Table Of ContentLecture Notes in Physics 897
Herbert Pfi ster
Markus King
Inertia and
Gravitation
The Fundamental Nature and Structure
of Space-Time
Lecture Notes in Physics
Volume 897
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Herbert Pfister (cid:129) Markus King
Inertia and Gravitation
The Fundamental Nature and Structure
of Space-Time
123
HerbertPfister MarkusKing
InstitutfürTheoretischePhysik FakultätEngineering
UniversitätTübingen HochschuleAlbstadt-Sigmaringen
Tübingen Albstadt
Germany Germany
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-15035-2 ISBN978-3-319-15036-9 (eBook)
DOI10.1007/978-3-319-15036-9
LibraryofCongressControlNumber:2015933272
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Thisisa bookonfoundationalissues ofclassical(non-quantum)physics.Itisnot
a systematic textbook,and it containsno exercises.Rather,we try to impartsome
knowledgeaboutthefundamentalnatureandstructureofthephysicalspacetimein
amostlynon-technical,neverthelesshopefullypreciseandconsistentlanguage.(A
fewmoretechnicaldetailsaredeferredtotwoappendices.)Somebasicknowledge
ofgeneralrelativityanddifferentialgeometrywouldcertainlybehelpful,especially
for Chap.2. In some places we also follow the historical evolution of views and
ideas, and we enrich the presentation by characteristic, sometimes provocative
quotations from the creators of the associated theories and models. An extensive
reference list documents the fact that all the claims and theorems formulated
here have a secure mathematical and/or observational basis, and it provides the
interestedandeducatedreaderwiththepossibilitytodigdeeperintospecialtopics.
In particular, the book aims to provide a (surely personal) selection of relevant
literatureonthefoundationsandthenatureofspacetime,andonethatisnotusually
foundin standardtextbooksonclassicalmechanicsorgeneralrelativity.Although
thebookwouldnotproperlyfallintothecategoryofpopularbooks,undergraduate
students should be able to read at least parts of the book (in particular Chap.1)
with understandingand profit. On the other hand, we hope that even professional
experts will find some new viewpoints and connections between different topics.
The selection of topics treated in detail (and the omission of other items) may be
somewhat unusual, but we think it is justified in view of the title of the book.
Wherever possible, we emphasize the mutual dependence and interplay between
differentsubjects.
Asthetitleofthebookpromises,acentralthemeisthephenomenonofinertia,
fromitshistoricintroductionbyGalileoandNewtontotheMachianhypothesisof
itsoriginincosmology,andthe(atleastpartial)observationalconfirmationofthis
astonishingfact.Itisquiteobviousthatthefirstpartofthebooktitleisadoptedfrom
thebook‘GravitationandInertia’(CiufoliniandWheeler1995),butinterchanging
thewordsbecause,presumablymorethaninanyotherbook,inertiaisadominating
themeofthepresentbook.Thesecondpartofthetitleistakenfromthebrillianttalk
byJ.EhlersattheTriesteconferencecelebratingDirac’sseventiethbirthday(Ehlers
v
vi Preface
1973).Inthisway,wewouldalsoliketohonorthetwoscientists,J.A.Wheelerand
J.Ehlers,whohavebeenthemostimportantmentorsforoneofthepresentauthors
(H.P.)inhisstrugglewithandenthusiasmforgeneralrelativity.
According to Einstein’s equivalence principle, inertia is intimately connected
with gravitation, and on the basis of this principle, through a tedious process,
Einstein eventually developed his relativistic theory of gravity, general relativity.
Thistheoryturnedouttobeacompletetriumphineveryrespect,anditstandstoday
unchanged,andwithoutasingleconflictwithexperiment,nearly100yearsafterits
creation. Chapters 2–4 deal with this theory but, as already stressed above,not in
thesenseofasystematicandcomplete,textbook-likepresentation,butbyfocusing
on some characteristic and fundamentaltopics within this immensely rich theory.
In Chap.2, we review the remarkable ‘derivation’ of the (pseudo-)Riemannian
spacetimestructurefromthepropertiesoftheelementaryobjects‘freeparticles’and
‘lightrays’,asinitiatedbyH.Weyl,andworkedoutindetailbyJ.Ehlers,F.Pirani,
A. Schild, and others, nowadays known as EPS axiomatics. Such a physically
satisfying deduction of the different levels of geometric structure of our ‘world’
ishardlyeverfound,eveninmoderntextbookpresentationsofEinstein’stheoryof
gravity.
Chapter 3 tries to give prominence to the very special structure of general
relativity,which manifestsitself in the many differentroutesleading to Einstein’s
field equations, in deep mathematical results (Cauchy problem, positive energy
theorem, singularity theorems), and in spectacular astrophysical predictions, e.g.,
the existence of black holes. General relativity also encompasses the whole of
classical physics, and therefore Einstein’s field equations are the most involved,
butalsotherichestequationsonecanthinkofinthisregime.We hopetotransmit
some of our own astonishment and fascination for the beauty, consistency, and
completenessofthistheory,somethingwhichnobodycouldforeseeatthe timeof
itscreation.
Chapter 4 returns in a way to the central topic of this book, inertia. In their
attempttofindthebasicsourceofthisphenomenon,E.Mach,B.andI.Friedlaender,
A.Föppl,A.Einstein,andothersmainlyconsideredrotatingsystems,inthetradition
of Newton’srotatingbucket. Within generalrelativity,this led to differentmodels
fortheso-calleddraggingoffreeparticlesandinertialsystemsbyaccelerating,and
particularly rotating heavy masses. Taken as a whole, such examples strengthen
the hypothesis of E. Mach that inertia is, in a non-causal way, ruled by the
overall masses of the universe. Modern precision experiments and observations
areinaccordancewiththisview,forwhichinterestingandstimulating,sometimes
provocativeformulationscan be found in moderntextbooksand research articles.
Aparticulareffectofmovingmassesingeneralrelativityshowsupinanew,non-
Newtonian ‘force’, called gravitomagnetism, which has recently been confirmed
by intricate satellite experiments, 90 years after its prediction by A. Einstein and
H.Thirring.
A decisiveseed forpartsofthisbook,particularlyChap.1,andforpartsofthe
researchof one of the authors(H.P.),grew outof the followingincident.In 1972,
as a young lecturer at the University of Tübingen, Germany, he had the duty (or
Preface vii
rathertheprivilege)todeliveramajorcourseontheoreticalmechanics(4haweek,
over a whole year). In preparing this course, he studied the standard textbooks
on mechanics and was thoroughly dissatisfied with the way the foundations of
mechanics,andinparticularNewton’sfirstlaw(thelawofinertia),werepresented
in most of these textbooks. A first shortcoming results from the fact how briefly,
superficially,and carelessly this law is often treated. Since this is usually the first
law which is presented to young physics students—and it is after all one of the
mostimportantanduniversallaws,withrelevanceinallareasofphysics,notonly
for mechanics—one would expect, also for pedagogical purposes, that it would
serveasamodelforathorough,clear,andlogicallyconvincingpresentationofthe
fundamentalfacts of nature. The difficulty, but also the general failure to achieve
this,wasclearlyexpressedlongagobyH.Hertzinhisfamousbookonmechanics
(Hertz1894):
Itisquitedifficulttopresenttheintroductiontomechanicstoanintelligentaudiencewithout
someembarrassment,withoutthefeelingthatoneshouldapologizehereandthere,without
thewishtopassquicklyoverthebeginnings.
Besidesthisfastandcarelesswayofpassingoverthebeginnings,anevenmore
serious deficit in many textbooks shows up in tautologies, circular arguments, a
missingdistinctionbetweendefinitionsandnon-trivialfactsofnature,andinsome
cases in a mixing between Newton’s first, second, and third laws. To make this
concrete, we quote from a well-established and highly recommended mechanics
textbook(Marion1965,p.58):
Thus,thefirstandsecondlawsarenotreally‘laws’intheusualsenseofthetermasused
inphysics;rather,theymaybeconsideredasdefinitions.Thethirdlaw,ontheotherhand,
isindeedalaw.Itisastatementconcerningtherealphysicalworldandcontainsallofthe
physics inNewton’s lawsof motion.[Then inafootnote] Thereasoning presented here,
viz.,thatthefirstandsecondlawsareactuallydefinitionsandthatthethirdlawcontainsthe
physics,isnottheonlypossibleinterpretation.LindsayandMargenau(1936)forexample,
presentthefirsttwolawsasphysicallawsandthenderivethethirdlawasaconsequence.
Andthisunspeakableformulation,whichisreallyaninsulttoNewton,istobe
found,notonly in the first editionof the bookfrom 1965,but literally unchanged
inthefourtheditionfrom1995.[Weadmitthatthereareafewtextbooksthatreally
takecareoverthepresentationofNewton’sbasiclaws,andavoidmostofthepitfalls.
Asanexample,wecallattentiontothetextbookStraumann(1987).]
Surely one cannot blame Newton for all the nonsense about Newton’s laws in
modern textbooks, even if from today’s perspective and with today’s knowledge
someofNewton’sconceptsandformulationsareunfortunateorevenuntenable,e.g.,
theconceptsof‘absolutespace’and‘absolutetime’.However,inordertoappreciate
Newton’sgeniusandhisprimacyintheformulationofthefoundationsofphysics,
onehastocompareNewton’sPrincipiawiththe(evenlesstenable,andmuchless
successful)attemptsofhisforerunnersandcontemporaries,asweshalldotosome
extentinSect.1.1.AndNewtonwaswellawareofthedifficultyofthesubjectandof
theprovisionalnatureofhisattempt,whenhewroteintheprefacetothePrincipia
(Newton1687,p.383):
viii Preface
Iearnestlyaskthateverythingbereadwithanopenmindandthatthedefectsinasubject
sodifficultmaybenotsomuchreprehendedasinvestigated,andkindlysupplementedby
newendeavorsofmyreaders.
But the unsurpassed success of Newton’s program in all applications pre-
vented later generations, and many of today’s textbook authors, from examining
Newton’s formulations critically and from eliminating its defects. For nearly 200
years, there was no real critical reflection and therefore no improvement on
Newton’s foundations of mechanics, until finally, beginning in the year 1870,
a growing number of researchers revisited these questions and reached, besides
a fundamental critique of Newton’s concepts of ‘absolute space’ and ‘absolute
time’ (particularly by E. Mach), a genuine clarification, and an elimination of
Newton’s absolute concepts, mainly due to C. Neumann and L. Lange, as we
shall analyze in detail in Sect.1.2. Sections 1.3 and 1.4 attempt to improve
on some deficiencies still present in the work of L. Lange, particularly in the
definitionofthebasicingredientsofthelawofinertia,‘freeparticles’,and‘straight
lines’.
ThegeometricalapproachtoEinstein’srelativistic spacetimetheory,emphasiz-
ingtheintrinsicspacetimestructure,startingwithWeylandculminatingintheEPS
axiomatics, is essentially based on a four-dimensional formulation of spacetime
as a differential manifold with geometrical structures (defined by tensor fields),
the second central theme of this book. In this respect, the book should also serve
as an introduction to the conceptual foundations of spacetime theories, and their
peculiarcharacterbasedon‘events’andthespatio-temporalrelationsbetweenthem,
and this from the perspective of Newton’s law of inertia. Section 1.5 develops
such a four-dimensional analysis of Newtonian physics, thereby bringing out the
fundamentalnatureofnon-relativisticspacetime(namely,theso-calledLeibnizian,
Galilean, and Newtonian accountsof space and time). In this respect, this section
servesasapreparationfor,andcontrastto,thecorrespondingstructuresingeneral
relativity, presented in Chap.2. The spacetime geometry of standard Newtonian
physics,consistentwithLange’sanalysisofthelawofinertia,isGalileanspacetime,
which is “presupposed in all standard accounts of Newtonian mechanics, even
thoughthispresuppositionisusuallytacitandunremarked”(Maudlin2012,p.64).
Newton’s law of gravitation, its reformulation in a four-dimensional geometric
setting (in the spirit of special and general relativity), its distinctive properties,
and some typical applications in astrophysics, e.g., rotating stars, are treated in
Sect.1.6. And here, surprisingly, even in this centuries old classical field, lie
dormantastrophysicallyrelevant,butmathematicallydifficultandhithertounsolved
problems.
Preface ix
WearedeeplyindebtedtoourcopyeditorStephenN.Lyleforthecarefuleditof
our manuscript, for the valuable comments, and for his judicious corrections and
improvement of grammar and style. We wish also to thank our editor Christian
CaronatSpringerforexcellentcooperationandsupportofthisbookprojectwithin
theLectureNotesinPhysicsseries.
Tübingen,Germany H.Pfister
November2014 M.King
This sketch of a merry-go-round shall serve as an example for the combined action of inertia
(centrifugalforceF )andgravity(forceF )fromeverydaylife.
c g
Notation
Inmostcasesweusetheso-calledgeometricunits,wherethegravitationalconstant
G andthelightvelocityc aresetto1.Four-dimensionalspacetimecoordinatesare
denotedbyGreekletters(cid:2);(cid:3);:::D0;1;2;3.Three-dimensionalspacecoordinates
aredenotedbyLatinlettersi;k;::: D1;2;3:TheMinkowskimetricisdenotedby
(cid:4) = diag(1;(cid:2)1;(cid:2)1;(cid:2)1/. The symbol A means antisymmetrization between
(cid:2)(cid:3) Œ(cid:2)(cid:3)(cid:5)
the indices, i.e., A D .A (cid:2)A /=2. Partial differentiation is written . / ,
Œ(cid:2)(cid:3)(cid:5) (cid:2)(cid:3) (cid:3)(cid:2) ;(cid:2)
andcovariantdifferentiationiswritten. /I(cid:2).OccasionallyX iswritteninsteadof
X(cid:2)fora4-vector,andxdenotesa3-vector.
