measured by our clocks must be multiplied,\nfor each particular place, by the above factor. Then, in terms of\nthe corrected measures so obtained, the particle will be found to\ndescribe a straight line with constant velocity although, in terms\nof our actual measures, it appears to fall with an acceleration.\n\n\n\n\n\n\n\n\n\nXXI\n\nTHE EQUIVALENCE HYPOTHESIS\n\nThe Discussion of This, With Its Difficulties and the Manner in Which\nEinstein Has Resolved Them, from the Essay by\n\nPROF. E. N. DA C. ANDRADE, ORDNANCE COLLEGE, WOOLWICH, ENGLAND\n\n\nHaving shown that, of several systems all moving with reference to one\nanother with uniform motion, no one is entitled to any preference over\nthe others, and having deduced the laws for such systems, Einstein\nwas confronted with a difficulty which had long been felt. A body\nrotating, which is a special case of an accelerated body, can be\ndistinguished from one at rest, without looking outside it, by the\nexistence of the so-called centrifugal forces.\n\nThis circumstance, which gives certain bodies an absolute or\npreferential motion, is unpalatable to the relativist; he would\nlike there to be no difference as regards forces [13] between the\ncase when the earth rotates with reference to outside bodies (the\nstars) considered as fixed, and the case when the earth is considered\nfixed and all outside bodies rotate around it. This point cannot be\ninvestigated by direct experiment; we can spin a top but we cannot\nkeep a top at rest and spin the world round it, to see if the forces\nare same.\n\nIn considering the problem of how to devise laws which should make all\nrotations relative, Einstein conceived the brilliant yet simple idea\nthat gravitation could be brought into the scheme as an acceleration\neffect, since both ordinary accelerational forces and gravitational\nforces are proportional to the same thing, the mass of a body. The\nimpossibility of separating the two kinds of effect can be easily\nseen by considering the starting of an elevator. When the elevator\nis quickly accelerated upwards we feel a downward pull, just as if\nthe gravitational pull had been increased, and if the acceleration\ncontinued to be uniform, bodies tested with a spring balance would\nall weigh more in the elevator than they did on firm ground. In a\nsimilar way the whole of the gravitational pull may be considered to\nbe an accelerational effect, the difficulty being to devise laws of\nmotion which will give the effects that we find by actual observation.\n\nBut it is obvious that we cannot, by ordinary mechanics, consider the\nearth as being accelerated in all directions, which we should have to\ndo, apparently, to account for the fact that the gravitational pull\nis always toward the center. [It is obvious that we cannot explain\ngravitation by assuming that the earth's surface is continually moving\noutward with an accelerated velocity.]227 So Einstein found that, as\nlong as we treat the problem by Euclid's geometry, we cannot reach\na satisfactory solution. But he found that to the four-dimensional\nspace made up of the three ordinary dimensions of space, together with\nthe time-dimension which we have already mentioned in discussing the\nspecial theory, may be attributed a peculiar geometry, the nature of\nwhich departs more and more from Euclidean geometry as we approach a\ngravitational body, and the net result of which is to make possible\nthe universal correspondence of gravitation and acceleration.\n\nThis modification of the geometry of space is often spoken of as the\n\"curvature of space,\" an expression which is puzzling, especially as\nthe space which is \"curved\" is four-dimensional time-space. But we\ncan get an idea of what is meant by considering figures, triangles\nsay, drawn on the surface, of a sphere. These triangles, although\ndrawn on a surface, will not have the same properties as triangles\ndrawn on flat paper--their three angles will not together equal right\nangles. They will be non-Euclidean. This is only a rough analogy, but\nwe can see that the curvature of the surface causes a departure from\nEuclidean geometry for plane figures, and consequently the departure\nfrom Euclidean laws extended to four dimensions may be referred to\nas caused by \"curvature of space.\"\n\nIt is difficult to imagine a lump of matter affecting the geometry of\nthe space round it. Once more we must use a rough illustration. Imagine\na very hot body, and that, knowing nothing of its properties, we\nhave to measure up the space round it with metal measuring-rods. The\nnearer we are to the body, the longer the rods will become, owing to\nthe expansion of the metal. When we measure out a square, one side\nof which is nearer the body than the opposite side, its angles will\nnot be right angles. If we knew nothing of the laws of heat we should\nsay that the body had made the space round it non-Euclidean.\n\nEinstein found, then, that by taking the properties of space, as given\nby measurement, to be modified in the neighborhood of masses of matter,\nhe could devise general laws according to which gravitational effects\nwould be produced, and there would be no absolute rotation. All\nforces will be the same whether a body rotates with everything\noutside it fixed, or the body is fixed, and everything rotates\nround it. All motion is then relative, and the theory is one of\n\"general relativity.\" The velocity of light is, however, no longer\nconstant, and its path is not a straight line, if it is passing near\ngravitating matter. This does not contradict the special theory,\nwhich did not allow for gravitation. Rather, the special theory is\na particular case to which the generalised theory reduces when there\nis no matter about, just as the Newtonian dynamics is a special case\nof the special theory, which we obtain when all velocities are small\ncompared to that of light.\n\n\n\n\n\n\n\n\n\nXXII\n\nTHE GENERAL THEORY\n\nFragments of Particular Merit on This Phase of the Subject\n\nBY VARIOUS CONTRIBUTORS\n\n\nWhen Dorothy was carried by the cyclone from her home in Kansas to the\nland of Oz, together with her uncle's house and her little dog Toto,\nshe neglected to lower the trap door over the hole in the floor which\nformerly led to the cyclone cellar and Toto stepped through. Dorothy\nrushed to the opening expecting to see him dashed onto the rocks below\nbut found him floating just below the floor. She drew him back into\nthe room and closed the trap.\n\nThe author of the chronicle of Dorothy's adventures explains that the\nsame force which held up the house held up Toto but this explanation\nis not necessary. Dorothy was now floating through space and house\nand dog were subject to the same forces of gravitation which gave\nthem identical motions. Dorothy must have pushed the dog down onto the\nfloor and in doing so must herself have floated to the ceiling whence\nshe might have pushed herself back to the floor. In fact gravitation\nwas apparently suspended and Dorothy was in a position to have tried\ncertain experiments which Einstein has never tried because he was\nnever in Dorothy's unique position.]188\n\n\n\nThe Principle of Equivalence, of which Einstein's suspended cage\nexperiment is the usual illustration, and upon which the generalized\ntheory of relativity is built, is thus stated by Prof. Eddington: \"A\ngravitational field of force is precisely equivalent to an artificial\nfield of force, so that in any small region it is impossible, by any\nconceivable experiment, to distinguish between them. In other words,\nforce is purely relative.\"\n\nThis may be otherwise stated by going back to our idea of a\nfour-dimensional world, the points of which represent the positions\nand times of events. If we mark in such a space-time the successive\npositions of an object we get a line, or curve, which represents the\nwhole history of the object, inasmuch as it shows us the position\nof the object at every time. The reader may imagine that all events\nhappen in one plane, so that only two perpendicular dimensions\nare needed to fix positions in space, with a third perpendicular\ndimension for time. He may then conceive, if he may not picture,\nan analogous process for four-dimensional space-time. These lines,\n\"tracks of objects through space-time,\" were called by Minkowski\n\"world-lines.\" We may now say that all the events we observe are the\nintersections of world-lines. The temperature at noon was 70\u00b0. This\nmeans that if I plot the world-line of the top of the mercury column\nand the world-line of a certain mark on the glass they intersect in\na certain point of space-time. All that we know are intersections of\nthese world-lines. Suppose now we have a large number of them drawn\nin our four-dimensional world, satisfying all known intersections,\nand let us suppose the whole embedded in a jelly. We may distort\nthis jelly in any way, changing our coordinates as we please, but we\nshall neither destroy nor create intersections of world-lines. It\nmay be proved that a change from one system of reference, to which\nobservations are referred, to any other system, moving in any way\nwith respect to the first system, may be pictured as a distortion of\nthe four-dimensional jelly. The laws of nature, therefore, being laws\nthat describe intersections, must be expressible in a form independent\nof the reference system chosen.\n\nFrom these postulates, Einstein was able to show such a formulation\npossible. His law may be stated very simply:--All bodies move through\nspace-time in the straightest possible tracks.\n\nThe fact that an easy non-mathematical explanation can not be given,\nof how this law is reached, or of just why the straightest track of\nMercury through space-time will give us an ellipse in space after we\nhave split space-time up into space and time, is no valid objection to\nthe theory. Newton's law that bodies attract with a force proportional\nto their masses and inversely proportional to the square of the\ndistance is simple, but no one has ever given an easy non-mathematical\nproof of how that law requires the path of Mercury to be an ellipse,\nwith the sun at a focus, instead of some other curve.]182\n\n\n\nOne of the grave difficulties we have in gaining a satisfactory\ncomprehension of Einstein's conceptions, is that they do not readily\nrelate themselves to our modes of geometrical thought. Within limits\nwe may choose our own geometry, but it may be at the cost of unwieldy\ncomplication. If we think with Newton in terms of Euclidean geometry\nand consider the earth as revolving around the sun, the motions of\nour solar system can be stated in comparatively simple terms. If,\non the other hand, we should persist in stating them, as Ptolemy\nwould have done, from the earth as a relatively stationary center,\nour formulas will become complicated beyond ready comprehension. For\nthis reason it is much simpler in applying the theory of relativity,\nand in considering and describing what actually happens in the physical\nuniverse, to use geometrical conceptions to which the actual conditions\ncan be easily related. We find such an instrument in non-Euclidean\ngeometry, wherein space will appear as though it were projected\nfrom a slightly concave mirror. It is in this sense that some speak\nof space as curved. The analogy is so suggestive it tempts one to\nlinger over it. Unless there were material objects within the range\nof the mirror, its conformation would be immaterial; the thought of\nthe space which the mirror, as it were, circumscribes, is dependent\nupon the presence of such material objects. The lines of light and of\nall other movement will not be quite \"straight\" from the view-point\nof Euclidean geometry. A line drawn in a universe of such a nature\nmust inevitably return upon itself. Nothing therefore, can ever pass\nout of this unlimitedly great but yet finite cosmos. But even now,\nsince our imaginary mirror is only very slightly concave, it follows\nthat for limited regions like the earth or even the solar system, our\nconception of geometry may well be rectilinear and Euclidean. Newton's\nlaw of gravitation will be quite accurate with only a theoretical\nmodification drawn from the theory of relativity.]82\n\n\n\nThe way in which a curvature of space might appear to us as a force\nis made plainer by an example. Suppose that in a certain room a\nmarble dropped anywhere on the floor always rolled to the center of\nthe room; suppose the same thing happened to a baseball, a billiard\nball, and a tennis ball. These results could be explained in two\nways; we might assume that a mysterious force of attraction existed\nat the center of the floor, which affected all kinds of balls alike;\nor we might assume that the floor was curved. We naturally prefer the\nlatter explanation. But when we find that in the neighborhood of a\nlarge material body all other bodies move toward it in exactly the\nsame manner, regardless of their nature or their condition, we are\naccustomed to postulate a mysterious attractive force (gravitation);\nEinstein, on the contrary, adopts the other alternative, that the\nspace around the body is curved.]223\n\n\n\nIn the ordinary \"analytical geometry,\" the position and motion of\nall the points considered is referred to a rigid \"body\" or \"frame of\nreference.\" This usually consists of an imaginary room of suitable\nsize. The position of any point is then given by three numbers, i.e.,\nits distances from one side wall and from the back wall and its height\nabove the floor. These three numbers can only give one point, every\nother point having at least one number different. In four-dimensional\ngeometry a fourth wall may be vaguely imagined as perpendicular to\nall three walls, and a fourth number added, giving the distance of\nthe \"point\" from this wall also. Since \"rigid\" bodies do not exist in\ngravitational fields the \"frame of reference\" must be \"non-rigid.\" The\nframe of reference in the Gaussian system need not be rigid, it can\nbe of any shape and moving in any manner, in fact a kind of jelly. A\n\"point\" or \"event\" in the four dimensioned world is still given by four\nnumbers but these numbers do not represent distances from anywhere;\nall that is necessary is that no two events shall have exactly the\nsame four numbers to represent them, and that two events which are\nvery close together shall be represented by numbers which differ only\nslightly from one another. This system assumes so little that it will\nbe seen to be very wide in its scope; although to the ordinary mind,\nwhat is gained in scope seems more than that lost in concreteness. This\ndoes not concern the mathematician, however, and by using this system\nhe gains his object, proving that the general laws of nature remain\nthe same when expressed in any Gaussian coordinate system whatever.]220\n\n\n\nEinstein enunciates a general principle that it is possible to find\na transformation of coordinate axes which is exactly equivalent\nto any force, and in particular one which is equivalent to the\nforce of gravitation. That is by concentrating our attention on\nthe transformation which is a purely mathematical operation we can\nafford to neglect the force completely. To get a better idea of this\nprinciple of equivalence as it is called, let us consider a relatively\nsimple example (which actually has nothing to do with gravitation,\nbut which will serve to make our notions clearer.) A person on the\nearth unconsciously refers all his experiences, i.e., the motions\nof the objects around him to a set of axes fixed in the earth on\nwhich he stands. However, we know that the earth is rotating about\nits axis, and his axes of reference are also rotating with respect to\nthe space about him. From the point of view of general relativity it\nis exactly because we do refer motions on the surface of the earth\nto axes rotating with the earth that we experience the so-called\ncentrifugal force of the earth's rotation, with which everyone is\nfamiliar. If we could find it convenient to transform from moving axes\nto fixed axes, the force would vanish, since it is exactly equivalent\nto the transformation from one set of axes to the other. However,\nwe find it unnatural to refer daily experiences to axes that are not\nplaced where we happen to be, and so we prefer to take the force and\nrotating axes instead of no force and fixed axes.]272\n\n\n\nWe seem to have a direct experience of force in our muscular\nsensations. By pushing or pulling we can set bodies in motion. It\nis natural to assume, that something similar occurs, when Nature set\nbodies in motion. But is this not a relic of animism? The savage and\nthe ancients peopled all the woods and skies with Gods and demons, who\ncarries on the activities of nature by their own bodily efforts. Today\nwe have dispossessed the demons, but the ghost of a muscular pull\nstill holds the planets in place.]141\n\n\n\nThe general theory is an extension of the special theory which enables\nthe law of gravitation to be deduced. Not in Newton's form, it is true,\nbut in a better form, that is, one that accounts for two important\nfacts otherwise not explained. But it is a far more general theory\nthat indicated above. It is a complete study of the relations between\nlaws expressed by means of any four coordinates (of which three space\nand one time is a special case), and the same laws expressed in the\nfour coordinates of a system having any motion whatever with respect\nto the first system. By restricting this general study in accordance\nwith certain postulates about the nature of the universe we live in,\nwe arrive at a number of conclusions which fit more closely with\nobserved facts that the conclusions drawn from Newton's theory.]221\n\n\n\n\n\n\n\n\n\nNOTES\n\n\n[1] It will be noted that Mr. Bolton pronounces the geometry of\nspace to be Euclidean in the absence of gravitational fields, not\nthat of space-time. This is in accord with what was pointed out on\npage 161.--Editor.\n\n[2] The author here comes perilously close to ascribing to this\n\"contraction\" the sort of physical reality which it does not\npossess. See page 96.--Editor.\n\n[3] Dr. Davis went rather fully into the algebra of the\nMichelson-Morley experiment. But Dr. Russell has covered the same\nground in a form somewhat more advantageous from the typographical\nviewpoint, and the point is not one which it is profitable to discuss\ntwice; so we eliminate this part of Dr. Davis' text.--Editor.\n\n[4] This statement is objectionable, as explained in Chapter\nIV.--Editor.\n\n[5] A. Einstein: Die Grundlage der allgemeinen\nRelativit\u00e4tstheorie. Ann. d. Physik. 4, vol. 49, page 776.\n\n[6] At this point we have again used the blue pencil on Dr. Davis'\ntext, his discussion of the three observational tests of the General\nTheory adding nothing to Dr. Pickering's.--The Editor.\n\n[7] Commander McHardy uses the term \"event\" in a sense somewhat\ndifferent from that seen in a majority of the essays. He reserves\nfor the four-dimensional element--the instant of time at a point in\nspace--the name \"point-event\"; and the term \"event\" he applies to a\ncollection of these forming, together, an observable whole. An actual\nphysical happening, like a railroad wreck or a laboratory experiment,\nit will be realized is of the latter sort, occupying an appreciable\nregion of space rather than a single point, and an appreciable interval\nof time rather than a single second. To the element, the \"point-event\"\nof Commander McHardy's essay, this bears the same relation that the\ngeometer's solid bears to his point. This comment is in no sense\nto be taken as criticism of Commander McHardy's terminology, which\nrather appeals to us; we make it merely to guard against confusion\nin the reader's mind.--Editor.\n\n[8] This paragraph is the result of an editorial revision\nof the author's text, designed to retain the substance of his\npresentation, while tying up what he has to say more definitely with\nthe preceding essays, and eliminating the distinction between finite\nand infinitesimal intervals, which we believe to be out of place in\nan essay of this character. We will not apologize to our mathematical\nreaders for having used finite and differential notation in the same\nequation, in violation of mathematical convention.--Editor.\n\n[9] Although gravitational force in a small region can be imitated\nor annulled by accelerating motion, there remains the disturbing\ninfluence of gravitational matter already referred to and expressed in\nthe fabric curvature. It is this that defines how unique tracks run,\nor rather, how bodies progress.--Author.\n\n[10] Not all gravitational fields may be transformed away by a\nproper choice of coordinates. If this were so, the space, whose\nnature is independent of any choice of coordinates, would always\nbe Euclidean.--Author.\n\n[11] Thus when it it said that a body contracts or that a clock runs\nslow when it is put in motion no actual physical change is implied. The\njudgment of different observers--one at rest with respect to the body\nand one not--are different.--Author.\n\n[12] The balance of Dr. Royds' essay is given to a discussion of\nthe phenomena of Mercury's perihelial advance, the deflection of\nlight under the gravitational field of the sun, and the shift in\nspectral lines, in connection with which alone Einstein's theory\nmakes predictions which are sufficiently at variance with those\nof Newtonian science to be of value in checking up the theory\nobservationally. In the interest of space conservation and in the\npresence of Dr. Pickering's very complete discussion of these matters\nwe omit Dr. Royds' statement.--Editor.\n\n[13] There is, on any view, no difference as regards observation of\nposition only.--Author.","title":"Einstein's Theories of Relativity and Gravitation: A selection of material from the essays submitted in the competition for the Eugene Higgins prize of $5,000","tradition":"philosophy-ethics","translated_text":null,"word_count":92370}