Foundations of General Relativity: From Einstein to Black Holes

394 Pages·2021·23.372 MB·English

by  Klaas Landsman

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FOUNDATIONS OF
GENERAL RELATIVITY
FROM EINSTEIN TO BLACK HOLES
KLAAS LANDSMAN
Foundations of General Relativity: From Einstein to Black Holes
Published by RADBOUD UNIVERSITY PRESS
Postbus 9102, 6500 HC Nijmegen, the Netherlands
www.radbouduniversitypress.nl | www.ru.nl/radbouduniversitypress
[email protected]
Cover image: © Edith de Jong
Cover design: Pumbo.nl
Print and distribution: Pumbo.nl
Version: 2021-09
ISBN: 978 90 831 789 29
DOI: http://doi.org/10.54195/EFVF4478
Free download at: www.radbouduniversitypress.nl
© Klaas Landsman September 2021
Institute for Mathematics, Astrophysics, and Particle Physics
Radboud University, Nijmegen, The Netherlands
This is an Open Access book published under the terms of Creative Commons Attribution-
Noncommercial-NoDerivatives International license (CC BY-NC-ND 4.0). This license allows
reusers to copy and distribute the material in any medium or format in unadapted form
only, for noncommercial purposes only, and only so long as attribution is given to the
creator, see https://creativecommons.org/licenses/by-nc-nd/4.0/.
Dedicated to Roger Penrose
iv
Preface v
Preface
This book grew out of lecture notes for my master’s courses on general relativity (GR) and
on singularities and black holes taught at Radboud University (Nijmegen). These notes were
originallyintended forourstudents withadouble bachelordegreein mathematicsandphysics,
butinitsfinalformthebookisintendedforallstudentsofGRofanyageandorientationwhohave
abackgroundincludingatleastfirstcoursesinspecialandgeneralrelativity,differentialgeometry,
andtopology.1 TherecenttextbookElementsofGeneralRelativitybyChrus´ciel(2019)would
makestudentssingularlywellpreparedforthisone,butalmostanyintroductionto GR,combined
with the typical mathematical background in manifolds etc. that is usually included in such
introductions,willdo. Thisbook,then,isasecond,mathematicallyorientedcourseingeneral
relativity, withextensivereferences andoccasional excursionsin the history and philosophyof
gravity,includingarelativelylengthyhistoricalintroduction. Assuch,itomitsstandardphysics
material like the classical tests etc. Furthermore, the material is developed in such a way that
throughthelasttwochaptersthereadermayacquireatasteofthemodernmathematicalstudyof
blackholesinitiatedbyPenrose,Hawking,andothers,sothatsuccessfulreadersmightbeable
tobeginreadingresearchpapersinthisdirection,especiallyinmathematicalphysicsandinthe
philosophyofphysics. Thisfocuscomeswithanintroductiontowhatiscalledcausaltheory,
butalas,italsoimpliesthatinordertokeepthebookmedium-sizedIhadtoomitapplications
likecosmologyandgravitationalwaves. InanycaseIhopethebookappealstomathematicians,
physicists,andphilosophers–perhapsevenhistorians–ofphysicsalike.
My own experience is that a really deep field such as GR (or quantum theory) can only be
learned by reading a large number of books saying the right things in different ways, as well
asbytalkingtogoodpeopleworking inthefield. Asa reader,myfirstencounterwith GR was
Einstein’sownexpositionRelativity: TheSpecialandGeneralTheory(Einstein,1921),whichis
stillinprint. Inthesummerof1981,havingjustgraduatedfromhighschool,thiswasfollowed
bytwobooksthatwerealittlemoredifficult,namelySpace-Time-Matter byWeyl(1922)and
TheMathematicalTheoryofRelativitybyEddington(1923),bothofwhicharenotonlyhighly
mathematicalbutalsoprofoundlyphilosophicalinspirit. Weylmakesthispointhimself:
At the same time it was my wish to present this great subject as an illustration of the
interminglingofphilosophical,mathematical,andphysicalthought,astudywhichisdear
to my heart. This could only be done by building up the theory systematically from the
foundationsandbyrestrictingattentionthroughouttotheprinciples. ButIhavenotbeen
able to satisfy these self-imposed requirements: the mathematician predominates at the
expenseofthephilosopher.2 (Weyl,1918,Preface)
Indeed, Weyl, Eddington and Einstein were natural philosophers in the spirit of the scientific
revolution,whosemixofphysics,mathematics,andphilosophywasthekeytoitssuccess. Hence
itseemshardlyacoincidencethatEinsteinwasNewton’ssuccessor,forifanyscientifictheory
haseverrepresentedthePhilosophiaeNaturalisPrincipiaMathematica,itmustbe GR.
1Logicallyspeaking,theGRmaterialisevendevelopedfromscratch,andindeedthefirstcourseinthisdirection
thatItaughtwasoptimisticallyofferedalsotomathematicsstudentswithoutanyphysicsbackground.Butexperience
showsthatthematerialmakeslittlesensewithoutsomepriorexposuretobothspecialandgeneralrelativity.
2‘Zugleich wollte ich an diesem Großem Thema ein Beispiel geben für die gegenseitige Durchdringung
philosophischen,mathematischenundphysikalischenDenkens,diemirsehramHerzenliegt;dieskonntenurdurch
einenvölliginsichgeschlossenenAufbauvonGrundaufgelingen,dersichdurchausaufdasPrinzipiellebeschränkt.
AberichhabemeineneigenenForderungenindieserHinsichtnichtvollGenügetunkönnen: derMathematiker
behieltaufKostendesPhilosophendasÜbergewicht.’ Translation: HenryL.Brose(Weyl,1922).
vi Preface
So even though at the time I understood almost nothing of the technical content of their
books,Einstein,Weyl,andEddingtonleftanindeliblemarkinthewaytheyapproachednatural
science through mathematics and philosophy. Still during that same long summer vacation
betweenhighschoolanduniversityin1981,whichIregardasoneofthehighpointsofmylife,I
alsoboughtGravitationbyMisner,Thorne&Wheeler(1973). ForawhileIconsideredthisthe
greatestbookwrittenonanytopicwhatsoever,3 andwhenMisner,wellinhiseightiesatthetime,
not only came to a talk I gave in one of Bub’s New Directions in the Foundations of Physics
conferencesinWashingtonDCbutevenaskedaquestion,afterhehadansweredpositivelyto
mycounter-questionifhewasCharlesMisnerIwaspetrifiedandunabletosayanything.4
MynextbookwasTheLargeScaleStructureofSpace-TimebyHawking&Ellis(1973),and
soon,untilGeneralRelativityandtheEinsteinEquationsbyChoquet-Bruhat(2009)andmost
recentlyTheGeometryofBlackHolesbyChrus´ciel(2020). Theseareallmasterpieceswritten
byfoundersofthefield;likemoststudentsandauthorsinmathematical GR Iamalsoindebtedto
Penrose (1972), O’Neill(1983) and Wald (1984). Furthermore, Earman(1995) set the stage in
the philosophy of physics. Other influences on thistext include Weinberg (1972), Kriele (1999),
Poisson(2004),Schoen(2009),Gourgoulhon(2012),Malament(2012),andMinguzzi(2019).
Thisbringsmetothequestionwhyanauthorwhosofarwrotelittleon GR isentitledtowrite
abookaboutthesubject–evenifithas beenanalmostlifelongpassion. Inthe first oftheJeeves
andWooster episodes(aboutanindolentEnglisharistocratandhisbutler),Wooster’sauntasks:
Doyouwork,MrWooster?
uponwhichWooster(i.e.thearistocrat),takenabackbyherquestion,mumbles:
Well,I’veknownafewpeoplewhowork.
I’ve known a few people who work, too (in GR, that is). The greatest of these, in my view, is
RogerPenrose,towhomthisbookisdedicatedinhonourofhispivotalroleinthecreationof
mathematicalrelativityandthemoderntheoryofsingularitiesandblackholes,5 combinedwitha
singularlackofpompandcircumstance,forascientistofhiscalibre. Inherrecentautobiography,
YvonneChoquet-Bruhat,whohasknownPenroseforover50years,putsitwell:
Inspiteofhissuccesses,heremainsamanwithoutpretension,openandfriendly. Hecame
tolisten,afewyearsago,toatalkIgaveataseminarinOxford. Afterwardswehadlunch
withafewcolleaguesandtheconversationturnedtothepublicationofhiscompleteworks.
Penrosesaid: ‘MyproblemistoknowifImustcorrectmymistakesbeforepublication.’ It
isagreatqualitytorecognizeamistake,evensmall. Fewhumanbeings,scientistsornot,
arereadytodoit. (Choquet-Bruhat,2018,chapter10)
Perhapsthekeytohissuccess,whichontheonehandseemstypicalformostgreatscientistsand
artistsbutontheotherhandseemsparadoxicalasapathtoinfluenceandeminence,isthis:
3Kaiser(2012)givesaninterestingperspectiveonGravitationanditshistory,whichconfirmsitsuniqueness.
4Nonetheless,InowseeabasicdrawbackofGravitation: withitsxxvi+1279pages,itleavesnoroomforthe
reader(exceptindoingtheexercices,whichIalldulydidinthenextfewyears),whoisoverwhelmedandcornered.
5A scientific biography of Penrose remains to be written (in 2019 Dennis Lemkuhl conducted a series of
interviewswithPenrose). Fornow,seee.g.Thorne(1994),Frauendiener(2000),Friedrich(2011),andEllis(2014).
Both the written AIP interview by Lightman (1989) and the videotaped interview by Turing’s biographer and
Penrose’sformerstudentHodges(2014)aregreatandintimateportraitsofPenrose.
Preface vii
Itwasimportantformealways,ifIwantedtoworkonaproblem,tothinkIhadadifferent
angleonitfromotherpeople. BecauseIwasn’tgoodatfollowingwhereeverybodyelse
went. Iwasn’tthekindofpersonwhocouldpickuptheprevalentargumentsandknowledge
ofthetime. Otherpeopleweregoodatthat. Theycouldsuckitalloutandputittogetherand
makeadvances. Iwasthekindofpersonwho’dhavesomekindofquirkywayoflookingat
somethingonmyown,whichIwouldhideawayandworkat. SoitmeantthatIhadtohave
somewayoflookingataproblemthatwasmyown.6
Here one would like to emphasize that the word ‘looking’, used twice, should be taken quite
literally: as he also emphasized elsewhere, Penrose is primarily a visual thinker. This is
exemplified mostfamously byhis invention ofthediagrams namedafter him,but itgoes back
a long way, including for example the “impossible figures” he created with his father, and his
interaction withthe Dutch artistMaurits Cornelis Escher(1898–1972).7 Penrose usuallydrew
hisownfiguresinaprofessional,yetplayfulandcharacteristicway,andeachofthemnotonly
makessomescientificpointbutisalsoapleasuretolookat. Afewarereproducedinthisbook.
I firstheard Penrose speakin Cambridge in1989 about hisrecent bookThe Emperor’s New
Mind: ConcerningComputers,MindsandTheLawsofPhysics,latersupersededbytheequally
controversialShadowsoftheMind: ASearchfortheMissingScienceofConsciousness(1994). I
gottoknowhimpersonallyduringaSevenPinesSymposiuminMinnesotain2005,wherethe
organizershadtheluminousideathatfamousandordinaryparticipantsshareanapartment. Iam
notsuretowhichoneof usthisarrangementwasinitiallymoreshocking,butwegotalongwell,
andheverykindlycametotheopeningconferenceofourinstituteIMAPPatNijmegen(2005)
asaspeaker(formingpartofastellarline-upincludingforexamplephysicistGerard’tHooft,
mathematicianDonZagier,andtheologianHansKüng),8 whereheexplainedthekeyideasof
his laterbook Cycles ofTime (2010). Having himall for myselffor 1.5 hours, Ithen drove him
tothefamousAmstelHotelinAmsterdam,themostexpensivehotelinthecountry,sinceIfelt
thatifthatistheplacewhereBobDylanandthelikestay,certainlyalsoRogerbelongedthere.9
He laterreturned tothe Netherlandsfor a mathematical physics conferenceand usuallycame to
mytalkswhenIwasvisitingOxfordandjoinedforlunchordinnerwheneverpossible.
Dominatingthepublicimage,StephenHawkingwasunquestionablyanotherkeyfigurein
mathematical relativity.10 I observed Hawking on an almost daily basis between 1989–1997,
whenIwasapostdocatDAMTPinCambridge,butIwasn’tinhisgroupandnevertalkedtohim
directly. Ididminglewithhiscirclethough,andinhaledacertainculturefromthis. Althoughin
thewakeofhisBriefHistoryofTime(1988)Stephenhadbythenbecomeascientificsuperstar,
itisonlyafterhisdeathin2018thatIreallycametoappreciatehisgeniusandhislife.11
HencethisbookhasbeenheavilyinfluencedbyHawkingandPenrose,andofcourseincludes
theirsingularity(i.e. incompleteness)theorems,but withoutbeingblindto otherdevelopments,
notablytheinitial-valueor PDE approachto GR,which,aswillbeexplainedindetailespecially
inconnectionwithcosmiccensorship,sometimesleadstoadifferentperspectiveonspace-time.
6QuotedfromtheAIPinterviewbyLightman(1989).
7SeeWright(2014)forthehistoryofPenrosediagrams;Wright(2013)explainsthelinkwithEscher. Penrose
admiredEscheratleastsince1954,when,asastudentparticipanttotheInternationalConferenceofMathematicians
inAmsterdam,hesawanEscherexhibition. In1962PenrosevisitedEscherathishomeinBaarn. SeePenrose
(2005),chapter2,andtheTVdocumentaryPenrose(2015). Seealso§1.9andfootnote486.
8AttheconferencedinnerwegaveeachspeakeranexpensiveJapanesewoodenpuzzle,whichweaskedthemto
solveasquicklyaspossible. Penrosewoneasily(which,ingoodspirits,greatlyannoyed’tHooftandZagier).
9OurfinancialstaffdidnotappreciatethisandIpaidforhisroom,withariverviewdoublingtheprize,myself.
10See§1.9forsomebriefhistoricalcommentsonthedevelopmentofmathematicalGR.
11SeeHawking(1999)foranunusuallyhonestaccountofthislife;thesecond(2007)editionismilder.
viii Preface
A complete coverage of the causal theory is both impossible in a work of this size and
undesirable for students looking for a first encounter, but fortunately it is also unnecessary in
view of the recent encyclopedic (Open Access) treatment by Minguzzi (2019), which always
lies open on my desk. Similarly, a complete description of the PDE approach to GR would
requirenot onlya verydifferentauthor (orrather ateamof authors),but alsomuch morespace
including preliminary material. So we are fortunate to have Ringström (2009) for those who
wantmorethantheveryfirstintroductiongivenhere,aswellasKlainerman&Nicolò(2003)and
Christodoulou(2008). Ihavetriedto dojusticetothemodernspiritofmathematicalrelativity,
whichischaracterizedbyamixofthecausalandthe PDE theoriesandculminatesinthecosmic
censorship and final state conjectures. The aim of this book is not at all to describe the latest
newsabout suchmatters, but merely toexplainwhat the discussions are about,and give students
and more senior readers not specializing in this area and entry point to the research literature.
Likewisefortheno-hairoruniquenesstheoremsforblackholesandblackholethermodynamics,
withwhichthebookends. Thusthebookstopsnotonlywhere(mathematicalorphilosophical)
researchpapersonclassical GR begin,butalsowherequantumaspectsofgravitybegin.
Finally,asmaybeexpectedmorefromaworkinthehumanitiesthaninmathematicalphysics
(betweenwhichthehistoryandphilosophyofphysicsresides),therearealmost700footnotes,
placedwherethename“footnote”suggeststheybelong. Theycontaincredits(e.g.forsomeof
theargumentsandderivationsIgive)andotherpointerstotheliterature,aswellasadditional
informationthatrefinesorqualifiesthemathematicsjustdiscussed,and/oraddsconceptualor
historicalinformationIfoundinteresting. Theymaybeskippedinprinciplebythosewhojust
wanttohearthemelody,buttheyseemtometobeessentialforenjoyingthefullsound.
Foramoredetailedsummaryofthisbooktheprospectivereaderisencouragedtotakealook
atthesynopsisandthetableofcontents,whichinthisorderimmediatelyfollowthispreface.
Ireceivedverykindhelpandfeedbackfromanumberofstudentsandcolleagues,ofwhom
IwouldliketomentionIbaiAsensioPol,JeremyButterfield,ErikCuriel,JeroenvanDongen,
JuliuszDoboszewski,JohnEarman,JanGłowacki,Evert-JanHekkelman,LeoGarciaHeveling,
MichelJanssen,DennisLemkuhl,MartinLesourd,SeraMarkoff,EttoreMinguzzi,JohnNorton,
BryanRoberts,QuintenRutgers,andJanSbierski. Mostchapterswerealsoreviewedduringthe
2020–2021Cambridge–LSEPhilosophyofPhysicsBootcamp,whichwasofgreathelp.
ThefinaleditofthisbookwasdoneduringJuly2021atalovelycottagebytheriverIJssel,
whichwecouldusethankstothegeneroushospitalityofourfriendsArendandEsthervander
Sluis. ThislastroundalsobenefitedfromtheonlineconferenceSingularitytheorems,causality,
andallthat: AtributetoRogerPenrose,heldinJune2021(organizedbyPiotrChrus´ciel,Greg
Galloway, Michael Kunzinger, Ettore Minguzzi, and Roland Steinbauer) where I could pick up
the latestnews and was also given the unexpectedhonour to speak. Mygreatest debt, however,
is to Edith de Jong, who contributed so much more than the beautiful cover art and various
drawingstothisbook,includingtheoneofPenroseonthededicationpageofthisbook.12
12ThisdrawingisbasedonaphotographofPenroseinGravitation(Misner,Thorne,andWheeler,1973,p.936).
Synopsis ix
Synopsis
Here is a brief summary of the chapters, which may also help potential readers as well as
instructors. My own experience is that chapters 2, 3, 4, 5, and 7 may form the basis of a
one-semester(master’s)courseentitledMathematicalstructure ofgeneral relativity.13 Thismay
be followed by another one-semester course called Singularities and black holes,14 based on
chapters6,§§8.1–8.4,9,andaselectionoftopicsfromchapter10. Foradvancedstudentswith
sufficientbackgroundinboth GR andmathematics,thelattercouldalsostandalone.15
Sincethisbook isalso,perhapseven largely,intended forself-studyandpleasure,it contains
no exercises. However, instructors (and even students with enough self-discipline) can easily
assign almost any derivation as an exercise (for themselves). Many difficult results are just
mentionedwithoutproof(alwayswithareference),andthesecouldserveasadvancedproblems.
1. Historical introduction. Based on recently completed research by historians of science,
the reader is introduced to Einstein’s “bumpy road” to his theory of general relativity.
Although GR may well be the most sublime of all scientific theories, created by a man
whois widely–perhapsexaggeratedly–seenas oneofthe supremegeniuses humanityhas
broughtabout,atleastthestoryofitsdiscoveryis“human,alltoohuman”. Ialsoinclude
alittlemathematicalhistory, involving Riemannandothers,aswellas abriefpictureof
mathematical GR untilabout1970. Iclosewithsomemusingsongeneralcovariance.
2. General differential geometry. This is a turbo introduction to manifolds and tensors,
intendedforreaderswhohavealreadyseensomebasictreatmentofthismaterial. Even
withinsomemodern,coordinate-freeapproachtodifferentialgeometry,bothabstractand
computationalaspectsof GR alsorequiretheuseofold-fashionedcoordinatesandindices.
3. Metric differential geometry. Here the pace slows down. In this brief chapter, which is
mainly a warm-up for the next two chapters, metrics, geodesics, connections, and the
Levi-Civita(i.e. metric)connection areintroduced. Thismaterial istotally standard, butI
havedonemybesttogivesomeperspectiveongeodesicsinLorentzianmanifolds.
4. Curvature. Thischaptermayhaveanunusualemphasisonsectionalcurvature,constant
curvature,andthenineteenthcenturyoriginsoftheabstractmoderntheoryinsubmanifolds
ofEuclidean space. In myexperience,this backgroundis especiallyhelpfulin understand-
ingtheGauss–WeingartenandGauss-Codazziequations,whichinturnareessentialforthe
derivationoftheconstraintequationsofGR. Inthesamespirit,thelastsectiondiscussesthe
classical“fundamentaltheoremforhypersurfaces”,whichgivesnecessaryandsufficient
conditionsfortheexistenceand(geometric)uniquenessofembeddingsofcurvedsurfaces
inflatspace. Thoughmuchsimpler,thistheoremresemblesthecorrespondingresultfor
theEinsteinequationsin§7.6,notablyregardingtheroleofconstraints.
5. Geodesics and causal structure. This chapter introduces the topological and geometric
techniques,largelydevelopedbyPenroseandothersinthe1960s,thatdemarcatemathe-
matical GR fromatheoreticalphysicstreatment. Asalreadymentioned,our discussion is
farfromcomplete,butitishopefullyenoughtoadvocateaspecificcausalwayofthinking.
13Chapter2shouldperhapsnotbediscussedindetail(whichmightrepelstudents);italonehasasummary(§2.7).
14Nonetheless,puttingchapter6beforechapter7inthisbookisalogicalchoice,sincethesingularitytheorems
donotrelyontheEinsteinequations. Thesecondhalfofchapter8containsadvancedandpartlyspeculative“retro”
materialthatIsimplyfindinteresting–especiallytheunresolvedproblemoftime–andcouldnotresistincluding.
15Natário(2021),whichIsawmuchtoolatetouseit,providesaone-semestercourseinallofmathematicalGR.