The British Society for the Philosophy of Science
Origin and Concept of Relativity (I) Author(s): G. H. Keswani Source: The British Journal for the Philosophy of Science, Vol. 15, No. 60 (Feb., 1965), pp. 286 306 Published by: Oxford University Press on behalf of The British Society for the Philosophy of Science Stable URL: http://www.jstor.org/stable/686536 Accessed: 19/05/2009 07:23 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=oup. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a notforprofit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org.
Oxford University Press and The British Society for the Philosophy of Science are collaborating with JSTOR to digitize, preserve and extend access to The British Journal for the Philosophy of Science.
http://www.jstor.org
(I) t ORIGINAND CONCEPTOF RELATIVITY G. H. KESWANI I* Introduction
asoneof the is rightlyregarded its future,relativitytheory WHATEVER has research of man. Yet, sufficient endeavours greatestintellectual notbeenmadeintoits origins. Sofaraswe areaware,it hasnoteven ' Theprinciple thephrase asto whousedanddiscussed beenestablished it in physics ' firstin the sensein whichwe understand of Relativity is physically body of a motion the xelative only that today,meaning andwhen.** ascertainable, In talkingof the origins,we meanthe specialtheory. In recent as a pariicularly yearsthe questionof originshasreceivedattention, exaniining After Whittakerl. by investigation resultof a provocative 28. iii. 64 t Received of paperis abouttheoriginof thespecialprinciple * PartsI and2 of thepresent of allmotion we shalldealwiththeideaof relativity papers Insubsequent relativity. of to theprinciple returning theoryof gravitation, in general,andwithEinstein's theoryof gravitation. of uniformmotionin thelightof Einstein's relativity Machandothershad Huyghens, Euler,Kant,Leibnitz, ** No doubt,Berkeley, of motion. Whatwe meanis a of relativity onthequestion andspeculated discussed no phenomena of motionforelectromagnetic of relativity of theprinciple discussion to of failures background withtheexperimental phenomena lessthanformechanical to ' aether.We owethisamplification detectmotionrelativeto the' luminiferous oneof thereferees. (I900of AetherandElectricity A Historyof theTheories 1(a)E. T. Whittaker, usedthe seemsto suggestthatPoincare ThomasNelson,I953. Whittaker I926), at ' forthefirsttimein theyearI904 in anaddress of relativity ' ThePrinciple phrase evenearlier.Rehadusedthephrase St.Louis,ibid.p. 30. Asweshallsee,Poincare of thatit wasdenigration J. L. Syngehasremarked conclusions, Whittaker's garding to the Contributions adding thatthiswordisalittletoostrong.' Whittaker's Einstein, ', Proc.Edin.Math.Soc.,I958, 2, partI, 46. Theoryof Relativity of Lorentz, somewellknowncontributions examines (b)ReneDugascritically Routledge& KeganPaul(English andPoincare.A Historyof Mechanics, Einstein, seepp.64S650. of I957). Particularly translation brieflyontheorigins views,MaxBorncommented by Whittaker's (c) Provoked heldinBernein theyearI955. onrelativity to a conference of thetheoryinanaddress in a bookbyMaxBornentitled:Physicsin My Generation, is reproduced Theaddress Press,I956. Pergamon 286
ORIGIN AND CONCEPT OF RELATIVITY
evidencefromvanoussources, Whittaker gavethecaseforPoincare andLorentz, almostto thecomplete exclusion of Einstein's nameandin sharpdiscordwithaccepted belie? Whittaker referred to thespecial theoryas' therelativity theoryofPoincare andLorentz ' andWhlttaker washisownman. Inthefaceof rather unpleasant reaction fromsome quarters, he maintained his judgmenteven laterirl a biographical memoirlof Einstein writtenon hisdeathin I955 We shallconsider thematterin fivesections: (a) WhatexactlywasPoincare's contribution priorto thepublicaiionof Einstein's paperof September I905 (submitted forpublicationon 30thJuneI905) ? 2 (b)Did Einstein knowof Poincare's contribution beforethe submissionof hispaper? Whichcontribution? (c) DidEinstein knowof Lorentz's mernoir3 of April/June, Igo4? (d)WhatdidPoincare, Lorentz, andEinstein themselves sayabout theanthorship of thetheory? (d) Gerald Holtonrefutestheconclusions of Whittaker:' OntheOriginsof the SpecialTheoryof Relativity,' Amer.J. Phys., I960, 28, 627. (e) Oneof theearliest examinations ratherbrief was madebyW. Pauli,who touched onthequestion oftheorigins hereandthereinanarticle onrelativity published intheyearI92I andreprinted inEnglish withnotesas: Theoryof Relativity,Pergamon Press,I958, pp. 3 and78 particularly. Pauliconcluded, ' Theformalgapsleftby Lorentz's workwerefilledbyPoincare.Hestatedtherelativity principle to begenerallyandrigorously valid. ItwasEinstein, finally,whoin a way completed thebasic formulation of thisdiscipline.' 1 Obituary by E. T. Whittaker in BiographicalMemoirsof Fellows of the Royal Society, I955, London, pp.3767. Thismemoircontains littlenewmaterial bearing onthequestion of prioritythat isnotthereinhisHistory, footnoteI(a)above,exceptingaremark onp.42totheeSect,' Einstein adopted Poincare's Principle of Relativity (usingPoincare's nameforit) '. 2 ' On theElectrodynamics of MovingBodies', Ann.derPhys., I905, I7, 89I: English translation in ThePrinciple ofRelativity, Dover,towhichweshallgivereferences here. 3 ' Electrodynamic Phenomena in a SystemMovingwithanyVelocitylessthan thatof Light', Proc.Acad.Sci.Amsterdam, I904, 6, 809,reproduced in rhe Principle of Relativity,Dover,to whichwe shallgive references here. Lorentz's memoirwas presented at the April,I904 meetingof the Academyandits Englishtranslation appeared inJuneI904. Whittaker, op.cit.,quotestheyearof publication in thetext andfootnoteonp. 3 I, asI903. Holton,ibid.,p. 63S,readsa meaning in thisslipof Whittaker allegingthat,' SinceWhittaker wasotherwise verycareful withvoluminouscitations of references, thisrepeated slip,whichdoublesthetimeintervalbetweenthe workof Lorentzandof Einsteinis not merelya mistake.'However, Whittaker quotesthecorrect year(I904) onp. 30,footnote4 andp.S3footnoteI, and obviously, mentionof theyearI903 elsewhere wasonlya mistake, pureandsimple. 287
G. H. KESWANI
(e) Didthethree Poincare, Lorents, andEinsteinmean thesame thingin theirideason relativity? Inwhatfollows,weshallrefertotheabovepaperof Einstein andthe earliermemoirof Lorents,onlyby thenamesof thetwo authors followedby thepertinent pagenumber of references 3 and4. Throughout, italicshavebeenusedby us for emphasis but no attempthasbeenmadeto showseparately the italicsusedby other authorsthemselves. 2 Poincare's Contribution
Theideaof relativity of motionhadbeensimmering in Poincare's mindatleastsincetheyearI 895whenhesaid,l' Et,eneffet,l'impossibilitede mettreen evidenceun mouvement relatifde la matierepar rapport a l'ether....' Thisis perhaps theearliest conjecture whichis thecentralideaof therelativity of motion,thatis, it is impossible to bringintoevidence thevelocityof a bodyrelativeto theaetllerorany absolute standard of rest. IntheyearI899, he said2similarly: 'je regarde commetrespro6alD1e quele phenomenes optiques nedependentquedesmouvements relatifs descorpsmateriels enpresence, sources luniineuses ou appareils optiqueset celanonpasauxquantites presde l'ordreducarreou ducubedel'aberration, maisrigoureusement.'
IntheyearI900 he queried3 againif theaetherreallyexistedadding thathebelieved thatexperiment couldrevealnothingmorethanrelative displacements. In the sameyear,he verynearlycameto namethe principle of relativity whenhe usedtheequivalent butphysically even moredescriptive andprecisephrase, ' Theprinciple ofrelative motion ',4 to signifythatit is possibleto ascertain onlythe relativemotionsof bodies. Soonafter,in theyeal I902, he analysed thisideacritically andat length,andappalently forthefirsttimeusedtheappellatiolls 'TheLaw 1 OeuvresdeHenriPoincare,t. ix, GauthierVillars, Paris,I954, p. 4I3. Henceforth we shallreferto thisvolumeof Poincare's worksasOeuvres. 2 A Sorbonne lecture of I 899reproduced inElectricite' et Optique,GauthierVillars, Paris,I954, p. 536. Whittaker, op.cit.,p. 30, in givinga translation of thispassage omitsthewords' ou ducube', anomission of littleconsequence. 3 Quoted in Whittaker, op.cit.,p. 30 4 Oeuvres,pp. 482483. Thewordsusedrepeatedly are,' Leprincipe dumouvementrelatif'. 288
ORIGIN AND CONCEPT OF RELATIVITY
of relativity'and' Theprincipleof relativity'lin hisbookScienceet Hypothese.
or the motion2 orthelawof relativity of relative Inliisprinciple phrases in I902 usedalltheseequivalent principle of relativityhe aswellas of positions ordisplacements therelativity Poincare asserted of in respect of uniform velocities.Allpointsin spaceareequivalent and saidPoincare, thatwecantakewithourinstruments3, thereadings distances only. A widerprinciple, tomeasure relative thusit ispossible axes thatof relativemotion,holdswhenthemotionof themovable linebutitisnottruewhen isuniform andinastraight (reference system) of insupport rotational,4 or(evenuniformly) themotion isaccelerated whena trainis quotesthefactof impactoccurring whichPoincare of flattening atthepolesof a rotating andthephenomena decelerated, time, pendulum.Regarding sphereandthe motionof Foucault's book 5 in thesame Poincare remarked time.... Notonlyhaveweno directintuitionof the 'There isnoabsolure equalityof two periods,butwe havenot evendirectintuitionof the places.' at two different of two eventsoccurrirlg sinlultaneity
inSeptember, I904, by werefollowed reflections Thesepenetrating before theInterdelivered byPoincare inanaddress similar reflections at St.Louisin U.S.A.6We of ArtsandSciences national Congress someminoromisatlengthpartlyto correct quotefromthisaddress betterthe butgenerally to understand translation, sionsin Whittaker's version mind. Thepagesreferto theEnglish processes of Poincare's 4. mentioned infootnote according to whichthelawsof physof relativity, p. 5 ' Theprinciple fixed, foranobserver shouldbethesame,whether icalphenomena of translaalongin uniformmovement carried orforanobserver tion....'
pp.76 and244. In Englishtrans.,Dover,I952 reprint, 1 ScienceandHypothesis, of thisbookis givenasI903 . ReneDugas,ibid. someplaces,theyearof publication asI902. by Flammarion p. 660,givestheyearof firstpublication Dover,pp. IIIII2 2 Scietlce andHypothesis, 3 Ibid.
p
77
of a Einsteinandothersalsoquotethe caseof flattening Ibid.pp. II3II4. of theGeneralTheoryof in thesamecontext. TheFoundations sphere intoanellipsoid Dover,p. I I2. ofRelativity, Retativity, Eng.trans.ThePrinciple in I898; seefootsaidso earlier Dover,p. go. Poincare 5 Science andHypothesis, note,ibid. appeared in rheMonistof by G.B. Halsted translation of theaddress 6 AnEnglish waspublished inBull.desSc. Theoriginalin French January I905, I5, No. I, I24. 4
Math.,I904,
28,
302.
289
G. H. KESWANI mentions It is to benotedthatPoincare
only uniform translational
motion.
thesignalfromthestationA, its B perceives ' Whenthe station clockshouldnot markthesamehourasthatof stationA at the by a conmomentof sendingthesignal,butthishouraugmented for Suppose, of thetransmission. theduration stantrepresenting example,thatthestationA sendsitssignalwhenitsclockmarks it whenitsclockmarks B perceives thehouro, andthatthestation if the slownessequalto t the hourt. The clocksareadjusted atldto verifyit, the of thetransmission, theduration represents stationB sendsin its turna signalwhenits clockmarkst. The timepiecesarethenadjusted.Andin fact,theymarkthesame whichis instant,buton onecondition, houratthesamephysical casetheduration arefixed. Inthecontrary thatthetwo stations sincethe willnotbethesameinthetwosenses, of thetransmission to meettheopticalperturbA forexamplemovesforward station B fliesaway,beforethe fromB, whilethestation ationemanating in that fromA. Thewatchesadjusted emanating perturbation thetruetime,theymarkwhatone donotmark,therefore, manner maycallthelocaltime,sothatoneof themgoesslowontheother. it. Allthe littlesincewe haveno meansof perceiving It matters whichhappenatA, forexamplewill be late,butall phenomena themwillnot whoascertains will beequallyso,andtheobserver of relativity it sincehiswatchis slow; soastheprinciple perceive heisat wouldhaveit, hewillhaveno meansof knowingwhether motion. restor in absolute hypothatdoesnot suffice,andcomplementary Unhappily, to admituniformcontraction it is necessary thesesarenecessary; in thesenseof themotion.' at rest of clocks Poincaredescribesthe methodof synchronisation two to applied when method this in discussing and (withina system) in relativemotioncomesto theconclusionthatone ofthe clocks clocks tliis findsthe principleof relativity goes slowerandnotwithstanding validfor thesystemsattachedto thetwo clocks. He alsoassertsthatit is thennecessaryto assumethatbodiesin motionundergocontraction. wouldariseanentirely if theyareconfirmed, alltheseresults, p. I6 ' From by this characterised all, above be, would which newmechanics, .' . . light of that surpass could velocity no fact,that Whittakerlquotes this passagesomewhatdifferentlymakingthe p.
28,
I0
versioninBull. desSc. Math.I904, op.cit.p. 3I. TheFrench 1E. T. Whittaker, unemechanique sortirait s'ilsseconfirmaient, as,' De touscesresultats, 3I6, reads 290
ORIGIN AND CONCEPT OF RELATIVITY
statement of theideascategorical.Hisrendering reads: ' Fromallthese resxlts theremust ariseanentirelynewkind of dynamics . . .'
TheoriginalFrench versionshowsthatHalsted's translation isaccurate. ' Shollldwe not also endeavour to obtaina moresatisfactory theoryof theelectrodynamics of bodiesin motion? Letustake,therefore, thetheoryof Lorentz, turnit in allsenses, modifyit littleby littleandperhaps everythingwill arrange itself.' p. 23 ' Perhaps likewise,we shouldconstruct a wholenewmechanics, that we onlysucceed in catching a glimpseof, whereinertiaincreasing withthevelocity,thevelocityof lightwouldbecomeanimpassablelimit' p.
I9
HereagainWhittaker'sl version' anewmechanics, where,theinertia increasing withthevelocity,thevelocityof lightwouldbecomealimit thatcouldnotbe exceeded ' is not qualified by Poincare's reservation2 ' thatwe onlysucceed in catching a glimpseof', whichis contained in theoriginaladdress in themiddleof thesentence. Theaddress contained no mathematics; it wasa popularlecture, ratherphilosophic andreflective.Althoughtheprinciple of relativity wassuccinctly enunciated, Poincareobviouslyhadsomedoubtsstill. Noticethe ' perhaps'mentionedtwiceandthe persistent moodof retraction at a feelingof insufficiency of theproposed doctrine.The philosophic moodandtheautonomous powerof wordsno doubtmade it possible fol himto carryhisformulation to a pointthatit cameto be verified. Buthe himselfhadmisgivings, definitely.Hisinsightand eloquence carriedhimwithanimpulsewhichhe couldnot yet fully sustain.It is significant thatforaboutninemonthshe didnot givea mathematical formto hisprinciple of relativity butthereweregood reasons forthishesitancy. In supportof this thesis,the followingadditional pessimistic remarksmadein theStLouisaddress maybenoted.Asbefore,thepages referto theEnglishversionin TheMonist. entierement nouvellequiseraitsurtoutcaracterisee parce faitqu'aucune vitessene pourrait depasser celledela lumiere,. . . ' 1E. T. Whittaker, op.cit.p. 3I 2 H. Poincare, Bull. des Sc. Math.,I904, 28, 324. Thewordsare,' Peutetre aussi devonsnous construire touteuneMecanique nouvellequenousne faisonsqu'entrevoir,ou, l'inertiecroissant avecla vitesse,la vitessede la lumieredeviendrait une imiteinfranchissable.' 29I
G. H. KESWANI
p. g ' We cometo theprinciple ofrelativity: thisnotonlyis confirmed by dailyexperience, notonlyis it a necessary consequence of the hypothesis of central forces,butit isimposed inanirresistible way uponourgoodsense,andyet it alsois battered.' p. I7 ' In the midstof so manyruins whatremains standing?The principleof leastactionis hithertointact. . .' (Inhis address Poincare commented onvariousprinciples suchastheprinciple of relativity bUt seemed to thinkthatthisprinciple didnot remain standing.)
Wemust,however, takenoteof thefollowing. (a)' Theprinciple of relativity' wasmentioned in theaddress andclearlydefined; a newmechanics inwhichtherrelocity of lightcouldnotbe surpassed waspredicted. (b)Themethodof synchronisation of clockswithlightsignals wasdescribed. (c)A more satisfactory theoryof ' the electrodynamics of movingbodies'wasconsidered necessary andit wasexpectedthatthiscouldbe secured by somemodification to Lorentz's theory. However, it wasnotlongbeforePoincare published a definitive paper.InJune I905 and stillbefore theappearance of Einstein's paper in September, Ig05,Poincarel published a paperentitled' Surla dynamique del'electron ' inwhichheestablished complete covariance of Maxwell's equations undera Lorentz transformation, including the correct transformation formulae forthecasewhenthespace, forwhich Maxwell's equations aregiven,isoccupied byelectric charges. Lorentz (p. I 5)had notsucceeded inestablishing covariance ofMaxwell's equatiOtlS involving chargedensity. Inthisrespect Poincare's paper wasa continuation ofLorentz's memoir ofI904 inwhichlleproposed thenew (Lorentz) transformation equations forspace andtime. Butmorethan this, Poincare immediately recognised the comlection betweenhis principle of relativity andthetransformation equations ofLorentz and remarked,2 ' Lorentz acherche acompleter etamodifier sonhypothese defaSon a lamettre enconcordance, avecle postulat del'impossibilite complete de la determination du mouvement absolu.'He sawtllat theLorentz transformation equations meantthatit wasimpossible to determine absolute motion.Itwasin thisverypaper thathenamed thetransformation equations afterLorentz.Henceforth weshallrefer H. Poincare, Cosnptes Rendus,I905, 2 Ibid.
p. I 504.
Oeuvres, p. 489 292
I40,
I504
ORIGIN
AND
CONCEPT
OF RELATIVITY
theseequations asL.T.E.(Lorentz transformation equations). The assertion thatL.T.E.forma groupalsooccurs hereforthefirsttime. Incidentally, Poincare suggested in thispaper thatgravitational effiects arepropogated withthevelocity oflight:indeed hetalked of gravitationalwaves! Soonafter,Poincare enlarged thisshortpaper intoalongeroneby thesamename. Thissecond paper,l written inJuly,Igo5,was,however,published inJanuary I906, after thepublication ofEinstein's paper. Theconnection between thepostulate ofrelativity andL.T.E. washere reiterated, theconception oftheLorents groupwasfurther developed, theterm' invariants dugroupe deLorents ' wasused2 forthefirsttime and(x,y, z, +/ I t) wereregarded ascoordinates in a4dimensional manifold. Forthesakeofhistorical interest wemaymention a thirdpaper by thesamename,published byPoincare 3 in I908 inwhich hecompared theprinciple ofrelativity toa ' pri1ciple ofequivalence ', introducing this phrase alsoapparently forthefirsttime. To summarise: AsfarbackasI895, Poincare theinnovator had conjectured thatit isimpossible todetectabsolute motion.InI900 he introduced ' Theprinciple of relative motion' whichhelatercalled by theequivalent terms' Thelawof relativity' and' Theprinciple of relativity' in hisbookScience andHypothesis published in I902. He furtller asserted inthisbookthatthereisnoabsolute timeandthatwe havenointuition of the' simultaneity ' of two' events'4 (mark the words)occurring attwodiSerent places.Ina lecture givenin I904, to
Poincarereiteratedthe principleof relativity, describedthe method of synchronisationof clocks with light signals,urged a more satisfactory theory of the electrodynamicsof moving bodies based on Lorents's ideas and predicteda new mechanicscharacterisedby the rule that the 1H. Poincare, ' Surladynamique del'electron ', Rendicontidel Circolotnatematico 2I, I29. Oeuvres,p. 494. ' Paris, juilletI905 ' iS mentioned atthe endof thepaper. di Palerno, I906, 2
Oeuvres,p. Oeuvres,p.
54I
567. Einstein laterusedthisphrase in a somewhat different sense. 4 Thenotion of an' event' ascharacterised bya giveninstant of time,a particular place and a physicaloccurrence thenandthere is indeedeven older. Years earlier RobertBrowning lamented ! 3
Neverthe timeandtheplace Andthelovedonealltogether. (Never the Timeand the Place) 293
G. H. KESWANI
inJuneI905 Thiswasfollowed besurpassed.l oflightcannot velocity ' in del'electron 4 Surladynamique paper entitled bya mathematical of detecting (impossibility betweenrelativity whichtheconnection a givenby Lorentz transformation motion)andtheLorents absolute wasrecognised. yearearlier wasnotonlythefirstto enunPoincare Inpoint of fact,therefore, worktllenecesinLorentz's buthealsodiscovered ciatetheprinciple, atonceleadsto thatthevelocityoflightcannotbesurpassed, postulate 1Poincare's meansthatc, thevelocity etc. Thispostulate formula, velocityaddition thecorrect is c itself. (Infact witha velocityv (< c)in thesamedirection of lightcompounded of v). thedirection thisis truewhatever v1andv2maybe of anytwovelocities fortheresultant formula Themostgeneral putin theform: +(V1,
V2)X (V1 + V2)
is madeyetreformandno assertion It is to benotedthatthisis analgebraic observable velocity. garding themaximum demands that Whenvl = c, theabovepostulate 0(c,
v2)x(c+v2)=
ie vl=s
c+ V2
c I + V2/C
v1andv2in the of anytwovelocities forthecomposition formula Now therequired tO I/(I/ + V21C) or in v1andv2and+ mustreduce mustbesymmetrical samedirection then, Obviously I/(I + V1/C) whenv1= c or v2= crespectively. I + V1V2/C2
of v1andv2is andtheresultant V1 +
V2
I + V1V2/C2
(vl+ v2)whenvl andv2arein thesame valuethantheclassical whichgivesa smaller .
,
( .lrectlon.
andthen different methodto thesamepurpose usesa somewhat E. T. Whittaker oftwovelocities.FromEuclidto forthecomposition derives L.T.E.fromtheformula 6I62. Cambridge, I949, pp. 4950, Eddington, higherthanc can' occur' (withintheframevelocities whether To thequestion Yes. A Sophistiisanemphatic answer P.W. Bridgman's theory), workofthespecial Wesleyan Univ.Press,I962, p. I08. Hewrites,' Ofcourse, ofRelativity, cate'sPris1ler formsimpleaddition inthesameframe,theclassical weremeasured if bothvelocities the inthestationaryframe willcontinue tohold. Inparticular, velocities ulaforrelative moving toaparticle movingto theleftwithvelocityo 75c relative velocityofaparticle willbe,despite hasbeenandalways to therightwithvelocityo 75cis I vS C, asit always higherthancdonotoccur. Sucharelative velocities statements thatrelative frequent If we measurement. notby directphysical by calculation, velocitycanbe obtained movingtotheleftwithvelocityo 75c on theparticle ameasuring apparatus stationed movingtotherightwithvelocityo 75 thevelocityoftheparticle andwithit measured lessthanc. Infact,we wouldobtaino g6 c.' c we wouldobtainsomething 294
ORIGIN AND CONCEPT OF RELATIVITY
sarymathematical formulation of the principle.All thishappened beforeEinstein's paperappeared. Thereis one difference thatmustbe noted. AlthoughPoincare statedhisideason therelativity of motionclearly,hehadsomedoubts whenthe principle wasappliedto accelerated motion. Therewere diSlculties, no doubt,whichseemedto demandan absoluteframe. Couldhe thenrejecttheaetheroutright? Yetthesedoubtsanddiflilculties cantakeawaylittlefromhiscontributionswhichstandindependently of hisdoubts. We shouldnotforget tliathe wasexploringrealmsof thoughtneverconceived before. Itappears to usthatthoselwhoconsider thatPoincare failedto takethe decisivestep,havenot fullyvaluedhis contribution andhavebeen severein theirjudgment.Hetooktherequired stepbutlookedround andsawthattherewereseriousdifficulties still; therewasa hesitancy butonethatcomesfroma fullness of knowledge.AfterallPoincare knewthenmorethananyoneelsewhatexactlywerethestakes. Poincare regarded relativity to beolllyempirically trueof uniform motion,butnotabsolutely true;it wasnotaltogether impossible forit to be violatedby someyet unknownexperiment.2Talkingof the We submitthatthefirstpartof theabovestatement is somewhat misleading.It isnotphysically meaningful tospeakof thevelocityof aparticle (movingtotheleft) retative toanother particle (movingtotheright)butasit (thevelocity)' occurs' in the framein whichtheobserver is stationary.ThusthefigureI5 C calculated aboveis a number arisingin analgebraic process, butnota resultof possible physical measurements. (Bridgman admitsthis.) ThepictureinBridgman's mindappears tohavebeensomething asfollows:I can seetwoparticles flyingapart withavelocityo 75crelative to me; sodoI notseethem gettingapartat a relativevelocityof I5 c? Theanswer is: No; oneor theother particle is movingawayfrommewithvelocityo 75c, whichis physically true,but thereis no meaning in anassertion aboutthevelocityof a particle relative to another particle butasthisrelative velocityof thetwoparticles occursin a thirdframe. 1Principally, we havein mindthe remarks of R. Taton,ReasonandChancein ScientHc Discovery, English trans., Hutchinson, I957, paragraph entitled, ' Poincare and thetheoryof relativity ', pp.I34I35. We shouldliketo saythatthejudgments in the paragraph mentioned arenot explicitlyrelatedto specificobservations on the contributions of Poincare andEinstein butareonlygeneralisations, apparently based Onsomepersonal 1mpresslons. 2 Such a viewwasheldalsobyH. A. Lorentz apparently upto theend. Hesaid, ' Thequestion whethertheprinciple of relativity holdsor not mustbe decidedby experiment.'Lectures on TheoreticatPhysics, vol.3, Macmillan, London,I93I, p. 255. H. Bondi,similarly, remarked recently, ' Thusspecial relativity isnotsomething that absolutely mustbetrue;it ismerelythestatement thatforalargerangeandvarietyof experiments, thepreferred velocityis irrelevant '. Proc.Roy. Soc.(A), I962, 270, 3II. 295 .
*
.
G. H. KE SWANI
possibility ofdetecting theabsolute motion oftheearth hesaid,l' Will thiseverbeaccomplished? I donotthinkso,andI shallexplain why; andyet it is not absurd....' 3 WasEinsteinawareof Poincare'sWork?
Einstein (p.38) introduces thenovelkeyphrase, ' Theprinciple of relativity ',intheverybeginning ofhispaper exactly inthewordsused byPoincare earlier inScienceand HypothesisandinhisStLouisaddress. Arewetoregard thisasamarvellous concurrence? ' Relativity ' wasa wordthenlittleusedinphysics, butEinstein usesitwithout anexplanation.
Andthatbrings usto thequestion whether Einstein wasaware of Poincare's earlier works.Itiscertain thathewas. Hehaddrunk deep atPoincare's Scienceet Hypotheseandforthiswehavetlleindependent testimony of CarlSeelig,2 Einstein's biographer. Thepagesgiven belowreferto footnote 2, p. 296. Einstein hadreadtheabovebook inthecompany ofSolovine andConrad Habicht under theauspices ofa privategathering called' Olympia'in BernebeforeSolovineleft Bernein I905 andHabicht earlier, in I904. p. 57 'AfterawhileSolovine suggested thattheyshouldreadusefulbooks together of anevening.... In additionto theseprogrammes included..., HenriPoitlcare's Lascience etL'Hypothese.... In 19?S Solovineleft Bernefor Pariswherehe wasemployed fromI909I9 assecretary andcollalDorator on theRevuePhilosopl1que.... 5960 ' Before Conrad HalDicht waselectedin 1904to thepostof mathematicsandphysicsteacher in theProtestant Educational Institute in Schiers (Graubunden)....' p. 6I ' Theroleplayed by " Olympia" astheychristened theirprivate academy, duringEinstein's firstyearsilsBerne,cannotbe underestimated initsinfluence andintellectual itnportance.Foramong thebooksdigested duringtheseregular evenings of discussion and reading,thosewhichmadea lastingimpression uponhimwere worksof ErnstMach,He>riPoincare's La Science et l'hypotheseConrad Habicht finallyproduced thislongawaited book froma bookshop....' .
p.
b
,
a letterdated6thMarch I905, to Habicht: 1 Science 1ld Hypothesis, Dover, p. I7I Carl Seelig, AlbertEinstein, StaplesPress,English trans., I956
2
296
ORIGIN AND CONCEPT OF RELATIVITY
of studyisstilla mereconcept:theelectrodynamics p. 75 ' Thefourth theory of space of the of a modification by the use moving bodies ofthisworkwillundoubtedly kinematispart andtime. Thepurely interest you.' bookScienceet Hypothesewasreadtogether Poincare's Obviously orwritingof andSolovinebeforethepublication Habicht, by Einstein, paperof I905. Einstein's Theexactdatewhenthisbookwasreadat the' Olympia'is not butthereis littledoubtleft givenbySeelig(itis hardlyto beexpected) wrotehis own paper. thatit wasbeforeEinstein by the chronology of which wassvellawareof Poincare.Theprivateacademy Einstein anxiously had been were members Einstein,Solovineand Habicht Scienceet Hypothese. waitingforPoincare's of learnttheprinciple thatEinstein to assume correct Itis,therefore, fromPoincare.l relativity inhispaperof I905 usedthemethodof WemaynotethatEinstein, of clockswithina systemexactlyas Poincaredid synchronisation postulate formPoincare's earlier in I904 andusedin a hardlydifferent thatthevelocityof lightcannotbe surpassed. paper. Forexample, thingsaboutEinstein's Therearesomecurious of movingbodies'? whydidhecallhispaper,' Ontheelectrodynamics of movingbodies wasvery In a sensethissubjectelectrodynamics wrotehis paper. However,the muchin theairat thetimeEinstein asexactlyor evenlargely papercannotbe regarded titleof Einstein's of the theoryof electroof its thesis. The fulfilment representative only through of movingbodiesin realitybecamepossible dynamics laterwork.In thewordsof Sommerfeld2: Minkowski7s had greatrespectfor Poincare.In fact,at oneplace,he 1Of course,Einstein Poincare. attitude towards reverent adopted a curiously to the volumeAlbert of variouscontributors Whilereplyingto the criticism I949 (seep. 677), of LivingPhilosophers, TheLibrary Scientist, EinsteitlPhilosopher canbeconsidered geometry whether aboutthequestion, Einstein putsthearguments pointof view,in theformof a discussion or falsifiable fromthephysical verif1able withthediscussion However,afterproceeding andReichenbach. betweenPoincare recoilsfromtheideaof spirit,Einstein fora whilein a purelyscientific irlthismanner debatewiththe chosenpairin imaginary in thisvoluntarily beingincluded Poincare in thisfashionbecausethe cannotbe continued ' Theconversation suddenremark, asthinker andauthordoesnot superiority writerforPoincare's respect of thepresent is substituted for nonpositivist ananonymous permitit; in whatfollows,therefore, '. Poincare Academic Press,I952, p. 280 Electrodyala>lics, 2 A. Sommerfeld, x
297
G. H. KESWANI
' EvenH. A. Lorentsdidnotquiteattainthefinalformin hispapers lDodies. not for magnetizable particularly (I903), in the Enzyklopadie of moving calledhispaperof I905 " On theelectrodynamics Einstein goalofhistheoryofrelaprincipal thismannera bodies"andindicatedin of the generalstructure upon the not enter tivity; howeverhe does the instead to himself but confines bodies ponderable for equations in I908, atlong electron.Minkowski arisingfortheisolated equations wasthefirstto solve of relativity, of theprinciple lastin fullpossession theproblemcompletely'.
formulearlierthata satisfactory however,hadindicated Poincare, with of movingbodieswasconnected ationof the electrodynamics theproblemof theaetheror theconceptof relativity. We let thismatterresttherepartlyin therealmof conjecture. 4 WasEinsteinawareof Lorentz'sMemoirof 1904?
all in AnnalenderPhysika sevenpapers, Einsteinhad published leadingjournalof physics duringthe periodI9OI5 andbeforethe paperandwas definitelywell in touch of the relativity publication ideasin physics,butasMaxBornlpointsout,thereis a withcurrent point ' Thestriking relativit,ypaper. peculiarityaboutthis conspicious literature.Itgives to previous nota singlereference is thatit contains of quitea newventure.Butthatis,of course,asI youtlleimpression havetriedto explain,not true.' work withprevious hopeto tracetheconnection Wecan,therefore, onlyfromthetextof thepaperbutbeforewe do thatlet us seewhat madein theI904 memoirof Lorentz. exactlywasthecontribution saidthatLorentsput forwardL.T.E.to establish It is sometimes equations.2Thiswasonlyapartof Lorentz's of Maxwell's covariance to showthatthe primarily programme.He devisedtheseequations forspaceandtimeweresuchasto givea nullresultof transformations originthatis howtheseequations experiments3: variousaetherdrift Pergamon Press,I956, 1MaxBorn,Physicsin My Generatiorl,
p. I93
andMagnetism, TheoryofElectricity see: J.Jeans,TheMathematical 2 Forexample I960, p. 600 Cambridge, equations weregivenby W. himselfpointedoutyearsago,similar 3 As Lorentz Voigtin I887 so thattheformof thewaveequation
a2+ a2+ a20_I a2+ aX2+ ay2 + aZ2 C2at2 X
thesamefor a movingsystem. ReneDugas,op. cit. p. 468,therefore, remained equations. transformation thattheyshouldbecalledVoigtLorentz suggests 298
ORIGIN AND CONCEPT OF RELATIVITY
(pp.III3 and29). Hethenshowedthatthemassof atedwithLorentz for withvelocity(p. 24), appliedthe equations an electronincreases then dataforelectrons of theexperimental toananalysis massvariation by areinfluenced of all particles thatmasses andconjectured available in the samemanner.This partof the work,surely, a translation of transformaasa mereattemptto findequations cannotbe regarded systemto anothersuchthatthe formof tion fromone coordinate unchanged. equations remains Maxwell's of for the variation the formulae IndeedEinsteinalsoestablished only,butwithhischarmasswithvelocityforthecaseof anelectron saw,'. . . theseresults physical insight,he(p.63)immediately acteristic points,becausea material as to themassarevalidfor all ponderable (inoursenseof material pointcanbemadeintoanelectron ponderable charge, nomatterhowsmall'. of anelectric theword)bytheaddition (Andwe couldnow say,no matterhow big!) How simple! The of thechargeof anelecon themagnitude analysis wasnotdependent their whatever trueof allmasses, are,therefore, tronandtheformulae charge. expressionl for the (pp.6263)got incorrect Einstein Incidentally, insteadof [mO/(Iv2/c2)2].Butfor mass', [mO/(Iv2lc2)] ' transverse [mO/(Iv2/c2)2]. mass' he gavethecorrectexpression the' longitudinal forboth. Lorentzgavecorrectexpressions knewof Lorentz's whetherEinstein We nowcometo thequestion there isa significant evidence but memoirof I904. Thereisno definite paper(p.60)to theeffect: remark in Einstein's vector(ux, ofvelocitiesthe thetheorem ofaddition ' Since asfollowsfrom uyX uz)is nothingelsethanthevelocityof theelectriccharge,measured in thesystemK, we havethe proofthat,onthebasisof ourkinematic theelectroofLorentz'stheoryof foundation thee]ectrodynamic principles, ofrelativity.' withtheprincipal of movingbodiesis inagreement dynamics
didnot sayexactlywhichtheoryor workof Lorentzhe Einstein wasin accordwiththe meantbutit is clearthatthetheoryin question kinematical of relaiivity,andthison the basisof Einstein's principle any wasnot claiming principles.It is to be notedagainthatEinstein advance butonlyanagreement.Now,noneof thetheories significant by himbeforehismemoirof I904 wasin agreeof Lorentzpublished (L.T.E.) of relativitybut the basicequations mentwiththe principle of of the memotrof I904 werestrictlyin accordwith the principle of Relativity,Dover,p. 63, and 1 See footnotesby A. Somlnerfeld,ThePrinciple ReneDugas,op. cit. p. 482 299
G. H. KESWANI
papers). remarks on thiswillfollowin subsequent relativity (further havehadin hismindanyothertheoryof Therefore, couldEinstein a yearearlier?He knewof Lorentz thanthe one propounded ', hehimof movingbodies ' Lorentz's theoryof theelectrodynamics a newtheoryalsoof theelectrodynamics selfwasputtingforward of ' a modification involving of movingbodies(thetitleof hispaper) thetheoryof spaceandtime', ashe putit in a letterlto hisfriend likelythatit couldbe I905. Itishardly on6thMarch Habicht written whichmight,afterall,havebeensuperseded anoldtheoryof Lorentz whowasstillaliveandattheheightofhis ofLorentz, byalatertheory wasto be in thistheoryof Lorentz powers.Moreover, intellectual ofEinstein's new' kineonthebasis ofrelativity agreement withtheprinciple didindeed andwe shallnowshowthatEinstein matical principles',
of toLorentz's memoir viewedinrelation establish suchanagreement principles ' Lorentz of lackof these' kinematical I904. Onaccount only equations covariance of Maxwell's hadbeenableto establish Lorentz hadnotdisit wasbecause partially.To be moreprecise, velocities thathe got forcompounditlg formula covered thecorrect in a formwhichis notfullythe equations of Maxnvell transformed follows thisformula ' system.Ofcourse, sameasforthe' stationary in the of Lorentz in thepossession fromL.T.E.,whichwerealready yearI904. remark thatatthe wemay,however, further, Beforewe proceed leaving madea statement (p.37)Eillstein of hispaper verybeginning workers had' showntothefirstorder thattheprevious theimpression andoptics (that)thesamelawsof electrodynanzics of smallquantities, of of reference forwhichtheequations willbe validforallframes abovedoesnot onp.60quoted mechanics holdgood' buthisstatement goes. Thisstatein sofarastheorderof accuracy claimanyadvance himself of fact,Lorentz onlyanagreement. Asa matter mentasserted (p.I5) thathistransformation ofhismemoir remarked atthebeginning of v, equations werevalidforsmallvaluesof v so that1,a function order', ofthesecond diflRered fromunity' nomorethanbya quantity p.27) (Lorentz, whenheshowed removed thislimitation buthehimself equalto unity,always. thatIwasindeed outthe (p.44) carried (p.I4) andEinstein bothLorentz Incidentally, thefirstonebeinga transition ' intwostages, ' Lorentz transformation replaced x by (xVt) in ' axes. Wehavetherefore, to the' moving atthemoregeneral (p.I4). Also,bothfirstarrive Lorentz's formulae CarlSeelig,op. cit. p. 75; alsoquotedearlierin tlle presentpaper. 3oo
ORIGIN AND CONCEPT OF RELATIVITY
x'pI(x vt)or:+(v) (x vt),etc.andshow formof transformation (p.27,47)thatIor+(v)mustbeunity. Wehave methods by diSerent takenI I. for electric As usual,Lorentz(p. I3) took Maxwell'sequations uZ,ug,uz) chargein emptyspacemovingwithvelocityu (components as: of coordinates ' system ' or ' stationary to the ' fixed referred divEp, divHO curl
H
=(at
curlE
+pu),

c
at '
where,E electricfieldstrength, and fieldstrength, H magnetic of electriccharges. p volumedensity in the charges as measured of velocityof theelectric Letthecomponents 'moving' system be ux, uy anduxandthe velocityof the 'moving' ' systembev in thedirecto the' stationary systemitselfwithreference (wehaveusedmoreor equations tionof U$. Usingthetransformation paper): lessthesamenotationasusedin Einstein's t'  :(t vxlc2)l LTE vt), y'y, z'z, (i) x':(x J * * V2/C2)i; where(I (ii)
p
Fpt,F2(UX
Uy:Uxtlz;
V)tXx>pHy
and(iii)X'  X, L' = L, y' _ (Y vN/c), M  (M+vZ/c), Z' :(Z+vM/c), N' :(N vY/c); of theelectric components where(X,Y,Z) and(L, M, N) arethespatial
frame(S),the inthestationary respectively fieldstrengths andmagnetic in themovingframe(S')being(X', Y', Z') quantities corresponding equations (p.I5) showedthatthetransformed and(L', M', N'), Lorentz are: div'Et =
(I
X2 ) p',
div'H
= o,
;>Ec curl'H' =  (a, +ptu'), curl'E' I
Primedsymbolsdiv',etc.,mean out iscarried thatdifferentiation withrespectto x', r' andz'.
DH'
C Dt
is unaltered It is obviousthatthe formof Maxwell'sequations butthe is chargefree if p'O, i.e. whenspaceunderconsideration equation transformed div E
(I 30I
62)p
,
etc
G. H. KESWANI
forthecasewhenchargeis present, is notof thesameformasthecorresponding equationfor the ' stationary' system. Lorentzdid not, therefore, establish completecovariance of Maxwell's equations.As we shallsee,Lorentz's equations fortransforming p, ux,uq,anduzwere notcorrect, because he didnotinvestigate thekinematic consequences of his (Lorentz) transformation equations.In particular, he didnot securethevelocityaddition formula, thoughthis is impliedin L.T.E., whichis required whentransformation of Maxwell's equations forthe caseof movingelectriccharges is underconsideration. Inthepaper,' Surla dynamique del'electron ' (thefirstmentioned shorter paperpublished inJuneI905), Poincare discovered thefollowing cortectequation fortransforming thechargedensity, soasto leavethe form of Maxwell'sequationsunchanged underhis transforrnation equation forcharge takenalongwithLorents's equations fortransforming E,H, x, r, Z andt. (However, Poincare didnotelaborate anyproof.) pt
(I _
p
X) p
Einstein proceeded asfollows. He firstconsidered thetwo equaforemptyspacein the' stationary ' system:
tions

curlH and curl
c at
c at
E.
Writtenin Cartesian formtheseare: X aN bM IaL aY az c
at
ay
az ' c at
flaz ay
Referredto the ' moving' systemof axes,thesearetransformed by L.T.E.aloneinto: I
abtXt  a t {:(N
vY)}
aa,{:(M+
8 z)},
Thisresultis easilyestablished asfollows: Iax Iraxat ax ax) obvlouslyc
at
ci
at
at
I { aX*F+
F { aX +
aX aY az Now a + > + a
+
ax
X*v
3
at
}
aX}
o in chargefree emptyspace. 302
etc.
_
R

o
ORIGIN AND CONCEPT OF RELATIVITY
Putting Bx aX
/ aY azA
ax
az y
at
+
< ay
l and

c
z aN aMX ,.
ay
I ,
az y
we have I DX
cat' r ay
az
6 )
j { ay(
Again
ay
, and
ay
*@C att
az
ayt{(N
vaz) c ay c azj
vbY
aM
raN
az
aZ(
6 )} @
, fromL.T.E. az{(M+CZ)}@
CY)}
(p.53)identified Einstein systems, oftwoinertial Onthebasisofequivalence to assumed equations withthefollowingcorresponding theseequations be of thesameformforthe' moving' system: aNt aMt I aL' ay' az' I ax' c att = ayt Bz' 'c at' azt ay, ,etc., for the electricandmagnetic equations andgot the transformation as: fieldstrengths
X' X, L'L' Y' :(Y
:(
c )
N':(N
 Y).
cN), M
Z' :(Z+M),
the (MorecorrectlyX' +(v)X,etc.,butconsidering fromX' to X etc., as Einstein inversetransformation 0(v)I.) pointedout,obviously Lorents(p. I4IS) gave the sameresults,implicitlyassumingthat shouldhavethe sameformin the ' moving' Maxwell'sequations system. forthetransfortheequations considered (p.sg60)further Einstein of (i) pu,withtheassistance p, andcurrent, mationof chargedensity, X, Y, Z) andH E (components fortransforming theaboveequations 3o3
l+uZp3
Now

etc.
<
G. H. KES WANI
L,M,N),(ii)theequation ax + a Y+ az = p and(iii)his newly discovered (p.50)formulae forcompoundi1ng velocities,1 viz. Ux(USC V)l(I U2V/C2), etc. In chargeoccupied space,the previous two setsof Maxwell's equations in the' stationary ' system are: ISaX ) aN aMIaL aY az
(components
Assuming thattheequations forthetransfornlation ofE andH areofthe same form forchargeoccupied space asforchargefree space, wefurther proceedthus: IaX I rax at ax ax) c at' c at *at' ' ax *at' _
I ax
aN
cat
by az
a
ax

ct t + ax } (Maxwell's equation forthe' stationc tary' system.
(
<cb
uZp
az)
t+t
and p
X
aM
faY
ax
*
.
aN
ca
aM
azl
ay
va
uZp+vp
va
c at'+ c(UZ8) p ={ay(N_vy)_
PuttingX = X',F (N _ v y) = N'
a
S
(M+vz)}
(M + vz) = M' asfior
:
chargefree space, wehave I
(axt v_+s(u
v)ptay)
aNt
aMt
az
(
tay
a
ay
a
az
a
az3
a Q
Ifthisequation hastohavethesameformastheoneforthe' stationary ' system, clearly uXp'F(ux
v)p.
Asstated above,Einstein (p.50)hadshownearlier inhispaperthat UX 
(UXV)/(I
UJCI}/C2).
1Thisformulaandthatfor theDopplereSectmayberegarded astheprincipal contribution ofEinstein torelativistic mechanics.Tosaythisisneitherto denythat thisis animportant contribution norto implythatthereis no othercontribution. 3o4
ORIGIN AND CONCEPT OF RELATIVITY
Einstein (p.60)therefore concluded that p
Ux8/62) p.
 :(I
It is easilyverifiedthattheremaining Maxwell's equations aretransformedcovariantly byL.T.E.takenalongwiththeaboveequations for transforming thecomponents of E, H, p andu. We thusseethatLorentssucceeded onlypartlyin establishing covariance of Maxwell'sequations.His equations for transforming E andH werecorrectand,giventhe correctkinematical equationfor corupounding velocities, hisanalysis alsoleadsto correctandcomplete covariance of Maxwell's equations.Armedwiththeformula,based or)L.T.E.,for compounding velocities,Einstein completed Lorentz's analysis for thetransformation of Maxwell's equations a1ldremarked thaton thebasisof hiskitlematicsnotfullydeveloped by Lorentz but followingfromhis equations (L.T.E.) theelectrodynamic theoryas propounded by Lorentz (p.I4I5) wasin agreement withtheprinciple of relativity.Thisremarkof Einsteinexactlydefinesthe advance madebyhimonLorentz's theorygiveninthepreceding memoir. And is thisunionof Lorentz's equations andtheprinciple of relativity nota marriage of the ideasof LorentzandPoincare?The oddsarethat Einstein solemnised thismarriage.We neednot go intothequestion whythepriestdidnotmakeit public. Moreover, thepriestrevealed secretsof physicalharmonywhichhad neverbeendreamtbefore. Einstein's workcannotbe regarded asmerebringingtogether of two previous works;it appeared sosoonaftertlletwoandhestruckoutin so manynewdirections thatit is probable thathewasonlylookingfor someconfirrrsation of hisaudacious ideasbeforepublishing them. We nzentioned a littleearlierthe formulaefor longitudinal and transverse massof an electrongivenby Lorentz(p. 24! andEinstein (p.63). Einstein usedthesame terrninology asLorentz andsaid(p.62), ' Takingtheordinary pointof viewwe nowinquireasto the" longitudinal"and" transverse" massof an electron.'Whosewasthis ordinary pointof view? NowMaxAbraharnl hadstillearlierusedtheterms' longitudinal ' and' transverse ' for the massesof an electronwithreference to the direction of it motionbuthegaveincorrect expressions forthesemasses evellin hispaperof I903, preceeding Lorentz's memoir. However,Einstein neitherrefersto the corrections to Abraham's MaxAbraham, 'Prinzipien derDynamik desElektrons ',Annalen derPhysik,I903, I0, I S?
3of
G. H. KESWANI
formulae, norexplains thesmalldiscrepancy betweenhisformulaand thatof Lorentz forthetransverse mass. Perhaps thiscanbecleared up. We mayaddthatthediSerence betweenLorentz's andEinstein's formulaeforthetransverse massis suchasmaygo tmdetected. (Tobeconcluded)
306