Asymptotic Analysis in General Relativity

381 Pages·2018·2.997 MB·English

by  Thierry Daudé, Dietrich Häfner, Jean-Philippe Nicolas

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
ManagingEditor:
ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat
www.cambridge.org/mathematics
325 LecturesontheRicciflow,P.TOPPING
326 ModularrepresentationsoffinitegroupsofLietype,J.E.HUMPHREYS
327 Surveysincombinatorics2005,B.S.WEBB(ed)
328 Fundamentalsofhyperbolicmanifolds,R.CANARY,D.EPSTEIN&A.MARDEN(eds)
329 SpacesofKleiniangroups,Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 Noncommutativelocalizationinalgebraandtopology,A.RANICKI(ed)
331 Foundationsofcomputationalmathematics,Santander2005,L.MPARDO,A.PINKUS,E.SU¨LI
&M.J.TODD(eds)
332 Handbookoftiltingtheory,L.ANGELERIHU¨GEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndEdition),A.KOCK
334 TheNavier–Stokesequations,N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability,A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules,I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory,J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory,N.YOUNG(ed)
339 GroupsStAndrews2005I,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory,J.B.CONREY,D.W.FARMER,F.MEZZADRI
&N.C.SNAITH(eds)
342 Ellipticcohomology,H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI,J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII,J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry,A.NEEMAN
346 Surveysincombinatorics2007,A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics,N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis,P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY
&A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY
&A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas,A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials,J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis,J.BLATH,P.MO¨RTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis,K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence,J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations,D.DELBOURGO
357 Algebraictheoryofdifferentialequations,M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory,M.R.BRIDSON,P.H.KROPHOLLER
&I.J.LEARY(eds)
359 Modulispacesandvectorbundles,L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARC´IA-PRADA&
S.RAMANAN(eds)
360 Zariskigeometries,B.ZILBER
361 Words:Notesonverbalwidthingroups,D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories,R.BAUTISTA,L.SALMERO´N&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008,F.CUCKER,A.PINKUS&M.J.TODD(eds)
364 Partialdifferentialequationsandfluidmechanics,J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009,S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems,B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena,G.BLOWER
368 GeometryofRiemannsurfaces,F.P.GARDINER,G.GONZA´LEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks,M.DRAIEF&L.MASSOULIE´
370 Theoryofp-adicdistributions,S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH
371 Conformalfractals,F.PRZYTYCKI&M.URBAN´SKI
372 Moonshine:Thefirstquartercenturyandbeyond,J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds)
373 Smoothness,regularityandcompleteintersection,J.MAJADAS&A.G.RODICIO
374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction,S.ALINHAC
375 Triangulatedcategories,T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds)
376 Permutationpatterns,S.LINTON,N.RUSˇKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY
378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 Realandcomplexsingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
381 Symmetriesandintegrabilityofdifferenceequations,D.LEVI,P.OLVER,Z.THOMOVA
&P.WINTERNITZ(eds)
Downloaded from https://www.cambridge.org/core. University of New England, on 04 Jan 2018 at 16:45:21, subject to the Cambridge Core
terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612
382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJ´ICˇEK
383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems,B.MARCUS,K.PETERSEN
&T.WEISSMAN(eds)
386 Independence-friendlylogic,A.L.MANN,G.SANDU&M.SEVENSTER
387 GroupsStAndrews2009inBathI,C.M.CAMPBELLetal.(eds)
388 GroupsStAndrews2009inBathII,C.M.CAMPBELLetal.(eds)
389 Randomfieldsonthesphere,D.MARINUCCI&G.PECCATI
390 Localizationinperiodicpotentials,D.E.PELINOVSKY
391 Fusionsystemsinalgebraandtopology,M.ASCHBACHER,R.KESSAR&B.OLIVER
392 Surveysincombinatorics2011,R.CHAPMAN(ed)
393 Non-abelianfundamentalgroupsandIwasawatheory,J.COATESetal.(eds)
394 Variationalproblemsindifferentialgeometry,R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds)
395 Howgroupsgrow,A.MANN
396 Arithmeticdifferentialoperatorsoverthep-adicintegers,C.C.RALPH&S.R.SIMANCA
397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology,J.BOLTE&F.STEINER(eds)
398 Mathematicalmodelsincontactmechanics,M.SOFONEA&A.MATEI
399 Circuitdoublecoverofgraphs,C.-Q.ZHANG
400 Densespherepackings:ablueprintforformalproofs,T.HALES
401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory,B.DENG,J.DU&Q.FU
402 Mathematicalaspectsoffluidmechanics,J.C.ROBINSON,J.L.RODRIGO&W.SADOWSKI(eds)
403 Foundationsofcomputationalmathematics,Budapest2011,F.CUCKER,T.KRICK,A.PINKUS
&A.SZANTO(eds)
404 Operatormethodsforboundaryvalueproblems,S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds)
405 Torsors,e´talehomotopyandapplicationstorationalpoints,A.N.SKOROBOGATOV(ed)
406 Appalachiansettheory,J.CUMMINGS&E.SCHIMMERLING(eds)
407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups,J.N.BRAY,D.F.HOLT
&C.M.RONEY-DOUGAL
408 Complexityscience:theWarwickmaster’scourse,R.BALL,V.KOLOKOLTSOV&R.S.MACKAY(eds)
409 Surveysincombinatorics2013,S.R.BLACKBURN,S.GERKE&M.WILDON(eds)
410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups,
T.CECCHERINI-SILBERSTEIN,F.SCARABOTTI&F.TOLLI
411 Modulispaces,L.BRAMBILA-PAZ,O.GARC´IA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds)
412 Automorphismsandequivalencerelationsintopologicaldynamics,D.B.ELLIS&R.ELLIS
413 Optimaltransportation,Y.OLLIVIER,H.PAJOT&C.VILLANI(eds)
414 AutomorphicformsandGaloisrepresentationsI,F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
415 AutomorphicformsandGaloisrepresentationsII,F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
416 Reversibilityindynamicsandgrouptheory,A.G.O’FARRELL&I.SHORT
417 Recentadvancesinalgebraicgeometry,C.D.HACON,M.MUSTAT¸Aˇ&M.POPA(eds)
418 TheBloch–KatoconjecturefortheRiemannzetafunction,J.COATES,A.RAGHURAM,A.SAIKIA
&R.SUJATHA(eds)
419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations,J.C.MEYER
&D.J.NEEDHAM
420 Arithmeticandgeometry,L.DIEULEFAITetal.(eds)
421 O-minimalityandDiophantinegeometry,G.O.JONES&A.J.WILKIE(eds)
422 GroupsStAndrews2013,C.M.CAMPBELLetal.(eds)
423 Inequalitiesforgrapheigenvalues,Z.STANIC´
424 Surveysincombinatorics2015,A.CZUMAJetal.(eds)
425 Geometry,topologyanddynamicsinnegativecurvature,C.S.ARAVINDA,F.T.FARRELL&J.-F.LAFONT(eds)
426 Lecturesonthetheoryofwaterwaves,T.BRIDGES,M.GROVES&D.NICHOLLS(eds)
427 RecentadvancesinHodgetheory,M.KERR&G.PEARLSTEIN(eds)
428 GeometryinaFre´chetcontext,C.T.J.DODSON,G.GALANIS&E.VASSILIOU
429 Sheavesandfunctionsmodulop,L.TAELMAN
430 RecentprogressinthetheoryoftheEulerandNavier–Stokesequations,J.C.ROBINSON,J.L.RODRIGO,
W.SADOWSKI&A.VIDAL-LO´PEZ(eds)
431 Harmonicandsubharmonicfunctiontheoryontherealhyperbolicball,M.STOLL
432 Topicsingraphautomorphismsandreconstruction(2ndEdition),J.LAURI&R.SCAPELLATO
433 RegularandirregularholonomicD-modules,M.KASHIWARA&P.SCHAPIRA
434 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems(2ndEdition),K.TAIRA
435 GradedringsandgradedGrothendieckgroups,R.HAZRAT
436 Groups,graphsandrandomwalks,T.CECCHERINI-SILBERSTEIN,M.SALVATORI&E.SAVA-HUSS(eds)
437 Dynamicsandanalyticnumbertheory,D.BADZIAHIN,A.GORODNIK&N.PEYERIMHOFF(eds)
438 Randomwalksandheatkernelsongraphs,M.T.BARLOW
439 Evolutionequations,K.AMMARI&S.GERBI(eds)
440 Surveysincombinatorics2017,A.CLAESSONetal.(eds)
441 Polynomialsandthemod2SteenrodalgebraI,G.WALKER&R.M.W.WOOD
442 Polynomialsandthemod2SteenrodalgebraII,G.WALKER&R.M.W.WOOD
443 Asymptoticanalysisingeneralrelativity,T.DAUDE´,D.HA¨FNER&J.-P.NICOLAS(eds)
444 Geometricandcohomologicalgrouptheory,P.H.KROPHOLLER,I.J.LEARY,C.MART´INEZ-PE´REZ&
B.E.A.NUCINKIS(eds)
Downloaded from https://www.cambridge.org/core. University of New England, on 04 Jan 2018 at 16:45:21, subject to the Cambridge Core
terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612
LondonMathematicalSocietyLectureNoteSeries:443
Asymptotic Analysis in General Relativity
Editedby
THIERRY DAUDE´
Universite´ deCergy-Pontoise,France
DIETRICH HA¨ FNER
Universite´ GrenobleAlpes,France
JEAN-PHILIPPE NICOLAS
Universite´ deBretagneOccidentale,France
Downloaded from https://www.cambridge.org/core. University of New England, on 04 Jan 2018 at 16:45:21, subject to the Cambridge Core
terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,
NewDelhi–110025,India
79AnsonRoad,#06–04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316649404
DOI:10.1017/9781108186612
©CambridgeUniversityPress2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2018
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloguinginPublicationdata
Names:Daude´,Thierry,1977–editor.|Ha¨fner,Dietrich,editor.|
Nicolas,J.-P.(Jean-Philippe),editor.
Title:Asymptoticanalysisingeneralrelativity/editedbyThierryDaude´
(Universite´deCergy-Pontoise),DietrichHa¨fner(Universite´GrenobleAlpes),
Jean-PhilippeNicolas(Universite´deBretagneOccidentale).
Othertitles:LondonMathematicalSocietylecturenoteseries;443.
Description:Cambridge,UnitedKingdom;NewYork,NY:
CambridgeUniversityPress,2017.|
Series:LondonMathematicalSocietylecturenoteseries;443|
Includesbibliographicalreferences.
Identifiers:LCCN2017023160|ISBN9781316649404(pbk.)|ISBN1316649407(pbk.)
Subjects:LCSH:Generalrelativity(Physics)–Mathematics.
Classification:LCCQC173.6.A832017|DDC530.11–dc23LCrecordavailableat
https://lccn.loc.gov/2017023160
ISBN978-1-316-64940-4Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Downloaded from https://www.cambridge.org/core. University of New England, on 04 Jan 2018 at 16:45:21, subject to the Cambridge Core
terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612
Contents
1 IntroductiontoModernMethodsforClassicaland
QuantumFieldsinGeneralRelativity page1
ThierryDaude´,DietrichHa¨fnerandJean-PhilippeNicolas
1.1 GeometryofBlackHoleSpacetimes 2
1.2 QuantumFieldTheoryonCurvedSpacetimes 3
1.3 ConformalGeometryandConformalTractorCalculus 4
1.4 AMinicourseinMicrolocalAnalysisandWavePropagation 5
References 6
2 GeometryofBlackHoleSpacetimes 9
LarsAndersson,ThomasBa¨ckdahlandPieterBlue
2.1 Introduction 9
2.2 Background 13
2.3 BlackHoles 28
2.4 SpinGeometry 43
2.5 TheKerrSpacetime 53
2.6 MonotonicityandDispersion 55
2.7 SymmetryOperators 65
2.8 ConservationLawsfortheTeukolskySystem 70
2.9 AMorawetzEstimatefortheMaxwellFieldonSchwarzschild 75
References 79
3 AnIntroductiontoConformalGeometryandTractor
Calculus,withaviewtoApplicationsinGeneralRelativity 86
SeanN.CurryandA.RodGover
3.1 Introduction 86
3.2 Lecture1:RiemannianInvariantsandInvariantOperators 91
v
Downloaded from https://www.cambridge.org/core. University of New England, on 04 Jan 2018 at 16:45:18, subject to the Cambridge Core
terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612
vi Contents
3.3 Lecture2:ConformalTransformationsandConformalCovariance 94
3.4 Lecture3:ProlongationandtheTractorConnection 104
3.5 Lecture 4: The Tractor Curvature, Conformal Invariants and
InvariantOperators 120
3.6 Lecture5:ConformalCompactificationofPseudo-Riemannian
Manifolds 128
3.7 Lecture6:ConformalHypersurfaces 145
3.8 Lecture7:GeometryofConformalInfinity 151
3.9 Lecture8:BoundaryCalculusandAsymptoticAnalysis 154
Appendix:ConformalKillingVectorFieldsandAdjointTractors 160
References 168
4 AnIntroductiontoQuantumFieldTheoryon
CurvedSpacetimes 171
ChristianGe´rard
4.1 Introduction 171
4.2 AQuickIntroductiontoQuantumMechanics 175
4.3 Notation 178
4.4 CCRandCARAlgebras 179
4.5 StatesonCCR/CARAlgebras 183
4.6 LorentzianManifolds 190
4.7 Klein–GordonFieldsonLorentzianManifolds 192
4.8 FreeDiracFieldsonLorentzianManifolds 197
4.9 MicrolocalAnalysisofKlein–GordonQuasi-FreeStates 201
4.10 ConstructionofHadamardStates 209
References 217
5 AMinicourseonMicrolocalAnalysisforWavePropagation 219
Andra´sVasy
5.1 Introduction 219
5.2 TheOverview 221
5.3 TheBasicsofMicrolocalAnalysis 230
5.4 PropagationPhenomena 285
5.5 ConformallyCompactSpaces 315
5.6 MicrolocalAnalysisintheb-Setting 349
References 371
Downloaded from https://www.cambridge.org/core. University of New England, on 04 Jan 2018 at 16:45:18, subject to the Cambridge Core
terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612
1
Introduction to Modern Methods for Classical
and Quantum Fields in General Relativity
ThierryDaude´,DietrichHa¨fnerandJean-PhilippeNicolas
The last few decades have seen major developments in asymptotic analysis
in the framework of general relativity, with the emergence of methods that,
untilrecently,werenotappliedtocurvedLorentziangeometries.Thishasled
notablytotheproofofthestabilityoftheKerr–deSitterspacetimebyP.Hintz
andA.Vasy[17].Anessentialfeatureofmanyrecentworksinthefieldisthe
useofdispersiveestimates;theyareatthecoreofmoststabilityresultsandare
also crucial for the construction of quantum states in quantum field theory,
domains that have a priori little in common. Such estimates are in general
obtained through geometric energy estimates (also referred to as vector field
methods)orviamicrolocal/spectralanalysis.Inourminds,thetwoapproaches
should be regarded as complementary, and this is a message we hope this
volume will convey succesfully. More generally than dispersive estimates,
asymptotic analysis is concerned with establishing scattering-type results.
Another fundamental example of such results is asymptotic completeness,
which, in many cases, can be translated in terms of conformal geometry as
the well-posedness of a characteristic Cauchy problem (Goursat problem) at
nullinfinity.Thishasbeenusedtodevelopalternativeapproachestoscattering
theory via conformal compactifications (see for instance F. G. Friedlander
[11]andL.MasonandJ.-P.Nicolas[22]).Thepresenceofsymmetriesinthe
geometricalbackgroundcanbeatremendoushelpinprovingscatteringresults,
dispersiveestimatesinparticular.Whatwemeanbysymmetryisgenerallythe
existence of an isometry associated with the flow of a Killing vector field,
though there exists a more subtle type of symmetry, described sometimes as
hidden,correspondingtothepresenceofKillingspinorsforinstance.Recently,
thevectorfieldmethodhasbeenadaptedtotakesuchgeneralizedsymmetries
intoaccountbyL.AnderssonandP.Bluein[2].
This volume compiles notes from the eight-hour mini-courses given at the
summerschoolonasymptoticanalysisingeneralrelativity,heldattheInstitut
1
Downloaded from https://www.cambridge.org/core. University of Exeter, on 30 Dec 2017 at 23:36:17, subject to the Cambridge Core terms
of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612.001
2 T.Daude´,D.Ha¨fnerandJ.-P.Nicolas
FourierinGrenoble,France,from16Juneto4July2014.Thepurposeofthe
summer school was to draw an up-to-date panorama of the new techniques
thathaveinfluencedtheasymptoticanalysisofclassicalandquantumfieldsin
generalrelativityinrecentyears.Itconsistedoffivemini-courses:
(cid:129) “Geometry of black hole spacetimes” by Lars Andersson, Albert Einstein
Institut,Golm,Germany;
(cid:129) “Anintroductiontoquantumfieldtheoryoncurvedspacetimes”byChristian
Ge´rard,Paris11University,Orsay,France;
(cid:129) “Anintroductiontoconformalgeometryandtractorcalculus,withaviewto
applicationsingeneralrelativity”byRodGover,AucklandUniversity,New
Zealand;
(cid:129) “TheboundedL2conjecture”byJe´re´mieSzeftel,Paris6University,France;
(cid:129) “Aminicourseonmicrolocalanalysisforwavepropagation”byAndra´sVasy,
StanfordUniversity,UnitedStatesofAmerica.
Among these, only four are featured in this book. The proof of the bounded
L2conjecturehavingalreadyappearedintwodifferentforms[20,21],Je´re´mie
Szeftelpreferrednottoaddyetanotherversionofthisresult;hislecturenotes
arethereforenotincludedinthepresentvolume.
1.1. GeometryofBlackHole Spacetimes
The notion of a black hole dates back to the 18th century with the works of
SimpsonandLaplace,butitfounditsmoderndescriptionwithintheframework
of general relativity. In fact the year after the publication of the general
theory of relativity by Einstein, Karl Schwarzschild [30] found an explicit
non-trivial solution of the Einstein equations that was later understood to
describe a universe containing nothing but an eternal spherical black hole.
The Kerr solution appeared in 1963 [19] and, with the singularity theorems
of Hawking and Penrose [15], black holes were eventually understood as
inevitabledynamicalfeaturesoftheevolutionoftheuniverseratherthanmere
mathematical oddities. The way exact black hole solutions of the Einstein
equationswerediscoveredwasbyimposingsymmetries.FirstSchwarzschild
looked for spherically symmetric and static solutions in four spacetime
dimensions, which reduces the Einstein equations to a non-linear ordinary
differentialequation(ODE).TheKerrsolutionappearswhenonerelaxesone
of the symmetries and looks for stationary and axially symmetric solutions.
RoyKerrobtainedhissolutionbyimposingonthemetrictheso-called“Kerr–
Schild” ansatz that corresponds to assuming a special algebraic property for
Downloaded from https://www.cambridge.org/core. University of Exeter, on 30 Dec 2017 at 23:36:17, subject to the Cambridge Core terms
of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612.001
ModernMethodsforClassicalandQuantumFieldsinGeneralRelativity 3
the Weyl tensor, namely that it has Petrov-type D, which is similar to the
conditionforapolynomialtohavetwodoubleroots.Thisalgebraicspeciality
oftheWeyltensorcanbeunderstoodasanothertypeofsymmetryassumption
about spacetime. This is a generalized symmetry that does not correspond
to an isometry generated by the flow of a vector field, but is related to the
existenceofaKillingspinor.TheKerrfamily,whichcontainsSchwarzschild’s
spacetime as the zero angular momentum case, is expected to be the unique
family of asymptotically flat and stationary (perhaps pseudo-stationary, or
locally stationary, would be more appropriate) black hole solutions of the
Einstein vacuum equations (there is a vast literature on this topic, see for
exampletheoriginalpaperbyD.Robinson[27],hisreviewarticle[28]andthe
recentanalyticapproachbyS.Alexakis,A.D.Ionescu,andS.Klainerman[1]).
Moreoveritisbelievedtobestable(thereisalsoanimportantliteratureonthis
question,thestabilityofKerr–deSitterblackholeswasestablishedrecentlyin
[17], though the stability of the Kerr metric is still an open problem). These
twoconjectures playacrucialroleinphysics whereitiscommonlyassumed
thatthelongtermdynamicsofablackholestabilizestoaKerrsolution.The
extendedlecturenotesbyLarsAndersson,ThomasBa¨ckdahl,andPieterBlue
take us through the many topics that are relevant to the questions of stability
and uniqueness of the Kerr metric, including the geometry of stationary and
dynamical black holes with a particular emphasis on the special features of
the Kerr metric, spin geometry, dispersive estimates for hyperbolic equations
andgeneralizedsymmetryoperators.ThetypeDstructureisanessentialfocus
of the course, with the intimate links between the principal null directions,
the Killing spinor, Killing vectors and tensors, Killing–Yano tensors and
symmetry operators. All these notions are used in the final sections where
some conservation laws are derived for the Teukolsky system governing the
evolution of spin n/2 zero rest-mass fields, and a new proof of a Morawetz
estimateforMaxwellfieldsontheSchwarzschildmetricisgiven.
1.2. QuantumFieldTheoryonCurvedSpacetimes
Inthe1980s,DimockandKaystartedaresearchprogramconcerningscatter-
ing theory for classical and quantum fields on the Schwarzschild spacetime;
see[9].TheirworkwasthenpushedfurtherbyBachelot, Ha¨fner,andothers,
leadinginparticulartoamathematicallyrigorousdescriptionoftheHawking
effect on Schwarzschild and Kerr spacetimes, see e.g. [4], [14]. In the
Schwarzschild case there exists a global timelike Killing vector field in the
exterioroftheblackholethatcanbeusedtodefinevacuumandthermalstates.
Downloaded from https://www.cambridge.org/core. University of Exeter, on 30 Dec 2017 at 23:36:17, subject to the Cambridge Core terms
of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108186612.001