Relativity: The Special and General TheoryAlbertEinstein1916ContentsI TheSpecialTheoryofRelativity 51 PhysicalMeaningofGeometricalPropositions 62 TheSystemofCo-ordinates 83 SpaceandTimeinClassicalMechanics 114 TheGalileanSystemofCo-ordinates 135 ThePrincipleofRelativityintheRestrictedSense 146 The Theoremof the Addition of Velocities Employed inClassicalMechanics 177 TheApparentIncompatabilityoftheLawofPropagationofLightwiththePrincipleofRelativity 188 OntheIdeaofTimeinPhysics 219 TheRelativityofSimulatneity 2410OntheRelativityoftheConceptionofDistance 2711TheLorentzTransformation 2912TheBehaviourofMeasuring-RodsandClocksinMotion 3313TheoremoftheAdditionofVelocities.TheExperimentofFizeau 3514TheHeuristicValueoftheTheoryofRelativity 38115GeneralResultsoftheTheory 4016ExperienceandtheSpecialTheoryofRelativity 4417MinkowskisFour-DimensionalSpace 48II ThenGeneralTheoryofRelativity 5118SpecialandGeneralPrincipleofRelativity 5219TheGravitationalField 5520TheEqualityof Inertial andGravitational MassasanAr-gumentfortheGeneralPostuleofRelativity 5721In What Respects Are the Foundations of Classical Mechan-icsandoftheSpecialTheoryofRelativityUnsatisfactory? 6022AFewInferencesfromtheGeneralPrincipleofRelativity 6223Behaviour of Clocks and Measuring-Rods on a RotatingBodyofReference 6524EuclideanandNon-EuclideanContinuum 6825GaussianCo-Ordinates 7126TheSpace-TimeContinuumoftheSpeicalTheoryofRela-tivityConsideredasaEuclideanContinuum 7427TheSpace-TimeContinuumoftheGeneral TheoryofRel-ativityisNotaEuclideanContinuum 7628ExactFormulationoftheGeneralPrincipleofRelativity 7929TheSolutionoftheProblemofGravitationontheBasisoftheGeneralPrincipleofRelativity 82III ConsiderationsontheUniverseasaWhole 8630CosmologicalDicultiesofNewtonsTheory 87231The Possibilityof a Finiteandyet UnboundedUni-verse 8932TheStructureofSpaceAccordingtotheGeneralTheoryofRelativity 93ASimpleDerivationof theLorentzTransformation(Supple-mentarytoSection11) 95BMINKOWSKIS FOUR-DIMENSIONAL SPACE (WORLD)(SUPPLEMENTARYTOSECTION17) 100CTheExperimental Conrmationof theGeneral TheoryofRelativity 102a. MotionofthePerihelionofMercury . . . . . . . . . . . . . . 103b. DeectionofLightbyaGravitationalField . . . . . . . . . . 104c. DisplacementofSpectralLinesTowardstheRed . . . . . . . 106DTheStructureofSpaceAccordingtotheGeneralTheoryofRelativity(SupplementarytoSection32) 1103(December,1916)The present book is intended,as far as possible,to give an exact insightintothetheoryofRelativitytothosereaderswho,fromageneralscienticandphilosophical pointof view, areinterestedinthetheory, butwhoarenot conversant with the mathematical apparatus of theoretical physics. Theworkpresumesastandardof educationcorrespondingtothatof auniver-sity matriculation examination, and, despite the shortness of the book, a fairamountofpatienceandforceofwillonthepartofthereader. Theauthorhassparedhimself nopainsinhisendeavourtopresentthemainideasinthesimplestandmostintelligibleform, andonthewhole, inthesequenceandconnectioninwhichtheyactuallyoriginated. Intheinterestof clear-ness, itappearedtomeinevitablethatI shouldrepeatmyself frequently,without paying the slightest attentionto the elegance ofthe presentation. Iadhered scrupulously to the precept of that brilliant theoretical physicist L.Boltzmann, accordingtowhommattersofeleganceoughttobelefttothetailorandtothecobbler. Imakenopretenceofhavingwithheldfromthereaderdicultieswhichareinherenttothesubject. Ontheotherhand, Ihavepurposelytreatedtheempirical physical foundationsofthetheoryinastep-motherlyfashion,sothatreadersunfamiliarwithphysicsmaynotfeel likethewandererwhowasunabletoseetheforestforthetrees. Maythebookbringsomeoneafewhappyhoursofsuggestivethought!December,1916A.EINSTEIN4PartITheSpecialTheoryofRelativity5Chapter1PhysicalMeaningofGeometricalPropositionsIn your schooldays most of you who read this book made acquaintance withthenoblebuildingofEuclidsgeometry, andyourememberperhapswithmorerespectthanlovethemagnicentstructure,ontheloftystaircaseofwhich you were chased about for uncounted hours by conscientious teachers.Byreasonofourpastexperience,youwouldcertainlyregardeveryonewithdisdainwhoshouldpronounceeventhemostout-of-the-waypropositionofthissciencetobeuntrue. Butperhapsthisfeelingofproudcertaintywouldleaveyouimmediatelyif someoneweretoaskyou: What, then, doyoumeanbytheassertionthatthesepropositionsaretrue?Letusproceedtogivethisquestionalittleconsideration.Geometrysetsoutformcertainconceptionssuchasplane,point,and straight line, with which we are able to associate more or less deniteideas, and from certain simple propositions (axioms) which, in virtue of theseideas, weareinclinedtoacceptastrue.Then, onthebasisof alogicalprocess, thejusticationofwhichwefeelourselvescompelledtoadmit, allremainingpropositions areshowntofollowfromthoseaxioms, i.e. theyareproven. Apropositionisthencorrect(true)whenithasbeenderivedintherecognisedmannerfromtheaxioms. Thequestionoftruthoftheindividualgeometricalpropositionsisthusreducedtooneofthetruthoftheaxioms. Nowithaslongbeenknownthatthelastquestionisnotonlyunanswerablebythemethodsof geometry, butthatitisinitself entirelywithoutmeaning. Wecannotaskwhetheritistruethatonlyonestraightlinegoes throughtwopoints. Wecanonlysaythat Euclideangeometrydeals withthings calledstraight lines,toeachof whichis ascribedthe6propertyof beinguniquelydeterminedbytwopointssituatedonit. Theconcepttruedoesnottallywiththeassertionsofpuregeometry,becausebythewordtrueweareeventuallyinthehabitofdesignatingalwaysthecorrespondencewitharealobject; geometry, however, isnotconcernedwith the relation of the ideas involved in it to objects of experience, but onlywiththelogicalconnectionoftheseideasamongthemselves.Itisnotdiculttounderstandwhy,inspiteofthis,wefeelconstrainedtocall thepropositionsof geometrytrue.Geometrical ideascorrespondtomoreorlessexactobjectsinnature,andtheselastareundoubtedlytheexclusive cause of the genesis of those ideas. Geometry ought to refrain fromsuchacourse, inordertogivetoitsstructurethelargestpossiblelogicalunity. The practice, for example, of seeinginadistancetwomarkedpositionsonapracticallyrigidbodyissomethingwhichislodgeddeeplyinourhabitofthought. Weareaccustomedfurthertoregardthreepointsasbeingsituatedonastraightline,iftheirapparentpositionscanbemadetocoincideforobservationwithoneeye, undersuitablechoiceofourplaceofobservation.If,inpursuanceofourhabitofthought,wenowsupplementthepropo-sitions of Euclidean geometry by the single proposition that two points on apractically rigid body always correspond to the same distance (line-interval),independently of any changes in position to which we may subject the body,the propositions of Euclidean geometry then resolve themselves into proposi-tions on the possible relative position of practically rigid bodies.1Geometrywhichhasbeensupplementedinthiswayisthentobetreatedasabranchof physics. Wecannowlegitimatelyaskastothetruthof geometricalpropositions interpreted in this way, since we are justied in asking whetherthese propositions are satised for those real things we have associated withthe geometrical ideas. In less exact terms we can express this by saying thatbythetruthofageometricalpropositioninthissenseweunderstanditsvalidityforaconstructionwithruleandcompasses.Of coursetheconvictionof thetruthof geometrical propositionsinthissenseisfoundedexclusivelyonratherincompleteexperience. Forthepresentweshall assumethetruthof thegeometrical propositions, thenatalaterstage(inthegeneral theoryof relativity)weshall seethatthistruthislimited,andweshallconsidertheextentofitslimitation.1It follows that a natural object is associated also with a straight line. Three points A,BandConarigidbodythuslieinastraightlinewhenthepointsAandCbeinggiven,BischosensuchthatthesumofthedistancesABandBCisasshortaspossible. Thisincompletesuggestionwillsuceforthepresentpurpose.7Chapter2TheSystemofCo-ordinatesOnthebasisofthephysicalinterpretationofdistancewhichhasbeenindi-cated, we are also in a position to establish the distance between two pointsonarigidbodybymeansofmeasurements. Forthispurposewerequireadistance (rod S) which is to be used once and for all, and which we employasastandardmeasure. If, now, AandBaretwopointsonarigidbody,wecanconstructthelinejoiningthemaccordingtotherulesofgeometry;then, startingfromA, wecanmarkothedistanceStimeaftertimeun-til wereachB. Thenumberof theseoperationsrequiredisthenumericalmeasure of the distance AB. This is the basis of all measurement of length.1Every description of the scene of an event or of the position of an objectinspaceisbasedonthespecicationof thepointonarigidbody(bodyof reference) withwhichthat event or object coincides. Thisappliesnotonlytoscientic description, but alsotoeverydaylife. If I analyse theplacespecicationTimesSquare,NewYork,2Iarriveatthefollowingre-sult. Theearthistherigidbodytowhichthespecicationofplacerefers;TimesSquare, NewYork,isawell-denedpoint, towhichanamehasbeenassigned,andwithwhichtheeventcoincidesinspace.3This primitive method of place specication deals only with places on thesurfaceofrigidbodies, andisdependentontheexistenceofpointsonthissurface which are distinguishable from each other. But we can free ourselvesfrom both of these limitations without altering the nature of our specicationofposition. If,forinstance,acloudishoveringoverTimesSquare,thenwecandetermineitspositionrelativetothesurfaceof theearthbyerectingapoleperpendicularlyontheSquare, sothat it reaches thecloud. Thelengthof thepolemeasuredwiththe standardmeasuring-rod, combinedwith the specication of the position of the foot of the pole, supplies us with8acompleteplacespecication. Onthebasisofthisillustration,weareabletoseethemannerinwhicharenementof theconceptionof positionhasbeendeveloped.a. We imagine the rigid body, to which the place specication is referred,supplementedinsuchamanner that the object whose positionwerequireisreachedby. thecompletedrigidbody.b. Inlocatingthepositionoftheobject,wemakeuseofanumber(herethelengthof thepolemeasuredwiththemeasuring-rod)insteadofdesignatedpointsofreference.c. We speak of the height of the cloud even when the pole which reachesthe cloud has not been erected. By means of optical observations of thecloudfromdierentpositionsontheground,andtakingintoaccountthepropertiesofthepropagationoflight,wedeterminethelengthofthepoleweshouldhaverequiredinordertoreachthecloud.From this consideration we see that it will be advantageous if, in the de-scriptionofposition,itshouldbepossiblebymeansofnumericalmeasuresto make ourselves independent of the existence of marked positions (possess-ingnames)ontherigidbodyof reference. Inthephysicsof measurementthisisattainedbytheapplicationoftheCartesiansystemofco-ordinates.This consists of threeplanesurfaces perpendicular toeachother andrigidlyattachedtoarigidbody. Referredtoasystemofco-ordinates, thescene of any event will be determined (for the main part) by the specicationof the lengths of the three perpendiculars or co-ordinates (x, y, z) whichcanbedroppedfromthesceneof theeventtothosethreeplanesurfaces.Thelengthsofthesethreeperpendicularscanbedeterminedbyaseriesofmanipulationswithrigidmeasuring-rodsperformedaccordingtotherulesandmethodslaiddownbyEuclideangeometry.In practice, the rigid surfaces which constitute the system of co-ordinatesare generally not available;furthermore, the magnitudes of the co-ordinatesarenotactuallydeterminedbyconstructionswithrigidrods, butbyindi-rectmeans. If theresultsof physicsandastronomyaretomaintaintheirclearness,thephysicalmeaningofspecicationsofpositionmustalwaysbesoughtinaccordancewiththeaboveconsiderations.4We thus obtain the following result: Every description of events in spaceinvolves the use of a rigid body to which such events have to be referred. Theresulting relationship takes for granted that the laws of Euclidean geometry9holdfordistances;thedistancebeingrepresentedphysicallybymeansoftheconventionoftwomarksonarigidbody.1Herewehaveassumedthatthereisnothingleftoveri.e. thatthemeasurementgivesawholenumber. This dicultyis got over bytheuseof dividedmeasuring-rods, theintroductionofwhichdoesnotdemandanyfundamentallynewmethod.2Einstein used Potsdamer Platz,Berlin in the original text. In the authorised trans-lation this was supplemented with Tranfalgar Square, London. We have changed this toTimes Square, New York, as this is the most well known/identiable location to Englishspeakersinthepresentday. [Notebythejanitor.]3Itisnotnecessaryheretoinvestigatefurtherthesignicanceof theexpressionco-incidenceinspace.Thisconceptionissucientlyobvioustoensurethatdierencesofopinionarescarcelylikelytoariseastoitsapplicabilityinpractice.4Arenement andmodicationof theseviews does not becomenecessaryuntil wecometodealwiththegeneraltheoryofrelativity,treatedinthesecondpartofthisbook.10Chapter3SpaceandTimeinClassicalMechanicsThe purpose of mechanics is to describe how bodies change their position inspacewithtime.Ishouldloadmyconsciencewithgravesinsagainstthesacredspiritof luciditywereItoformulatetheaimsof mechanicsinthisway, without serious reection and detailed explanations. Let us proceed todisclosethesesins.It is not clear what is to be understood here by position and space. Istand at the window of a railway carriage which is travelling uniformly,anddropastoneontheembankment,withoutthrowingit. Then,disregardingtheinuenceoftheairresistance,Iseethestonedescendinastraightline.Apedestrianwhoobservesthemisdeedfromthefootpathnoticesthatthestonefalls toearthinaparaboliccurve. I nowask: Dothepositionstraversedbythestonelieinrealityonastraightlineoronaparabola?Moreover, whatismeantherebymotioninspace? Fromtheconsidera-tionsoftheprevioussectiontheanswerisself-evident. Intherstplaceweentirelyshunthevaguewordspace,ofwhich,wemusthonestlyacknowl-edge, we cannot form the slightest conception, and we replace it by motionrelativetoapracticallyrigidbodyof reference.Thepositionsrelativetothebodyofreference(railwaycarriageorembankment)havealreadybeendenedindetailintheprecedingsection. Ifinsteadofbodyofreferenceweinsertsystemofco-ordinates,whichisausefulideaformathematicaldescription,weareinapositiontosay: Thestonetraversesastraightlinerelative toasystemof co-ordinates rigidlyattachedtothe carriage, butrelative to a system of co-ordinates rigidly attached to the ground (embank-ment) it describes aparabola. Withtheaidof this exampleit is clearly11seenthat there is nosuchthingas anindependentlyexistingtrajectory(lit. path-curve1), butonlyatrajectoryrelativetoaparticularbodyofreference.Inordertohaveacompletedescriptionofthemotion, wemustspecifyhowthe bodyalters its positionwithtime; i.e. for everypoint onthetrajectoryitmustbestatedatwhattimethebodyissituatedthere. Thesedatamustbesupplementedbysuchadenitionof timethat, invirtueofthisdenition,thesetime-valuescanberegardedessentiallyasmagnitudes(resultsof measurements)capableof observation. If wetakeourstandonthegroundof classical mechanics, wecansatisfythisrequirementforourillustrationinthe followingmanner. We imagine twoclocks of identicalconstruction; the man at the railway-carriage window is holding one of them,and the man on the footpath the other. Each of the observers determines theposition on his own reference-body occupied by the stone at each tick of theclock he is holding in his hand. In this connection we have not taken accountof the inaccuracy involved by the niteness of the velocity of propagation oflight. Withthisandwithaseconddicultyprevailinghereweshall havetodealindetaillater.1Thatis,acurvealongwhichthebodymoves.12Chapter4TheGalileanSystemofCo-ordinatesAsiswellknown,thefundamentallawofthemechanicsofGalilei-Newton,whichisknownasthelawofinertia, canbestatedthus: Abodyremovedsucientlyfarfromotherbodiescontinuesinastateofrestorofuniformmotionina straight line. This lawnot onlysays something about themotionof thebodies, butitalsoindicatesthereference-bodiesorsystemsofcoordinates, permissibleinmechanics, whichcanbeusedinmechanicaldescription. Thevisiblexedstarsarebodiesforwhichthelawof inertiacertainlyholdstoahighdegreeofapproximation. Nowifweuseasystemofco-ordinateswhichisrigidlyattachedtotheearth,then,relativetothissystem, everyxedstardescribesacircleof immenseradiusinthecourseof anastronomical day, aresultwhichisopposedtothestatementof thelawofinertia. Sothatifweadheretothislawwemustreferthesemotionsonly to systems of coordinates relative to which the xed stars do not moveinacircle. Asystemof co-ordinatesof whichthestateof motionissuchthatthelawof inertiaholdsrelativetoitiscalledaGalileiansystemofco-ordinates. The laws of the mechanics of Galei-Newton can be regardedasvalidonlyforaGalileiansystemofco-ordinates.13Chapter5ThePrincipleofRelativityintheRestrictedSenseIn order to attain the greatest possible clearness,let us return to our exam-pleoftherailwaycarriagesupposedtobetravellinguniformly. Wecallitsmotionauniformtranslation(uniformbecauseitisof constantvelocityanddirection,translationbecausealthoughthecarriagechangesitsposi-tionrelativetotheembankmentyetitdoesnotrotateinsodoing). Letusimaginearavenyingthroughtheairinsuchamannerthatitsmotion,asobserved from the embankment, is uniform and in a straight line. If we wereto observe the ying raven from the moving railway carriage. we should ndthat the motion of the raven would be one of dierent velocity and direction,butthatitwouldstill beuniformandinastraightline. Expressedinanabstractmannerwe maysay: Ifamassm ismovinguniformlyinastraightlinewithrespect toaco-ordinatesystemK, thenit will alsobemovinguniformlyandinastraightlinerelativetoasecondco-ordinatesystemK

providedthatthelatterisexecutingauniformtranslatorymotionwithre-specttoK. Inaccordancewiththediscussioncontainedintheprecedingsection,itfollowsthat:If Kis aGalileianco-ordinate system. theneveryother co-ordinatesystemK

isaGalileianone, when, inrelationtoK, itisinaconditionof uniformmotionof translation. RelativetoK

themechanical laws ofGalilei-NewtonholdgoodexactlyastheydowithrespecttoK.Weadvanceastepfarther inour generalisationwhenweexpress thetenetthus: If, relativetoK, K

isauniformlymovingco-ordinatesystemdevoid of rotation, then natural phenomena run their course with respect toK

accordingtoexactlythesamegeneral lawsaswithrespecttoK. This14statementiscalledtheprincipleofrelativity(intherestrictedsense).Aslongasonewasconvincedthatallnaturalphenomenawerecapableofrepresentationwiththehelpofclassicalmechanics,therewasnoneedtodoubtthevalidityof thisprincipleof relativity. Butinviewof themorerecent development of electrodynamics and optics it became more and moreevident that classical mechanics aords aninsucient foundationfor thephysical description of all natural phenomena. At this juncture the questionofthevalidityoftheprincipleofrelativitybecameripefordiscussion, anditdidnotappearimpossiblethattheanswertothisquestionmightbeinthenegative.Nevertheless,therearetwogeneralfactswhichattheoutsetspeakverymuchinfavourof thevalidityof theprincipleof relativity. Eventhoughclassical mechanics does not supply us with a suciently broad basis for thetheoreticalpresentationofallphysicalphenomena, stillwemustgrantitaconsiderable measure of truth, since it supplies us with the actual motionsof theheavenlybodies withadelicacyof detail littleshort of wonderful.Theprincipleofrelativitymustthereforeapplywithgreataccuracyinthedomainofmechanics. Butthataprincipleofsuchbroadgeneralityshouldholdwithsuchexactnessinonedomainof phenomena, andyetshouldbeinvalidforanother,isapriorinotveryprobable.Wenowproceedtothesecondargument, towhich, moreover, weshallreturnlater. Iftheprincipleofrelativity(intherestrictedsense)doesnothold, thentheGalileianco-ordinatesystems K, K

, K

, etc., whicharemovinguniformlyrelativetoeachother, will notbeequivalentforthede-scriptionof natural phenomena. Inthiscaseweshouldbeconstrainedtobelievethatnatural lawsarecapableofbeingformulatedinaparticularlysimple manner,and of course only on condition that,from amongst all pos-sibleGalileianco-ordinatesystems, weshouldhavechosenone (K0)of aparticular state of motionas our bodyof reference. We shouldthenbejustied(becauseofitsmeritsforthedescriptionofnaturalphenomena)incallingthissystemabsolutelyatrest,andall otherGalileiansystemsKin motion. If, for instance, our embankment were the system K0then ourrailwaycarriagewouldbeasystemK, relativetowhichlesssimplelawswouldholdthanwithrespecttoK0. ThisdiminishedsimplicitywouldbeduetothefactthatthecarriageKwouldbeinmotion(i.e. really)withrespecttoK0. Inthegeneral lawsof naturewhichhavebeenformulatedwithreferencetoK,themagnitudeanddirectionofthevelocityofthecar-riage would necessarily play a part. We should expect, for instance, that thenoteemittedbyanorganpipeplacedwithitsaxisparallel tothedirectionof travel wouldbedierentfromthatemittedif theaxisof thepipewere15placedperpendiculartothisdirection.Nowinvirtue of its motioninanorbit roundthe sun, our earthiscomparablewitharailwaycarriagetravellingwithavelocityof about30kilometres per second. If the principle of relativity were not valid we shouldthereforeexpectthatthedirectionof motionof theearthatanymomentwouldenterintothelawsofnature,andalsothatphysicalsystemsintheirbehaviour would be dependent on the orientation in space with respect to theearth. Forowingtothealterationindirectionofthevelocityofrevolutionof theearthinthecourseof ayear, theearthcannot beat rest relativetothehypothetical systemK0throughout thewholeyear. However, themost careful observations havenever revealedsuchanisotropicpropertiesinterrestrial physical space, i.e. a physical non-equivalence of dierentdirections. This is verypowerful argument infavour of the principle ofrelativity.16Chapter6TheTheoremoftheAdditionofVelocitiesEmployedinClassicalMechanicsLet us supposeour oldfriendtherailwaycarriagetobetravellingalongtherailswithaconstant velocityv, andthatamantraversesthelengthof thecarriageinthedirectionof travel withavelocityw. Howquicklyor,inotherwords,withwhatvelocityWdoesthemanadvancerelativetotheembankment duringtheprocess? Theonlypossibleanswer seems toresultfromthefollowingconsideration: Ifthemanweretostandstillforasecond, hewouldadvancerelativetotheembankmentthroughadistancevequal numericallytothevelocityof thecarriage. As aconsequenceofhiswalking, however, hetraversesanadditional distancewrelativetothecarriage, andhencealsorelativetotheembankment, inthis second, thedistance w being numerically equal to the velocity with which he is walking.Thus in total be covers the distance W= v +wrelative to the embankmentin the second considered. We shall see later that this result, which expressesthetheoremof theadditionof velocitiesemployedinclassical mechanics,cannot bemaintained; inotherwords, thelawthat wehavejust writtendown does not hold in reality. For the time being, however, we shall assumeitscorrectness.17Chapter7TheApparentIncompatabilityoftheLawofPropagationofLightwiththePrincipleofRelativityThereishardlyasimplerlawinphysicsthanthataccordingtowhichlightispropagatedinemptyspace. Everychildatschool knows, orbelievesheknows, that this propagationtakes placeinstraight lines withavelocityc = 300, 000km./sec. Atalleventsweknowwithgreatexactnessthatthisvelocityisthesameforall colours, becauseif thiswerenotthecase, theminimumof emissionwouldnot beobservedsimultaneouslyfor dierentcoloursduringtheeclipseofaxedstarbyitsdarkneighbour. Bymeansof similarconsiderationsbasedonobservationsof doublestars, theDutchastronomer De Sitter was also able to show that the velocity of propagationoflightcannotdependonthevelocityofmotionofthebodyemittingthelight. The assumption that this velocity of propagation is dependent on thedirectioninspaceisinitselfimprobable.Inshort, let us assume that the simple lawof the constancyof thevelocityoflightc(invacuum)isjustiablybelievedbythechildatschool.Whowouldimaginethat this simplelawhas plungedtheconscientiouslythoughtful physicist into the greatest intellectual diculties?Let us considerhowthesedicultiesarise.Of coursewemust refer theprocess of thepropagationof light (andindeedeveryotherprocess)toarigidreference-body(co-ordinatesystem).Assuchasystemletusagainchooseourembankment. Weshall imagine18the air above it to have beenremoved. If a rayof light be sent alongthe embankment, we see fromthe above that the tipof the raywill betransmittedwiththevelocitycrelativetotheembankment. Nowlet ussupposethatourrailwaycarriageisagaintravellingalongtherailwaylineswiththevelocityv, andthatitsdirectionisthesameasthatoftherayoflight,butitsvelocityofcoursemuchless. Letusinquireaboutthevelocityofpropagationoftherayoflightrelativetothecarriage. Itisobviousthatwecanhereapplytheconsiderationof theprevioussection, sincetherayoflightplaysthepartofthemanwalkingalongrelativelytothecarriage.Thevelocitywofthemanrelativetotheembankmentisherereplacedbythevelocityoflightrelativetotheembankment. wistherequiredvelocityoflightwithrespecttothecarriage,andwehavew = c v.The velocity of propagation ot a ray of light relative to the carriage thuscomescutsmallerthanc.But this result comes intoconict withtheprincipleof relativitysetforth in Section V. For, like every other general law of nature, the law of thetransmissionoflightinvacuo[invacuum]must, accordingtotheprincipleof relativity,be the same for the railway carriage as reference-body as whentherailsarethebodyofreference. But,fromouraboveconsideration,thiswouldappeartobeimpossible. Ifeveryrayoflightispropagatedrelativeto the embankment with the velocity c, then for this reason it would appearthatanotherlawofpropagationoflightmustnecessarilyholdwithrespecttothecarriagearesultcontradictorytotheprincipleofrelativity.Inviewofthisdilemmathereappearstobenothingelseforitthantoabandon either the principle of relativity or the simple law of the propagationof lightinvacuo. Thoseof youwhohavecarefullyfollowedtheprecedingdiscussionarealmostsuretoexpectthatweshouldretaintheprincipleofrelativity, whichappeals soconvincinglytothe intellect because it is sonatural and simple. The law of the propagation of light in vacuo would thenhave to be replaced by a more complicated law conformable to the principleof relativity. Thedevelopmentof theoretical physicsshows, however, thatwecannotpursuethiscourse. Theepoch-makingtheoreticalinvestigationsof H. A. Lorentz on the electrodynamical and optical phenomena connectedwith moving bodies show that experience in this domain leads conclusively toatheoryofelectromagneticphenomena, ofwhichthelawoftheconstancyof the velocityof light invacuois anecessaryconsequence. Prominenttheoretical physicists were theref ore more inclined to reject the principle of19relativity,inspiteofthefactthatnoempiricaldatahadbeenfoundwhichwerecontradictorytothisprinciple.At this juncture the theory of relativity entered the arena. As a result ofan analysis of the physical conceptions of time and space, it became evidentthat inrealilythereis not theleast incompatibilitiybetweentheprincipleofrelativityandthelawofpropagationoflight,andthatbysystematicallyholdingfasttoboththeselawsalogicallyrigidtheorycouldbearrivedat.Thistheoryhasbeencalledthespecial theoryofrelativitytodistinguishitfromtheextendedtheory, withwhichweshall deal later. Inthefollow-ingpagesweshall presentthefundamental ideasof thespecial theoryofrelativity.20Chapter8OntheIdeaofTimeinPhysicsLightninghasstrucktherailsonourrailwayembankmentattwoplacesAand B far distant from each other. I make the additional assertion that thesetwolightningashesoccurredsimultaneously. IfIaskyouwhetherthereissenseinthisstatement,youwillanswermyquestionwithadecidedYes.But if I now approach you with the request to explain to me the sense of thestatement more precisely, you nd after some consideration that the answertothisquestionisnotsoeasyasitappearsatrstsight.After some time perhaps the following answer would occur to you: Thesignicanceofthestatementisclearinitselfandneedsnofurtherexplana-tion; ofcourseitwouldrequiresomeconsiderationifIweretobecommis-sioned to determine by observations whether in the actual case the two eventstook place simultaneously or not. I cannot be satised with this answer forthefollowingreason. Supposingthatasaresultofingeniousconsiderationsan able meteorologist were to discover that the lightning must always strikethe places A and B simultaneously, then we should be faced with the task oftestingwhetherornotthistheoreticalresultisinaccordancewiththereal-ity. Weencounterthesamedicultywithallphysicalstatementsinwhichtheconceptionsimultaneousplaysapart. Theconceptdoesnotexistforthephysicistuntilhehasthepossibilityofdiscoveringwhetherornotitisfullledinanactualcase. Wethusrequireadenitionofsimultaneitysuchthatthisdenitionsuppliesuswiththemethodbymeansofwhich, inthepresent case, he can decide by experiment whether or not both the lightningstrokes occurred simultaneously. As long as this requirement is not satised,Iallowmyselftobedeceivedasaphysicist(andofcoursethesameapplies21ifIamnotaphysicist),whenIimaginethatIamabletoattachameaningtothestatementof simultaneity. (I wouldaskthereadernottoproceedfartheruntilheisfullyconvincedonthispoint.)After thinking the matter over for some time you then oer the followingsuggestionwithwhichtotestsimultaneity. Bymeasuringalongtherails,theconnectinglineABshouldbemeasuredupandanobserverplacedatthemid-pointMofthedistanceAB.Thisobservershouldbesuppliedwithan arrangement (e.g. two mirrors inclined at 90) which allows him visuallytoobservebothplacesAandBatthesametime. Iftheobserverperceivesthetwoashesoflightningatthesametime,thentheyaresimultaneous.Iamverypleasedwiththissuggestion,butforallthatIcannotregardthematterasquitesettled,becauseIfeelconstrainedtoraisethefollowingobjection:Your denition would certainly be right, if only I knew that the light bymeans of which the observer at M perceives the lightning ashes travels alongthelengthA MwiththesamevelocityasalongthelengthB M.But an examination of this supposition would only be possible if we alreadyhadatourdisposalthemeansofmeasuringtime. Itwouldthusappearasthoughweweremovinghereinalogicalcircle.After further considerationyoucast asomewhat disdainful glance atmeandrightlysoandyoudeclare:I maintainmyprevious denitionnevertheless, because inrealityitassumesabsolutelynothingaboutlight. Thereisonlyonedemandtobemade of the denition of simultaneity, namely, that in every real case it mustsupplyuswithanempirical decisionastowhetherornottheconceptionthat has to be dened is fullled. That my denition satises this demand isindisputable. That light requires the same time to traverse the path A MasforthepathB Misinrealityneitherasuppositionnorahypothesisabout the physical nature of light, but a stipulation which I can make of myownfreewillinordertoarriveatadenitionofsimultaneity.It is clear that this denitioncanbe usedtogive anexact meaningnotonlytotwoevents, buttoasmanyeventsaswecaretochoose, andindependentlyof thepositionsof thescenesof theeventswithrespecttothe body of reference1(here the railway embankment). We are thus led alsoto a denition of time in physics. For this purpose we suppose that clocksof identical construction are placed at the points A, B, and C of the railwayline(co-ordinatesystem)andthattheyaresetinsuchamannerthatthepositions of their pointers are simultaneously (in the above sense) the same.Under these conditions we understand by the time of an event the reading(position of the hands) of that one of these clocks which is in the immediate22vicinity(inspace)of theevent. Inthismanneratime-valueisassociatedwitheveryeventwhichisessentiallycapableofobservation.Thisstipulationcontainsafurtherphysical hypothesis, thevalidityofwhichwill hardlybedoubtedwithoutempirical evidencetothecontrary.Ithasbeenassumedthatall theseclocksgoat thesamerate if theyareofidentical construction. Statedmoreexactly: Whentwoclocksarrangedatrestindierentplacesofareference-bodyaresetinsuchamannerthataparticular positionof thepointers of theoneclockis simultaneous (intheabovesense)withthesameposition,ofthepointersoftheotherclock,thenidenticalsettingsarealwayssimultaneous(inthesenseoftheabovedenition).1Wesupposefurther,that,whenthreeeventsA,B,andCoccurindierentplacesinsuch a manner that A is simultaneous with B and B is simultaneous with C (simultaneousinthesenseof theabovedenition), thenthecriterionforthesimultaneityof thepairof eventsA, Cisalsosatised. Thisassumptionisaphysical hypothesisaboutthetheofpropagationoflight: itmustcertainlybefullledifwearetomaintainthelawoftheconstancyofthevelocityoflightinvacuo.23Chapter9TheRelativityofSimulatneityUptonowour considerations havebeenreferredtoaparticular bodyofreference, whichwe have styledarailwayembankment.We suppose averylong train travelling along the rails withthe constantvelocityv and inthe direction indicated in Fig. 9.1. People travelling in this train will with avantageviewthetrainasarigidreference-body(co-ordinatesystem);theyregardall eventsinreferencetothetrain. Theneveryeventwhichtakesplacealongthelinealsotakesplaceataparticularpointofthetrain. Alsothe denition of simultaneity can be given relative to the train in exactly thesamewayaswithrespecttotheembankment. Asanatural consequence,however,thefollowingquestionarises:Figure9.1:Embankment

TrainA M Bv-v-M

-Aretwoevents(e.g. thetwostrokesof lightningAandB)whicharesimultaneouswithreferencetotherailwayembankment alsosimultaneousrelativelytothetrain? Weshall showdirectlythattheanswermustbeinthenegative.24WhenwesaythatthelightningstrokesAandBaresimultaneouswithrespect to be embankment, we mean: the rays of light emitted at the placesAandB, wherethelightningoccurs, meeteachotheratthemid-pointMof thelengthA Bof theembankment. ButtheeventsAandBalsocorrespondtopositions AandBonthetrain. Let M

bethemid-pointofthedistanceA Bonthetravellingtrain. Justwhentheashes(asjudgedfromtheembankment)of lightningoccur, thispointM

naturallycoincideswiththepointMbutitmovestowardstherightinthediagramwiththevelocityvofthetrain. IfanobserversittinginthepositionM

inthetraindidnotpossessthisvelocity, thenhewouldremainpermanentlyatM,andthelightraysemittedbytheashesoflightningAandBwouldreachhimsimultaneously, i.e. theywouldmeetjustwhereheissituated.Nowinreality(consideredwithreferencetotherailwayembankment)heishasteningtowardsthebeamoflightcomingfromB, whilstheisridingonaheadofthebeamoflightcomingfromA.HencetheobserverwillseethebeamoflightemittedfromBearlierthanhewillseethatemittedfromA.Observers who take the railway train as their reference-body must thereforecome to the conclusion that the lightning ash B took place earlier than thelightningashA.Wethusarriveattheimportantresult:Events which are simultaneous with reference to the embankment are notsimultaneouswithrespecttothetrain, andviceversa(relativityof simul-taneity). Everyreference-body(co-ordinatesystem)hasitsownparticulartime;unlesswearetoldthereference-bodytowhichthestatementoftimerefers,thereisnomeaninginastatementofthetimeofanevent.Nowbeforetheadventof thetheoryof relativityithadalwaystacitlybeenassumedinphysicsthat thestatement of timehadanabsolutesig-nicance, i.e. thatitisindependentof thestateof motionof thebodyofreference. Butwehavejustseenthatthisassumptionisincompatiblewiththemostnatural denitionof simultaneity; if wediscardthisassumption,thentheconictbetweenthelawofthepropagationoflightinvacuoandtheprincipleofrelativity(developedinSection6)disappears.Wewereledtothatconictbytheconsiderationsof Section6, whichare now no longer tenable. In that section we concluded that the man in thecarriage, whotraversesthedistancewpersecondrelativetothecarriage,traversesthesamedistancealsowithrespecttotheembankmentineachsecond of time. But, accordingtotheforegoingconsiderations, thetimerequired by a particular occurrence with respect to the carriage must not beconsideredequaltothedurationofthesameoccurrenceasjudgedfromtheembankment(asreference-body). Henceitcannotbecontendedthatthemaninwalkingtravelsthedistancewrelativetotherailwaylineinatime25whichisequaltoonesecondasjudgedfromtheembankment.Moreover, the considerations of Section6are basedonyet asecondassumption, which, inthelightofastrictconsideration, appearstobear-bitrary, although it was always tacitly made even before the introduction ofthetheoryofrelativity.26Chapter10OntheRelativityoftheConceptionofDistanceLetusconsidertwoparticularpointsonthetrain1travellingalongtheem-bankmentwiththevelocityv, andinquireastotheirdistanceapart. Wealreadyknowthatitisnecessarytohaveabodyofreferenceforthemea-surement of adistance, withrespect towhichbodythe distance canbemeasured up. It is the simplest plan to use the train itself as reference-body(co-ordinate system). Anobserver inthe trainmeasures the interval bymarkingohismeasuring-rodinastraightline(e.g. alongtheoorofthecarriage)asmanytimesasisnecessarytotakehimfromtheonemarkedpointtotheother. Thenthenumberwhichtellsushowoftentherodhastobelaiddownistherequireddistance.It is adierent matter whenthedistancehas tobejudgedfromtherailwayline. Herethefollowingmethodsuggestsitself. IfwecallA

andB

thetwopointsonthetrainwhosedistanceapartisrequired, thenbothofthesepointsaremovingwiththevelocityvalongtheembankment. IntherstplacewerequiretodeterminethepointsAandBoftheembankmentwhich are just being passed by the two points A

and B

at a particular timetjudged from the embankment. These points A and B of the embankmentcanbedeterminedbyapplyingthedenitionof timegiveninSection8.ThedistancebetweenthesepointsAandBisthenmeasuredbyrepeatedapplicationoftheemeasuring-rodalongtheembankment.A priori it is by no means certain that this last measurement will supplyus with the same result as the rst. Thus the length of the train as measuredfrom the embankment may be dierent from that obtained by measuring inthe train itself. This circumstance leads us to a second objection which must27be raised against the apparently obvious consideration of Section 6. Namely,if the man in the carriage covers the distance win a unit of timemeasuredfromthetrain,thenthisdistanceasmeasuredfromtheembankmentisnotnecessarilyalsoequaltow.1e.g. themiddleoftherstandofthehundredthcarriage.28Chapter11TheLorentzTransformationThe results of the last three sections show that the apparent incompatibilityofthelawofpropagationoflightwiththeprincipleofrelativity(Section7)hasbeenderivedbymeansofaconsiderationwhichborrowedtwounjusti-ablehypothesesfromclassicalmechanics;theseareasfollows:a. The time-interval (time) betweentwoevents is independent of theconditionofmotionofthebodyofreference.b. Thespace-interval (distance) betweentwopoints of arigidbodyisindependentoftheconditionofmotionofthebodyofreference.Ifwedropthesehypotheses,thenthedilemmaofSection7disappears,becausethetheoremof theadditionof velocitiesderivedinSection6be-comes invalid. The possibility presents itself that the law of the propagationoflightinvacuo maybe compatible withthe principle ofrelativity,andthequestionarises: Howhavewetomodifytheconsiderationsof Section6inordertoremovetheapparentdisagreementbetweenthesetwofundamentalresultsofexperience? Thisquestionleadstoageneral one. Inthediscus-sionofSection6wehavetodowithplacesandtimesrelativebothtothetrainandtotheembankment. Howarewetondtheplaceandtimeofaneventinrelationtothetrain, whenweknowtheplaceandtimeoftheeventwithrespecttotherailwayembankment?Isthereathinkableanswertothis questionof suchanaturethat thelawof transmissionof light invacuodoesnotcontradicttheprincipleofrelativity? Inotherwords: Canweconceiveof arelationbetweenplaceandtimeof theindividual eventsrelativetobothreference-bodies,suchthateveryrayoflightpossessesthevelocityof transmissioncrelativetotheembankmentandrelativetothe29train?Thisquestionleadstoaquitedenitepositiveanswer,andtoaper-fectly denite transformation law for the space-time magnitudes of an eventwhenchangingoverfromonebodyofreferencetoanother.Figure11.1:KxzyK

x

z

y

---Before we deal with this, we shall introduce the following incidental con-sideration. Uptothepresentwehaveonlyconsideredeventstakingplacealongtheembankment, whichhadmathematicallytoassumethefunctionofastraightline. InthemannerindicatedinSection2wecanimaginethisreference-bodysupplementedlaterallyandinaverticaldirectionbymeansof aframeworkof rods, sothataneventwhichtakesplaceanywherecanbelocalisedwithreferencetothis framework. Similarly, wecanimaginethetraintravellingwiththevelocityvtobecontinuedacrossthewholeofspace, sothateveryevent, nomatterhowfaroitmaybe, couldalsobelocalisedwithrespecttothesecondframework. Withoutcommittinganyfundamental error, we can disregard the fact that in reality these frameworkswouldcontinuallyinterferewitheachother,owingtotheimpenetrabilityofsolidbodies. Ineverysuchframeworkweimaginethreesurfacesperpen-diculartoeachothermarkedout, anddesignatedasco-ordinateplanes(co-ordinatesystem). Aco-ordinatesystemKthencorrespondstotheembankment, and a co-ordinate system K

to the train. An event, whereveritmayhavetakenplace, wouldbexedinspacewithrespecttoKbythe30threeperpendicularsx, y, zontheco-ordinateplanes, andwithregardtotime by a time value t. Relative to K

,the same event would be xed in re-spectofspaceandtimebycorrespondingvaluesx

, y

, z

, t

,whichofcoursearenotidentical withx, y, z, t. Ithasalreadybeensetforthindetail howthesemagnitudesaretoberegardedasresultsofphysicalmeasurements.Obviouslyourproblemcanbeexactlyformulatedinthefollowingman-ner. Whatarethevaluesx

, y

, z

, t

, ofaneventwithrespecttoK

, whenthemagnitudes x, y, z, t, of thesameevent withrespect toKaregiven?Therelationsmustbesochosenthatthelawof thetransmissionof lightinvacuois satisedfor oneandthesamerayof light (andof courseforeveryray)withrespecttoKandK

. Fortherelativeorientationinspaceof the co-ordinate systems indicated in the diagram (Fig 11.1), this problemissolvedbymeansoftheequations:x

=x vt_I v2c2y

= yz

= zt

=t vc2x_I v2c2ThissystemofequationsisknownastheLorentztransformation.1Ifinplaceofthelawoftransmissionoflightwehadtakenasourbasisthe tacit assumptions of the older mechanics as to the absolute character oftimesandlengths, theninsteadof theaboveweshouldhaveobtainedthefollowingequations:x

= x vty

= yz

= zt

= tThissystemofequationsisoftentermedtheGalileitransformation.TheGalileitransformationcanbeobtainedfromtheLorentztransformationbysubstitutinganinnitelylargevalueforthevelocityoflightcinthelattertransformation.31Aided by the following illustration, we can readily see that, in accordancewith the Lorentz transformation, the law of the transmission of light in vacuoissatisedbothforthereference-bodyKandforthereference-bodyK

. Alight-signal is sent along the positive x-axis, and this light-stimulus advancesinaccordancewiththeequationx = ct,i.e. with the velocity c. According to the equations of the Lorentz transfor-mation,thissimplerelationbetweenxandtinvolvesarelationbetweenx

andt

. Inpointoffact, ifwesubstituteforxthevaluectintherstandfourthequationsoftheLorentztransformation,weobtain:x

=(c v)t_I v2c2t

=(I vc)t_I v2c2fromwhich,bydivision,theexpressionx

= ct

immediatelyfollows. IfreferredtothesystemK

, thepropagationoflighttakes placeaccordingtothis equation. Wethus seethat thevelocityoftransmissionrelativetothereference-bodyK

isalsoequaltoc. Thesameresultisobtainedforraysof lightadvancinginanyotherdirectionwhat-soever. Of causethisisnotsurprising, sincetheequationsof theLorentztransformationwerederivedconformablytothispointofview.1AsimplederivationoftheLorentztransformationisgiveninAppendixI.32Chapter12TheBehaviourofMeasuring-RodsandClocksinMotionPlaceametre-rodinthex

-axisofK

insuchamannerthatoneend(thebeginning)coincideswiththepointx

= 0whilsttheotherend(theendoftherod)coincideswiththepointx

=I. Whatisthelengthofthemetre-rodrelativelytothesystemK? Inordertolearnthis, weneedonlyaskwherethebeginningof therodandtheendof therodliewithrespecttoKataparticulartimetofthesystemK. BymeansoftherstequationoftheLorentztransformationthevaluesofthesetwopointsatthetimet = 0canbeshowntobex(beginingofrod)= 0I v2c2x(endofrod)= 1I v2c2thedistancebetweenthepointsbeing_I v2/c2.But the metre-rod is moving with the velocity v relative to K. It thereforefollowsthatthelengthof arigidmetre-rodmovinginthedirectionof itslengthwithavelocityvis_I v2/c2ofametre.The rigid rod is thus shorter when in motion than when at rest,and themorequicklyitismoving,theshorteristherod. Forthevelocityv= cwe33shouldhave_I v2/c2= 0,andforstiIIgreatervelocitiesthesquare-rootbecomesimaginary. Fromthisweconcludethatinthetheoryofrelativitythevelocitycplays thepart of alimitingvelocity, whichcanneither bereachednorexceededbyanyrealbody.OfcoursethisfeatureofthevelocitycasalimitingvelocityalsoclearlyfollowsfromtheequationsoftheLorentztransformation,forthesebecamemeaninglessifwechoosevaluesofvgreaterthanc.If,onthecontrary,wehadconsideredametre-rodatrestinthex-axiswithrespecttoK, thenweshouldhavefoundthatthelengthof therodasjudgedfromK

wouldhavebeen_I v2/c2;thisisquiteinaccordancewiththeprincipleofrelativitywhichformsthebasisofourconsiderations.APriori it is quite clear that we must be able to learn something aboutthephysical behaviourofmeasuring-rodsandclocksfromtheequationsoftransformation,forthemagnitudesz, y, x, t,arenothingmorenorlessthanthe results of measurements obtainable bymeans of measuring-rods andclocks. IfwehadbasedourconsiderationsontheGalileiantransformationweshouldnothaveobtainedacontractionof therodasaconsequenceofitsmotion.Letusnowconsideraseconds-clockwhichispermanentlysituatedattheorigin(x

=0)ofK

. t

=0andt

=Iaretwosuccessiveticksofthisclock. TherstandfourthequationsoftheLorentztransformationgiveforthesetwoticks:t = 0andt

=I_I v2c2AsjudgedfromK, theclockismovingwiththevelocityv; asjudgedfromthisreference-body,thetimewhichelapsesbetweentwostrokesoftheclockisnotonesecond,butI_I v2c2seconds, i.e. asomewhatlargertime. Asaconsequenceof itsmotiontheclock goes more slowly than when at rest. Here also the velocity c plays thepartofanunattainablelimitingvelocity.34Chapter13TheoremoftheAdditionofVelocities.TheExperimentofFizeauNow in practice we can move clocks and measuring-rods only with velocitiesthat are small compared with the velocity of light;hence we shall hardly beabletocomparetheresultsoftheprevioussectiondirectlywiththereality.But, on the other hand, these results must strike you as being very singular,andforthatreasonIshall nowdrawanotherconclusionfromthetheory,one which can easily be derived from the foregoing considerations, and whichhasbeenmostelegantlyconrmedbyexperiment.InSection6wederivedthetheoremoftheadditionofvelocitiesinonedirectioninthe formwhichalsoresults fromthe hypotheses of classicalmechanicsThis theorem can also be deduced readily horn the Galilei trans-formation(Section11). Inplaceofthemanwalkinginsidethecarriage,weintroduceapointmovingrelativelytotheco-ordinatesystemK

inaccor-dancewiththeequationx

= wt

.BymeansoftherstandfourthequationsoftheGalileitransformationwecanexpressx

andt

intermsofxandt,andwethenobtainx = (v +w)t.35This equation expresses nothing else than the law of motion of the pointwithreferencetothesystemK(ofthemanwithreferencetotheembank-ment). We denote this velocity by the symbol W, and we then obtain, as inSection6,W= v +w (13.1)Butwecancarryoutthisconsiderationjustaswellonthebasisofthetheoryofrelativity. Intheequationx

= wt

(13.2)wemustthenexpressx

andt

intermsofxandt, makinguseoftherstand fourth equations of the Lorentz transformation. Instead of the equation13.1wethenobtaintheequationW=v +wI +vwc2whichcorrespondstothetheoremofadditionforvelocitiesinonedirectionaccording to the theory of relativity. The question now arises as to which ofthese two theorems is the better in accord with experience. On this point weaxe enlightened by a most important experiment which the brilliant physicistFizeau performed more than half a century ago, and which has been repeatedsince then by some of the best experimental physicists,so that there can benodoubtaboutitsresult. Theexperimentisconcernedwiththefollowingquestion. Lighttravelsinamotionlessliquidwithaparticularvelocityw.Howquicklydoesittravel inthedirectionofthearrowinthetubeT(seetheaccompanyingdiagram, Figure13.1)whentheliquidabovementionedisowingthroughthetubewithavelocityv?Figure13.1:T-vInaccordancewiththeprincipleofrelativityweshallcertainlyhavetotakeforgrantedthatthepropagationof lightalwaystakesplacewiththesamevelocitywwithrespect totheliquid, whetherthelatterisinmotionwithreferencetootherbodiesornot. Thevelocityof lightrelativetothe36liquid and the velocity of the latter relative to the tube are thus known, andwerequirethevelocityoflightrelativetothetube.Itisclearthatwehavetheproblemof Section6againbeforeus. Thetube plays the part of the railway embankment or of the co-ordinate systemK,the liquid plays the part of the carriage or of the co-ordinate system K

,and nally, the light plays the part of the man walking along the carriage, orofthemovingpointinthepresentsection. IfwedenotethevelocityofthelightrelativetothetubebyW, thenthisisgivenbytheequation13.1or13.2,accordingastheGalileitransformationortheLorentztransformationcorresponds tothe facts. Experiment1decides infavour of equation13.2derivedfromthetheoryof relativity, andtheagreement is, indeed, veryexact. According to recent and most excellent measurements by Zeeman, theinuenceofthevelocityofowvonthepropagationoflightisrepresentedbyformula13.2towithinonepercent.Nevertheless we must now draw attention to the fact that a theory of thisphenomenonwasgivenbyH. A. Lorentzlongbeforethestatementof thetheoryof relativity. Thistheorywasof apurelyelectrodynamical nature,andwasobtainedbytheuseofparticularhypothesesastotheelectromag-netic structure of matter. This circumstance, however, does not in the leastdiminishtheconclusiveness of theexperiment as acrucial test infavourof thetheoryof relativity, fortheelectrodynamicsof Maxwell-Lorentz, onwhichtheoriginal theorywasbased, innowayopposesthetheoryof rel-ativity. Ratherhasthelatterbeendevelopedtromelectrodynamicsasanastoundinglysimplecombinationandgeneralisationofthehypotheses,for-merlyindependentofeachother,onwhichelectrodynamicswasbuilt.1Fizeau found W= w+v_I In2_, where n =cwis the index of refraction of the liquid.Ontheotherhand,owingtothesmallnessofvwc2ascomparedwithI,wecanreplace(B)intherstplacebyW=(w + v)_I vwc2_, ortothesameorderof approximationbyw +v_I In2_,whichagreeswithFizeausresult.37Chapter14TheHeuristicValueoftheTheoryofRelativityOur train of thought in the foregoing pages can be epitomised in the followingmanner. Experiencehasledtotheconvictionthat, ontheonehand, theprincipleof relativityholdstrueandthat ontheother handthevelocityof transmissionof lightinvacuohastobeconsideredequal toaconstantc. Byunitingthesetwopostulatesweobtainedthelawof transformationfortherectangularco-ordinatesx, y, zandthetimetof theeventswhichconstitutetheprocessesofnature. InthisconnectionwedidnotobtaintheGalilei transformation, but, dieringfromclassical mechanics, theLorentztransformation.Thelawoftransmissionoflight,theacceptanceofwhichisjustiedbyouractualknowledge,playedanimportantpartinthisprocessofthought.Once in possession of the Lorentz transformation,however,we can combinethiswiththeprincipleofrelativity,andsumupthetheorythus:Every general law of nature must be so constituted that it is transformedinto a law of exactly the same form when, instead of the space-time variablesx, y, z, toftheoriginal coordinatesystemK, weintroducenewspace-timevariablesx

, y

, z

, t

ofaco-ordinatesystemK

. Inthisconnectionthere-lationbetweentheordinaryandtheaccentedmagnitudesisgivenbytheLorentztransformation. Orinbrief: General lawsofnatureareco-variantwithrespecttoLorentztransformations.Thisisadenitemathematical conditionthatthetheoryof relativitydemands of a natural law, and in virtue of this, the theory becomes a valuableheuristicaidinthesearchforgeneral lawsof nature. If ageneral lawofnatureweretobefoundwhichdidnotsatisfythiscondition, thenatleast38one of the twofundamental assumptions of the theorywouldhave beendisproved. Letusnowexaminewhatgeneral resultsthelattertheoryhashithertoevinced.39Chapter15GeneralResultsoftheTheoryItisclearfromourpreviousconsiderationsthatthe(special)theoryofrel-ativityhasgrownoutof electrodynamicsandoptics. Intheseeldsithasnot appreciablyalteredthepredictions of theory, but it has considerablysimpliedthetheoretical structure, i.e. thederivationoflaws, andwhatisincomparablymoreimportantithasconsiderablyreducedthenumberof independenthypotheseformingthebasisof theory. Thespecial theoryofrelativityhasrenderedtheMaxwell-Lorentztheorysoplausible,thatthelatterwouldhavebeengenerallyacceptedbyphysicistsevenifexperimenthaddecidedlessunequivocallyinitsfavour.Classical mechanicsrequiredtobemodiedbeforeit couldcomeintoline withthe demands of the special theoryof relativity. For the mainpart, however, thismodicationaectsonlythelawsforrapidmotions, inwhichthevelocitiesof mattervarenotverysmall ascomparedwiththevelocityof light. We have experience of suchrapidmotions onlyinthecaseofelectronsandions;forothermotionsthevariationsfromthelawsofclassical mechanics are too small to make themselves evident in practice. Weshallnotconsiderthemotionofstarsuntilwecometospeakofthegeneraltheoryofrelativity. Inaccordancewiththetheoryofrelativitythekineticenergyofamaterial pointofmassmisnolongergivenbythewell-knownexpressionmv22butbytheexpression40mc2_I v2c2Thisexpressionapproachesinnityasthevelocityvapproachestheve-locityof light c. Thevelocitymust thereforealways remainless thanc,howevergreatmaybetheenergiesusedtoproducetheacceleration. Ifwedevelopthe expressionfor the kinetic energyinthe formof aseries, weobtainmc2+mv22+38mv4c2+ Whenv2/c2issmall comparedwithunity, thethirdof thesetermsisalwayssmallincomparisonwiththesecond,whichlastisaloneconsideredin classical mechanics. The rst term mc2does not contain the velocity, andrequiresno considerationifweareonlydealingwiththequestionasto howtheenergyofapoint-mass; dependsonthevelocity. Weshall speakofitsessentialsignicancelater.Themostimportantresultof ageneral charactertowhichthespecialtheory of relativity has led is concerned with the conception of mass. Beforetheadventofrelativity,physicsrecognisedtwoconservationlawsoffunda-mental importance, namely, thelawof thecanservationof energyandthelawof theconservationof mass thesetwofundamental laws appearedtobequiteindependentof eachother. Bymeansof thetheoryof relativitytheyhavebeenunitedintoonelaw. Weshallnowbrieyconsiderhowthisunicationcameabout,andwhatmeaningistobeattachedtoit.Theprincipleof relativityrequiresthatthelawof theconcervationofenergyshouldholdnotonlywithreferencetoaco-ordinatesystemK,butalso withrespect to everyco-ordinate systemK

whichis ina state ofuniformmotionof translationrelativetoK, or, briey, relativetoeveryGalileiansystemof co-ordinates. Incontrasttoclassical mechanics; theLorentz transformation is the deciding factor in the transition from one suchsystemtoanother.By means of comparatively simple considerations we are led to draw thefollowingconclusionfromthesepremises, inconjunctionwiththefunda-mental equationsoftheelectrodynamicsofMaxwell: Abodymovingwiththevelocityv, whichabsorbs1anamountof energyE0intheformof ra-diationwithoutsueringanalterationinvelocityintheprocess, has, asaconsequence,itsenergyincreasedbyanamount41E0_I v2c2Inconsiderationoftheexpressiongivenaboveforthekineticenergyofthebody,therequiredenergyofthebodycomesouttobe_m+E0c2_c2_I v2c2Thusthebodyhasthesameenergyasabodyofmass_m+E0c2_moving with the velocity v. Hence we can say: If a body takes up an amountofenergyE0,thenitsinertialmassincreasesbyanamountE0c2theinertial massof abodyisnot aconstant but variesaccordingtothechangeintheenergyofthebody. Theinertial massofasystemofbodiescan even be regarded as a measure of its energy. The law of the conservationofthemassofasystembecomesidenticalwiththelawoftheconservationof energy, andisonlyvalidprovidedthatthesystemneithertakesupnorsendsoutenergy. Writingtheexpressionfortheenergyintheformmc2+E0_I v2c2weseethat thetermmc2, whichhas hithertoattractedour attention, isnothingelsethantheenergypossessedbythebody2beforeitabsorbedtheenergyE0.Adirectcomparisonofthisrelationwithexperimentisnotpossibleatthe present time (1920; see3Note, p. 48), owing to the fact that the changesinenergyE[0] towhichwecanSubjectasystemarenotlargeenoughtomakethemselvesperceptibleasachangeintheinertialmassofthesystem.E0c242istoosmall incomparisonwiththemassm, whichwaspresentbeforethealterationof the energy. It is owing to this circumstance that classicalmechanicswasabletoestablishsuccessfullytheconservationof massasalawofindependentvalidity.Let me addanal remarkof afundamental nature. The success oftheFaraday-Maxwellinterpretationofelectromagneticactionatadistanceresultedinphysicistsbecomingconvincedthattherearenosuchthingsasinstantaneous actions at a distance (not involving an intermediary medium)of the type of Newtons lawof gravitation. Accordingtothe theoryofrelativity, actionatadistancewiththevelocityof lightalwaystakestheplaceof instantaneousactionatadistanceorof actionatadistancewithan innite velocity of transmission. This is connected with the fact that thevelocitycplaysafundamentalroleinthistheory. InPartIIweshallseeinwhatwaythisresultbecomesmodiedinthegeneraltheoryofrelativity.1E0is the energy taken up, as judged from a co-ordinate system moving with the body.2Asjudgedfromaco-ordinatesystemmovingwiththebody.3TheequationE= mc2hasbeenthoroughlyprovedtimeandagainsincethistime.43Chapter16ExperienceandtheSpecialTheoryofRelativityTowhatextentisthespecialtheoryofrelativitysupportedbyexperience?This questionis not easilyansweredfor thereasonalreadymentionedinconnectionwiththefundamentalexperimentofFizeau. Thespecialtheoryof relativityhascrystallisedoutfromtheMaxwell-Lorentztheoryof elec-tromagneticphenomena. Thus all facts of experiencewhichsupport theelectromagnetictheoryalsosupport thetheoryof relativity. As beingofparticularimportance,Imentionherethefactthatthetheoryofrelativityenablesustopredicttheeectsproducedonthelightreachingusfromthexedstars. Theseresults areobtainedinanexceedinglysimplemanner,andtheeectsindicated,whichareduetotherelativemotionoftheearthwith reference to those xed stars are found to be in accord with experience.Werefertotheyearlymovementoftheapparentpositionofthexedstarsresultingfromthemotionof theearthroundthesun(aberration), andtotheinuenceof theradial componentsof therelativemotionsof thexedstarswithrespecttotheearthonthecolourof thelightreachingusfromthem. Thelattereectmanifestsitselfinaslightdisplacementofthespec-trallinesofthelighttransmittedtousfromaxedstar,ascomparedwiththepositionof thesamespectral lineswhentheyareproducedbyater-restrialsourceoflight(Dopplerprinciple). Theexperimentalargumentsinfavour of the Maxwell-Lorentz theory, which are at the same time argumentsinfavourofthetheoryofrelativity, aretoonumeroustobesetforthhere.Inrealitytheylimitthetheoretical possibilitiestosuchanextent, thatnoothertheorythanthatof Maxwell andLorentzhasbeenabletoholditsownwhentestedbyexperience.44ButtherearetwoclassesofexperimentalfactshithertoobtainedwhichcanberepresentedintheMaxwell-Lorentztheoryonlybytheintroductionofanauxiliaryhypothesis, whichinitselfi.e. withoutmakinguseofthetheoryofrelativityappearsextraneous.Itisknownthatcathoderaysandtheso-called-raysemittedbyra-dioactivesubstancesconsistofnegativelyelectriedparticles(electrons)ofverysmall inertiaandlargevelocity. Byexaminingthedeectionoftheseraysundertheinuenceof electricandmagneticelds, wecanstudythelawofmotionoftheseparticlesveryexactly.Inthetheoretical treatment of theseelectrons, wearefacedwiththedicultythatelectrodynamictheoryof itself isunabletogiveanaccountof their nature. For sinceelectrical masses of onesignrepel eachother,the negative electrical masses constitutingthe electronwouldnecessarilybe scattered under the inuence of their mutual repulsions,unless there areforces of another kind operating between them, the nature of which has hith-ertoremainedobscuretous.1Ifwenowassumethattherelativedistancesbetweentheelectrical massesconstitutingtheelectronremainunchangedduringthemotionoftheelectron(rigidconnectioninthesenseofclassicalmechanics), we arrive at a law of motion of the electron which does not agreewith experience. Guided by purely formal points of view, H. A. Lorentz wasthe rst to introduce the hypothesis that the form of the electron experiencesacontractioninthedirectionofmotioninconsequenceofthatmotion. thecontractedlengthbeingproportionaltotheexpressionI v2c2.This,hypothesis,whichisnotjustiablebyanyelectrodynamicalfacts,supplies us then with that particular law of motion which has been conrmedwithgreatprecisioninrecentyears.The theory of relativity leads to the same law of motion, without requir-ing any special hypothesis whatsoever as to the structure and the behaviouroftheelectron. WearrivedatasimilarconclusioninSection13inconnec-tionwiththeexperimentof Fizeau, theresultof whichisforetoldbythetheory of relativity without the necessity of drawing on hypotheses as to thephysicalnatureoftheliquid.Thesecondclassoffactstowhichwehavealludedhasreferencetothequestionwhetherornotthemotionoftheearthinspacecanbemadeper-ceptibleinterrestrialexperiments. WehavealreadyremarkedinSection5that all attempts of this nature led to a negative result. Before the theory of45relativitywasputforward,itwasdiculttobecomereconciledtothisneg-ative result, for reasons now to be discussed. The inherited prejudices abouttime and space did not allow any doubt to arise as to the prime importanceof the Galileian transformation for changing over from one body of referencetoanother. NowassumingthattheMaxwell-Lorentzequationsholdforareference-bodyK, wethenndthattheydonotholdforareference-bodyK

movinguniformlywithrespecttoK,ifweassumethattherelationsoftheGalileiantransformstionexistbetweentheco-ordinatesof KandK

.It thusappearsthat, of all Galileianco-ordinatesystems, one(K) corre-spondingtoaparticularstateof motionisphysicallyunique. Thisresultwas interpreted physically by regarding Kas at rest with respect to a hypo-thetical ther of space. On the other hand, all coordinate systems K

movingrelativelytoKweretoberegardedasinmotionwithrespecttothether.TothismotionofK

againstthether(ther-driftrelativetoK

)wereattributedthemorecomplicatedlawswhichweresupposedtoholdrelativetoK

. Strictlyspeaking,suchanther-driftoughtalsotobeassumedrel-ativetotheearth,andforalongtimetheeortsofphysicistsweredevotedtoattemptstodetecttheexistenceofanther-driftattheearthssurface.In one of the most notable of these attempts Michelson devised a methodwhichappearsasthoughitmustbedecisive. Imaginetwomirrorssoar-ranged on a rigid body that the reecting surfaces face each other. A ray oflight requires a perfectly denite time T to pass from one mirror to the otherandbackagain, ifthewholesystembeatrestwithrespecttothether. Itis found by calculation, however, that a slightly dierent time T

is requiredfor this process, if the body, together with the mirrors, be moving relativelytothether. Andyetanotherpoint: itisshownbycalculationthatforagivenvelocityvwithreferencetothether, thistimeT

isdierentwhenthebodyismovingperpendicularlytotheplanesofthemirrorsfromthatresultingwhenthemotionisparallel totheseplanes. Althoughtheesti-mateddierencebetweenthesetwotimesisexceedinglysmall, MichelsonandMorleyperformedanexperiment involvinginterferenceinwhichthisdierenceshouldhavebeenclearlydetectable. Buttheexperimentgaveanegativeresultafactveryperplexingtophysicists. LorentzandFitzGer-ald rescued the theory from this diculty by assuming that the motion of thebody relative to the ther produces a contraction of the body in the directionof motion, the amount of contraction being just sucient to compensate forthediereaceintimementionedabove. ComparisonwiththediscussioninSection 11 shows that also from the standpoint of the theory of relativity thissolutionofthedicultywastherightone. Butonthebasisofthetheoryof relativity the method of interpretation is incomparably more satisfactory.46Accordingtothistheorythereisnosuchthingasaspeciallyfavoured(unique)co-ordinatesystemtooccasiontheintroductionof thether-idea,andhencetherecanbenother-drift, nor anyexperiment withwhichtodemonstrate it. Here the contraction of moving bodies follows from the twofundamental principles of the theory,without the introduction of particularhypotheses; and as the prime factor involved in this contraction we nd, notthe motion in itself, to which we cannot attach any meaning, but the motionwithrespecttothebodyofreferencechosenintheparticularcaseinpoint.Thusforaco-ordinatesystemmovingwiththeearththemirrorsystemofMichelson and Morley is not shortened, but it is shortened for a co-ordinatesystemwhichisatrestrelativelytothesun.1The general theory of relativity renders it likely that the electrical masses of an electronareheldtogetherbygravitationalforces.47Chapter17MinkowskisFour-DimensionalSpaceThe non-mathematician is seized by a mysterious shuddering when he hearsof four-dimensional things, by a feeling not unlike that awakened bythoughtsoftheoccult. Andyetthereisnomorecommon-placestatementthanthattheworldinwhichweliveisafour-dimensional space-timecon-tinuum.Space is a three-dimensional continuum. By this we mean that it is pos-sible to describe the position of a point (at rest) by means of three numbers(co-ordinales) x, y, z, andthat thereis anindenitenumber of points intheneighbourhoodof thisone, thepositionof whichcanbedescribedbyco-ordinatessuchasx1, y1, z1, whichmaybeasnearaswechoosetotherespectivevaluesof theco-ordinatesx, y, z, of therstpoint. Invirtueofthelatterpropertywespeakofacontinuum,andowingtothefactthattherearethreeco-ordinateswespeakofitasbeingthree-dimensional.Similarly, the worldof physical phenomena whichwas briey calledworld by Minkowski is naturally four dimensional in the space-time sense.Foritiscomposedofindividual events, eachofwhichisdescribedbyfournumbers, namely, threespaceco-ordinatesx, y, z, andatimeco-ordinate,the time value t. The world is in this sense also a continuum;for to everyeventthereareasmanyneighbouringevents(realisedoratleastthink-able) as we care to choose, the co-ordinates x1, y1, z1, t1of which dier by anindenitelysmallamountfromthoseoftheeventx, y, z, toriginallyconsid-ered. That we have not been accustomed to regard the world in this sense asafour-dimensionalcontinuumisduetothefactthatinphysics,beforetheadvent of the theory of relativity, time played a dierent and more indepen-48dent role,as compared with the space coordinates. It is for this reason thatwehavebeeninthehabitof treatingtimeasanindependentcontinuum.Asamatteroffact, accordingtoclassical mechanics, timeisabsolute, i.e.itisindependentofthepositionandtheconditionofmotionofthesystemofco-ordinates. WeseethisexpressedinthelastequationoftheGalileiantransformation(t

= t)Thefour-dimensional modeof considerationof theworldis naturalonthetheoryof relativity, sinceaccordingtothis theorytimeis robbedof itsindependence. Thisisshownbythefourthequationof theLorentztransformation:t

=t vc2x_I v2c2Moreover, according to this equation the time dierence t

of two eventswith respect to K

does not in general vanish, even when the time dierencet

ofthesame events withreferenceto Kvanishes. Pure space-distanceof two events with respect to Kresults in time-distance of the same eventswith respect to K

. But the discovery of Minkowski, which was of importancefortheformaldevelopmentofthetheoryofrelativity, doesnotliehere. Itis to be found rather in the fact of his recognition that the four-dimensionalspace-time continuum of the theory of relativity, in its most essential formalproperties, showsapronouncedrelationshiptothethree-dimensional con-tinuum of Euclidean geometrical space.1In order to give due prominence tothisrelationship,however,wemustreplacetheusualtimeco-ordinatetbyanimaginarymagnitudeIctproportionaltoit. Undertheseconditions,thenaturallawssatisfyingthedemandsofthe(special)theoryofrelativityassume mathematical forms, in which the time co-ordinate plays exactly thesameroleasthethreespaceco-ordinates. Formally,thesefourco-ordinatescorrespondexactlytothethreespaceco-ordinatesinEuclideangeometry.Itmustbecleareventothenon-mathematicianthat, asaconsequenceofthis purelyformal additiontoour knowledge, thetheoryperforcegainedclearnessinnomeanmeasure.These inadequate remarks can give the reader only a vague notion of theimportantideacontributedbyMinkowski. Withoutitthegeneral theoryofrelativity,ofwhichthefundamentalideasaredevelopedinthefollowingpages, would perhaps have got no farther than its long clothes. Minkowskiswork is doubtless dicult of access to anyone inexperienced in mathematics,but since it is not necessary to have a very exact grasp of this work in ordertounderstandthefundamental ideas of either thespecial or thegeneral49theoryof relativity, I shall leaveit hereat present, andrevert toit onlytowardstheendofPart2.1Cf. thesomewhatmoredetaileddiscussioninAppendixII.50PartIIThenGeneralTheoryofRelativity51Chapter18SpecialandGeneralPrincipleofRelativityThebasalprinciple,whichwasthepivotofallourpreviousconsiderations,was the special principle of relativity, i.e. the principle of the physicalrelativityof all uniformmotion. Let as once more analyse its meaningcarefully.It was at all times clear that, fromthe point of viewof the ideaitconveystous, everymotionmustbeconsideredonlyasarelativemotion.Returningtotheillustrationwehavefrequentlyusedof theembankmentandtherailwaycarriage,wecanexpressthefactofthemotionheretakingplaceinthefollowingtwoforms,bothofwhichareequallyjustiable:a. Thecarriageisinmotionrelativetotheembankment,b. Theembankmentisinmotionrelativetothecarriage.In(a)theembankment,in(b)thecarriage,servesasthebodyofrefer-ence in our statement of the motion taking place. If it is simply a question ofdetectingorofdescribingthemotioninvolved,itisinprincipleimmaterialtowhatreference-bodywereferthemotion. Asalreadymentioned, thisisself-evident, butitmustnotbeconfusedwiththemuchmorecomprehen-sivestatementcalledtheprincipleof relativity,whichwehavetakenasthebasisofourinvestigations.The principle we have made use of not onlymaintains that we mayequallywell choosethecarriageortheembankmentasourreference-bodyforthedescriptionofanyevent(forthis,too,isself-evident). Ourprincipleratherassertswhatfollows: If weformulatethegeneral lawsof natureastheyareobtainedfromexperience,bymakinguseof52a. theembankmentasreference-body,b. therailwaycarriageasreference-body,thenthesegeneral laws of nature(e.g. thelaws of mechanics or thelawof thepropagationof lightinvacuo)haveexactlythesameforminbothcases. Thiscanalsobeexpressedasfollows: Forthephysical descriptionof natural processes, neitherof thereferencebodiesK, K

isunique(lit.speciallymarkedout)ascomparedwiththeother. Unliketherst, thislatterstatementneednotof necessityholdapriori; itisnotcontainedinthe conceptions of motion and reference-body and derivable from them;onlyexperiencecandecideastoitscorrectnessorincorrectness.Up to the present, however, we have by no means maintained the equiv-alenceof all bodies of referenceKinconnectionwiththeformulationofnatural laws. Our course was more on the following Iines. In the rst place,westartedoutfromtheassumptionthatthereexistsareference-bodyK,whoseconditionofmotionissuchthattheGalileianlawholdswithrespecttoit: Aparticleleft toitself andsucientlyfar removedfromall otherparticlesmovesuniformlyinastraightline. WithreferencetoK(Galileianreference-body)thelawsof natureweretobeassimpleaspossible. ButinadditiontoK, all bodiesof referenceK

shouldbegivenpreferenceinthissense, andtheyshouldbeexactlyequivalenttoKfortheformulationofnaturallaws,providedthattheyareinastateofuniformrectilinearandnon-rotarymotionwithrespecttoK; all thesebodiesof referencearetoberegardedasGalileianreference-bodies. Thevalidityof theprincipleofrelativitywasassumedonlyforthesereference-bodies, butnotforothers(e.g. thosepossessingmotionofadierentkind). Inthissensewespeakofthespecialprincipleofrelativity,orspecialtheoryofrelativity.Incontrasttothiswewishtounderstandbythegeneral principleofrelativitythe followingstatement: All bodies of reference K, K

, etc.,areequivalentforthedescriptionofnaturalphenomena(formulationofthegenerallawsofnature),whatevermaybetheirstateofmotion. Butbeforeproceeding farther, it ought to be pointed out that this formulation must bereplaced later by a more abstract one, for reasons which will become evidentatalaterstage.Sincetheintroductionofthespecialprincipleofrelativityhasbeenjus-tied, everyintellectwhichstrivesaftergeneralisationmustfeel thetemp-tationtoventurethesteptowardsthegeneralprincipleofrelativity. Butasimple and apparently quite reliable consideration seems to suggest that, forthe present at any rate, there is little hope of success in such an attempt; Let53us imagine ourselves transferred to our old friend the railway carriage, whichistravellingatauniformrate. Aslongasitismovingunifromly,theoccu-pant of the carriage is not sensible of its motion, and it is for this reason thathecanwithoutreluctanceinterpretthefactsofthecaseasindicatingthatthe carriage is at rest, but the embankment in motion. Moreover, accordingto the special principle of relativity, this interpretation is quite justied alsofromaphysicalpointofview.If the motion of the carriage is now changed into a non-uniform motion,asforinstancebyapowerful applicationof thebrakes, thentheoccupantof thecarriageexperiencesacorrespondinglypowerful jerkforwards. Theretarded motion is manifested in the mechanical behaviour of bodies relativeto the person in the railway carriage. The mechanical behaviour is dierentfromthat of thecasepreviouslyconsidered, andfor this reasonit wouldappear to be impossible that the same mechanical laws hold relatively to thenon-uniformlymovingcarriage,asholdwithreferencetothecarriagewhenatrestorinuniformmotion. AtalleventsitisclearthattheGalileianlawdoesnotholdwithrespecttothenon-uniformlymovingcarriage. Becauseof this, we feel compelled at the present juncture to grant a kind of absolutephysical reality to non-uniform motion, in opposition to the general principleof relatvity. But in what follows we shall soon see that this conclusion cannotbemaintained.54Chapter19TheGravitationalFieldf wepickupastoneandthenletitgo, whydoesitfall totheground?The usual answer to this question is: Because it is attracted by the earth.Modernphysicsformulatestheanswerratherdierentlyforthefollowingreason. As a result of the more careful study of electromagnetic phenomena,we have come to regard action at a distance as a process impossible withouttheinterventionof someintermediarymedium. If, forinstance, amagnetattracts apieceof iron, wecannot becontent toregardthis as meaningthatthemagnetactsdirectlyontheironthroughtheintermediateemptyspace, butweareconstrainedtoimagineafterthemannerof Faradaythat the magnet always calls into being something physically real in the spacearound it,that something being what we call a magnetic eld. In its turnthismagneticeldoperatesonthepieceof iron, sothatthelatterstrivestomovetowards themagnet. Weshall not discuss herethejusticationforthisincidental conception, whichisindeedasomewhatarbitraryone.We shall onlymentionthat withits aidelectromagnetic phenomenacanbetheoreticallyrepresentedmuchmoresatisfactorilythanwithoutit, andthisappliesparticularlytothetransmissionofelectromagneticwaves. Theeectsofgravitationalsoareregardedinananalogousmanner.Theactionof theearthonthestonetakesplaceindirectly. Theearthproducesinitssurroundingagravitational eld, whichactsonthestoneandproducesitsmotionoffall. Asweknowfromexperience,theintensityof theactiononabodydimishesaccordingtoaquitedenitelaw, asweproceedfartherandfartherawayfromtheearth. Fromourpointof viewthismeans: Thelawgoverningthepropertiesof thegravitational eldinspacemustbeaperfectlydeniteone, inordercorrectlytorepresentthediminutionofgravitationalactionwiththedistancefromoperativebodies.55Itissomethinglikethis: Thebody(e.g. theearth)producesaeldinitsimmediateneighbourhooddirectly; theintensityanddirectionof theeldatpointsfartherremovedfromthebodyarethencedeterminedbythelawwhichgovernsthepropertiesinspaceofthegravitationaleldsthemselves.In contrast to electric and magnetic elds, the gravitational eld exhibitsamostremarkableproperty, whichisoffundamental importanceforwhatfollows. Bodies which are moving under the sole inuence of a gravitationaleld receive an acceleration, which does not in the least depend either on thematerialoronthephysicalstateofthebody. Forinstance, apieceofleadandapieceofwoodfallinexactlythesamemannerinagravitationaleld(invacuo), whentheystartofromrestorwiththesameinitial velocity.Thislaw,whichholdsmostaccurately,canbeexpressedinadierentforminthelightofthefollowingconsideration.AccordingtoNewtonslawofmotion,wehave(Force)=(inertialmass) (acceleration),where the inertial mass is a characteristic constant of the accelerated body.Ifnowgravitationisthecauseoftheacceleration,wethenhave(Force)=(gravitationalmass) (intensityofthegravitationaleld),wherethegravitational massislikewiseacharacteristicconstantforthebody. Fromthesetworelationsfollows:(acceleration) =gravitationalmassinertialmassintensityofthegravitationaleldIf now, as we nd from experience, the acceleration is to be independentof the nature and the condition of the body and always the same for a givengravitational eld, thentheratioof thegravitational totheinertial massmustlikewisebethesameforall bodies. Byasuitablechoiceof unitswecanthusmakethisratioequal tounity. Wethenhavethefollowinglaw:Thegravitationalmassofabodyisequaltoitsinertiallaw.It is true that this important law had hitherto been recorded in mechan-ics, butithadnotbeeninterpreted. Asatisfactoryinterpretationcanbeobtained only if we recognise the following fact: The same quality of a bodymanifestsitselfaccordingtocircumstancesasinertiaorasweight(lit.heaviness). Inthefollowingsectionweshall showtowhat extent thisis actuallythecase, andhowthis questionis connectedwiththegeneralpostulateofrelativity.56Chapter20TheEqualityofInertialandGravitationalMassasanArgumentfortheGeneralPostuleofRelativityWeimaginealargeportionofemptyspace, sofarremovedfromstarsandother appreciablemasses, that wehavebeforeus approximatelythecon-ditionsrequiredbythefundamental lawof Galilei. ItisthenpossibletochooseaGalileianreference-bodyforthispartofspace(world),relativetowhichpointsatrestremainatrestandpointsinmotioncontinueperma-nentlyinuniformrectilinear motion. As reference-bodylet us imagineaspaciouschestresemblingaroomwithanobserverinsidewhoisequippedwithapparatus. Gravitationnaturallydoesnotexistforthisobserver. Hemustfastenhimselfwithstringstotheoor,otherwisetheslightestimpactagainst the oor will cause him to rise slowly towards the ceiling of the room.To the middle of the lid of the chest is xed externally a hook with ropeattached, andnowabeing(what kindof abeingis immaterial tous)beginspullingatthiswithaconstantforce. Thechesttogetherwiththeobserver then begin to move upwards with a uniformly accelerated motion.Incourseoftimetheirvelocitywillreachunheard-ofvaluesprovidedthatwe are viewing all this from another reference-body which is not being pulledwitharope.But how does the man in the chest regard the Process?The accelerationof thechestwill betransmittedtohimbythereactionof theoorof thechest. Hemustthereforetakeupthispressurebymeansof hislegsif he57doesnotwishtobelaidoutfull lengthontheoor. Heisthenstandinginthechestinexactlythesamewayasanyonestandsinaroomofahomeonourearth. Ifhereleasesabodywhichhepreviouslyhadinhisland,theaccelertionofthechestwillnolongerbetransmittedtothisbody, andforthisreasonthebodywill approachtheoorof thechestwithanacceler-atedrelativemotion. Theobserverwill furtherconvincehimself thattheacceleration of the body towards the oor of the chest is always of the samemagnitude, whatever kind of body he may happen to use for the experiment.Relying on his knowledge of the gravitational eld (as it was discussed inthe preceding section), the man in the chest will thus come to the conclusionthat he and the chest are in a gravitational eld which is constant with regardto time. Of course he will be puzzled for a moment as to why the chest doesnotfallinthisgravitationaleld. justthen,however,hediscoversthehookinthemiddleof thelidof thechestandtheropewhichisattachedtoit,andheconsequentlycomestotheconclusionthatthechestissuspendedatrestinthegravitationaleld.Oughtwetosmileatthemanandsaythatheerrsinhisconclusion?Idonotbelieveweoughttoifwewishtoremainconsistent;wemustratheradmitthathismodeof graspingthesituationviolatesneitherreasonnorknown mechanical laws. Even though it is being accelerated with respect totheGalileianspacerstconsidered,wecanneverthelessregardthechestasbeingatrest. Wehavethusgoodgroundsforextendingtheprincipleofrelativitytoincludebodiesofreferencewhichareacceleratedwithrespecttoeachother, andasaresult wehavegainedapowerful argument for ageneralisedpostulateofrelativity.We must note carefully that the possibility of this mode of interpretationrests onthe fundamental propertyof the gravitational eldof givingallbodiesthesameacceleration,or,whatcomestothesamething,onthelawof the equality of inertial and gravitational mass. If this natural law did notexist, themanintheacceleratedchestwouldnotbeabletointerpretthebehaviourof thebodiesaroundhimonthesuppositionof agravitationaleld, and he would not be justied on the grounds of experience in supposinghisreference-bodytobeatrest.Supposethatthemaninthechestxesaropetotheinnersideofthelid, andthatheattachesabodytothefreeendoftherope. Theresultofthiswill betostrechtheropesothatitwill hangverticallydownwards.Ifweaskforanopinionofthecauseoftensionintherope,themaninthechestwill say: Thesuspendedbodyexperiencesadownwardforceinthegravitational eld, andthisisneutralisedbythetensionoftherope; whatdetermines the magnitude of the tension of the rope is the gravitational mass58of the suspended body. On the other hand, an observer who is poised freelyin space will interpret the condition of things thus: The rope must perforcetake part in the accelerated motion of the chest, and it transmits this motiontothebodyattachedtoit. Thetensionoftheropeisjustlargeenoughtoeecttheaccelerationof thebody. Thatwhichdeterminesthemagnitudeofthetensionoftheropeistheinertialmassofthebody.Guidedbythisexample,weseethatourextensionoftheprincipleofrelativityimpliesthenecessityofthelawoftheequalityofinertialandgravitationalmass. Thuswehaveobtainedaphysicalinterpretationofthislaw.Fromourconsiderationof theacceleratedchestweseethatageneraltheoryofrelativitymustyieldimportantresultsonthelawsofgravitation.Inpointoffact,thesystematicpursuitofthegeneralideaofrelativityhassuppliedthe laws satisedbythe gravitational eld. Before proceedingfarther,however,I must warn the reader against a misconception suggestedby these considerations. A gravitational eld exists for the man in the chest,despitethefactthattherewasnosucheldfortheco-ordinatesystemrstchosen. Nowwemighteasilysupposethattheexistenceofagravitationaleld is always only an apparent one. We might also think that, regardless ofthe kind of gravitational eld which may be present, we could always chooseanother reference-body such that no gravitational eld exists with referenceto it. This is by no means true for all gravitational elds, but only for thoseof quitespecial form. It is, for instance, impossibletochooseabodyofreference such that, as judged from it, the gravitational eld of the earth (initsentirety)vanishes.We cannowappreciate whythat argument is not convincing, whichwebroughtforwardagainstthegeneral principleof relativityattheendofSection18. It is certainlytruethat theobserver intherailwaycarriageexperiencesajerkforwardsasaresultoftheapplicationofthebrake, andthat he recognises, in this the non-uniformity of motion (retardation) of thecarriage. Butheiscompelledbynobodytoreferthisjerktoarealaccel-eration (retardation) of the carriage. He might also interpret his experiencethus: Mybodyof reference(thecarriage) remains permanentlyat rest.Withreferencetoit,however,thereexists(duringtheperiodofapplicationofthebrakes)agravitational eldwhichisdirectedforwardsandwhichisvariable with respect to time. Under the inuence of this eld, the embank-menttogetherwiththeearthmovesnon-uniformlyinsuchamannerthattheiroriginalvelocityinthebackwardsdirectioniscontinuouslyreduced.59Chapter21InWhatRespectsAretheFoundationsofClassicalMechanicsandoftheSpecialTheoryofRelativityUnsatisfactory?We have alreadystatedseveral times that classical mechanics starts outfrom the following law: Material particles suciently far removed from othermaterialparticlescontinuetomoveuniformlyinastraightlineorcontinuein a state of rest. We have also repeatedly emphasised that this fundamentallaw can only be valid for bodies of reference Kwhich possess certain uniquestatesofmotion,andwhichareinuniformtranslationalmotionrelativetoeachother. Relativetootherreference-bodiesKthelawisnotvalid. Bothinclassical mechanics andinthespecial theoryof relativitywethereforedierentiatebetweenreference-bodies Krelativetowhichtherecognisedlaws of naturecanbesaidtohold, andreference-bodies Krelativetowhichtheselawsdonothold.Butnopersonwhosemodeofthoughtislogical canrestsatisedwiththis condition of things. He asks: How does it come that certain reference-bodies (or their states of motion) aregivenpriorityover other reference-bodies (ortheirstates ofmotion)?Whatis the reasonforthis Preference?In order to show clearly what I mean by this question,I shall make use of acomparison.Iamstandinginfrontofagasrange. Standingalongsideofeachother60ontherangearetwopanssomuchalikethatonemaybemistakenfortheother. Botharehalf full of water. I noticethat steamis beingemittedcontinuouslyfromtheonepan, butnotfromtheother. Iamsurprisedatthis, evenifIhaveneverseeneitheragasrangeorapanbefore. ButifInownoticealuminoussomethingof bluishcolourundertherstpanbutnotundertheother, Iceasetobeastonished, evenif Ihaveneverbeforeseenagasame. ForIcanonlysaythatthisbluishsomethingwill causetheemissionof thesteam, oratleastpossiblyitmaydoso. If, however,Inoticethebluishsomethinginneithercase, andifIobservethattheonecontinuouslyemits steamwhilst the other does not, thenI shall remainastonishedanddissatiseduntil I have discoveredsome circumstance towhichIcanattributethedierentbehaviourofthetwopans.Analogously, Iseekinvainforareal somethinginclassical mechanics(orinthespecialtheoryofrelativity)towhichIcanattributethedierentbehaviour of bodies considered with respect to the reference systems KandK.1Newtonsawthisobjectionandattemptedtoinvalidateit,butwithoutsuccess. ButE. Machrecognseditmostclearlyofall, andbecauseofthisobjectionheclaimedthatmechanicsmustbeplacedonanewbasis. Itcanonly be got rid of by means of a physics which is conformable to the generalprincipleof relativity, sincetheequationsof suchatheoryholdforeverybodyofreference,whatevermaybeitsstateofmotion.1The objectionis of importance more especially whenthe state of motion of thereference-bodyis of suchanaturethat it does not requireanyexternal agencyfor itsmaintenance,e.g. inthecasewhenthereference-bodyisrotatinguniformly.61Chapter22AFewInferencesfromtheGeneralPrincipleofRelativityThe considerations of Section 20 show that the general principle of relativityputs us in a position to derive properties of the gravitational eld in a purelytheoretical manner. Letussuppose, forinstance, thatweknowthespace-timecourseforanynatural processwhatsoever, asregardsthemannerinwhichittakesplaceintheGalileiandomainrelativetoaGalileianbodyof referenceK. Bymeansof purelytheoretical operations(i.e. simplybycalculation) we are then able to nd how this known natural process appears,asseenfromareference-bodyK

whichisacceleratedrelativelytoK. ButsinceagravitationaleldexistswithrespecttothisnewbodyofreferenceK, ourconsiderationalsoteachesushowthegravitational eldinuencestheprocessstudied.Forexample, welearnthatabodywhichisinastateofuniformrecti-linearmotionwithrespecttoK(inaccordancewiththelawof Galilei)isexecutinganacceleratedandingeneral curvilinearmotionwithrespecttotheacceleratedreference-bodyK

(chest). Thisaccelerationorcurvaturecorrespondstotheinuenceonthemovingbodyof thegravitational eldprevailingrelativelytoK. Itisknownthatagravitational eldinuencesthemovementof bodiesinthisway, sothatourconsiderationsuppliesuswithnothingessentiallynew.However, weobtainanewresultof fundamental importancewhenwecarry out the analogous consideration for a ray of light. With respect to theGalileianreference-bodyK, sucharayof lightistransmittedrectilinearly62withthevelocityc. Itcaneasilybeshownthatthepathof thesamerayof lightisnolongerastraightlinewhenweconsideritwithreferencetotheacceleratedchest(reference-bodyK

). Fromthisweconclude, that, ingeneral,raysoflightarepropagatedcurvilinearlyingravitationalelds. Intworespectsthisresultisofgreatimportance.Inthe rst place, it canbe comparedwiththe reality. Althoughadetailedexaminationofthequestionshowsthatthecurvatureoflightraysrequired by the general theory ofrelativity is only exceedinglysmall forthegravitational eldsatourdisposal inpractice, itsestimatedmagnitudeforlight rays passing the sun at grazing incidence is nevertheless 1.7 seconds ofarc. Thisoughttomanifestitself inthefollowingway. Asseenfromtheearth, certain xed stars appear to be in the neighbourhood of the sun, andarethuscapableof observationduringatotal eclipseof thesun. Atsuchtimes, thesestarsoughttoappeartobedisplacedoutwardsfromthesunbyanamountindicatedabove, ascomparedwiththeirapparentpositionintheskywhenthesunissituatedatanotherpartof theheavens. Theexaminationofthecorrectnessorotherwiseofthisdeductionisaproblemofthegreatestimportance,theearlysolutionofwhichistobeexpectedofastronomers.1In the secon