Table Of ContentRELATIVITYPRINCIPLESANDTHEORIES
FROMGALILEOTOEINSTEIN
Relativity Principles
and Theories from Galileo
to Einstein
OLIVIER DARRIGOL
ResearchDirector,CentreNationaldelaRechercheScientifique:SPHere
3
3
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CONTENTS
Preface viii
Conventionsandnotations xiii
1 Rethinkingmotionintheseventeenthcentury 1
1.1 Galileo’sscienceofmotion 2
1.2 BeeckmanandDescartesonfreefall 8
1.3 Descartes’sworld 10
1.4 Newton’slawsofmotion 13
1.5 Huygens’smechanics 19
Conclusions 23
2 DerivingNewton’ssecondlawfromrelativityprinciples 28
2.1 Rationalmechanicsintheeighteenthcentury 29
2.2 Nineteenth-centuryFrenchtextbooks 36
2.3 Principlesanddeductions 42
Conclusions 47
3 Thespace–time–inertiatangle 52
3.1 FromHuygenstoKant 53
3.2 Criticisminthelastthirdofthenineteenthcentury 61
3.3 Themeasurementoftime 75
Conclusions 79
4 Theopticsofmovingbodies 84
4.1 Thespeedoflight 85
4.2 Thecorpuscularapproach 88
4.3 Stellaraberrationsinthewavetheory 92
4.4 TheFresneldrag 102
4.5 Towardanopticalrelativity 108
Conclusions 114
5 Theelectrodynamicsofmovingbodies 118
5.1 Earlyelectrodynamics 119
5.2 Germanactionatadistance 121
5.3 Britishfieldtheories 124
5.4 MaxwellinGermany 133
5.5 Effectsofabsolutemotion 137
vi CONTENTS
5.6 Theseparationofetherandmatter 141
Conclusions 152
6 Poincaré’srelativitytheory 157
6.1 Criticalteaching 159
6.2 FortheLorentzjubilee 165
6.3 Insidetheelectron 170
6.4 Thepostulateofrelativity 175
Conclusions 184
7 TherelativitytheoryofEinstein,Minkowski,andLaue 188
7.1 TheyoungEinstein’sventuresinelectrodynamics 191
7.2 AlternativestoLorentz’stheory 195
7.3 Einstein’srelativitytheory 202
7.4 Earlyreception1905–1908 212
7.5 Constructingarelativisticelectron 222
7.6 OutsideGermany 229
Conclusions 231
8 FromRiemanntoRicci 236
8.1 Gauss’scurvedsurfaces 239
8.2 Riemann’scurvature 244
8.3 Non-Euclideangeometries 250
8.4 Theabsolutedifferentialcalculus 256
Conclusions 267
9 MostlyEinstein:Togeneralrelativity 269
9.1 Heuristicarguments(1906–1911) 272
9.2 Thestatictheoryof1912 280
9.3 TheZürichnotebook 289
9.4 TheEntwurf theoryof1913 298
9.5 Thescalartheory 302
9.6 Bridledcovariance 305
9.7 Justifiedtransformationsandadaptedcoordinates 309
9.8 November1915 314
Conclusions 326
10 Meshandmeasureinearlygeneralrelativity 341
10.1 AGaussianpreliminary 344
10.2 Einstein’sGrundlageof1916 348
10.3 Thegravitationalredshift 360
10.4 Thegravitationaldeflectionoflight 368
CONTENTS vii
10.5 TheadvanceofMercury’sperihelion 375
Conclusions 386
11 Epilogue 389
11.1 Actorsandstages 389
11.2 Mechanicalrelativity 392
11.3 Opticalrelativity 398
11.4 Electrodynamicrelativity 400
11.5 Specialrelativity 404
11.6 Generalrelativity 407
Abbreviations 416
References 417
Index 465
PREFACE
[The theory of relativity] did not at all originate in a revolutionary act but as
anaturaldevelopmentofalinethatcanbetracedthroughcenturies.¹(Albert
Einstein,1921)
Therelativitytheoriesoftheearlytwentiethcenturydeeplyalteredourmostbasicnotions
ofspace,time,andmotion.Theymadetimedependontheobserver,theyentwinedspace
andtime,andtheyreducedgravitationtoinertialmotioninacurvedspace–time.Histori-
anshavetriedtoexplaintheradicalnessofthesechangesinvariousways:throughEinstein’s
peculiar genius, through contemporary developments in philosophy or technology, and
through a major crisis in physics around 1900. These three explanations correspond to
three alleged preconditions of radical change: singular minds, external causation, and
internalcrisis.Theinternalcrisisapproachisperhapsthemostconvincing,becausephilo-
sophicalandtechnologicalconsiderationsmayhavebeeninternalizedbeforetheyserved
thefoundersofrelativity,andbecausethesingularityofthesefoundersmayboildownto
theirabilitytodetectandsolvecrises.Butthisapproachcanonlybeafirstapproximation,
andafullerhistoryshouldcombineinternal,external,andbiographicalelements.
Whether or not they come close to this ideal, the received histories of relativity are
short-term, fragmented histories. Special and general relativity are usually treated sepa-
rately,andthemaximaltimespanisaboutacenturylong(intheinternalcrisisapproach),
with a focus on the late nineteenth and early twentieth century. There are exceptions:
JürgenRennhascomparedtheemergenceofrelativitytheorywiththeemergenceofearly
modernmechanics;JeanEisenstaedthasdevelopedanalogiesbetweeneighteenth-century
Newtonianopticsandgeneralrelativity;severalphilosophers,includingRobertoTorretti
andRobertDiSalle,haveexploredthevariousconceptsofspaceandtimeinthehistory
ofphysicsfromAristotletorelativity;andonephysicist,JulianBarbour,hasprobedthe
long-termhistoryofdynamicsfromaMachianviewpoint.Buttheirmainintentionwas
nottodemonstrategenuine,long-termhistoricalconnections:itwastoinstructontheva-
rietyofpossibleconcepts,toreflectontheirsimilaritiesanddifferences,or(inRenn’scase)
toillustrateagenerictheoryofconceptualdevelopment.²
Incontrast,thisbookpresentsalong-term,multi-approachhistoryofrelativityfrom
GalileotoEinstein,inthevariouscontextsofmechanics,mathematics,philosophy,astron-
omy,optics,andelectrodynamics.Onemightfearthatanyattempttoconnecteventsthree
¹Einstein[1921],p.431.
²Torretti1978;Renn1993;Barbour2001;Eisenstaedt2005b;DiSalle2006.AlsoworthmentioningareMarie-
AntoinetteTonnelat’sHistoireduprincipederelativité(Tonnelat1971)forananalysisofthesuccessivemeanings
ofrelativityfromantiquitytogeneralrelativity;andRaffaellaToncelli’sdissertationontheroleofprinciplesinthe
constructionofrelativitytheory(Toncelli2010),whichalsohasaverylargescope.
PREFACE ix
centuriesapartwouldbesuperficialandartificial,andthiswasmyownopinionuntilIre-
alized,afterafewpartialpre-histories,thatgenuinehistoricalconnectionsexistedbetween
thevarioususesofrelativityprinciplesacrossthreecenturies.Iobservedtwokindsofcon-
nectedness. First, at least until the early twentieth century physicists had a much longer
memory than they have nowadays. They read and exploited literature written centuries
earlier,whereastodayscientiststendtodisregardanythingolderthanafewyears.Second,
therewereindirecthistoricalconnectionsthroughchainsofsuccessiveborrowings.Inboth
casesitisessential,inaproperlyhistoricalstudy,totakeintoaccountthechangeofcontext
intheborrowingprocess,evenwhentheborrowerisnotawareofthischange.Otherwise
we would misunderstand the successive systems of thought, and we would misrepresent
thetransferofknowledgefromonesystemtoanother.³
When developed with sufficient care, this long-term view has several advantages. It
showsthatquestionsaboutthenatureofspace,time,andinertiatraversedthehistoryof
physicsfromtheearlymodernperiodtotherelativitytheories,althoughtheanswersgiven
to these questions varied considerably. It explains how the relativity principle emerged
as a true, constructive principle as early as the seventeenth century, how it came to be
named so in the mid-nineteenth century to mark its constructive power in mechanics,
andhowthisnameandconceptreachedtheyoungEinstein.Itdemonstratesdeepanalo-
gies between some authors’ dismissal of Newton’s absolute space and Einstein’s later
dismissal of the electromagnetic ether. It establishes an indirect historical link between
acorollaryofNewton’sPrincipiaandEinstein’sequivalenceprinciple,throughaFrench
principleofaccelerativerelativity.Itsituatesbothprinciplesinaspace–time–inertiatangle
thatoriginatedinGalileo’sandNewton’stheoriesofmotionandevolvedintoEinstein’s
finalreductionofgravitationtoinertia,throughintermediatestepsintheeighteenthand
nineteenthcentury.Also,thereareadvantagesintreatingthehistoriesofspecialandgen-
eral relativity in the same volume: this highlights the continuity of Einstein’s endeavors,
and shows the importance of Minkowski’s and Laue’s versions of the special theory in
Einstein’squestforarelativisticfieldtheoryofgravitation.
Anylong-termhistoryfacesdifficultiesinnamingbasicconceptsandprinciplesacross
time.Today,weusuallydefinetherelativityprinciplethroughthecompleteequivalenceof
allinertialreferenceframes.Itisimpossibleoratleastdangeroustoapplythisdefinition
uniformlyfromGalileotoEinstein.Thename“referenceframe”firstappearedin1884,
under James Thomson’s pen. The concept earlier existed under different names, guises,
and extensions: a metonymical boat for Galileo, a “space” for Newton, and a “system
ofaxes”forLaplace.Statementsofrelativityhadavariablestatus:anempiricalfactfor
Galileo,atheoremforNewton,andagenuineprincipleforHuygens,Laplace,andlater
Frenchauthors.Thereweredifferenttheoremsandprinciplesofrelativity(hencetheplural
inmytitle)accordingtotheimpliedclassofequivalentframes:inertialframesandacceler-
atedframesforNewton,Laplace,andBélanger;justinertialframesforHuygens,Poincaré,
andtheearlyEinstein;andanyacceleratedframeandevenaframing“mollusk”forthe
later Einstein. Until the late nineteenth century, relativity principles and theorems were
³BycontextIheremeanintellectualandexperimentalcontext,althoughthechangesinthisnarrowercontext
usuallyimplythebroadersocio-culturalcontext.
