GRAVITATIONAL COLLAPSE AND SPACETIME SINGULARITIES Physicalphenomenainastrophysicsandcosmologyinvolvegravitationalcollapse in a fundamental way. The final fate of a massive star when it collapses under its own gravity at the end of its life cycle is one of the most important ques- tionsingravitationtheoryandrelativisticastrophysics, andisthefoundationof blackhole physics. General relativity predicts that continual gravitational collapse gives rise to a spacetime singularity, which may be hidden inside an event horizon or visi- ble to external observers. This book investigates these issues, and shows how such visible ultra-dense regions arise naturally and generically as an outcome of dynamicalgravitationalcollapse.Quantumgravitymaytakeoverinsuchregimes to resolve the classical spacetime singularity. The quantum effects from a visible extreme gravity region could then propagate to external observers, providing a usefullaboratoryforquantumgravity,andimplyinginterestingconsequencesfor ultra-high energy astrophysical phenomena in the universe. Thisvolumewillbeofinteresttograduatestudentsandacademicresearchers in gravitation physics and fundamental physics, as well as in astrophysics and cosmology. It includes a detailed review of recent research into gravitational collapse, and several examples of collapse models are worked out in detail. Pankaj S. Joshi conducts research at the Tata Institute of Fundamental Research, Mumbai. His research interests include gravitation physics, spacetime structure and quantum gravity, and cosmology and relativistic astrophysics. He has published many research papers and books in these areas, and has held vis- itingfacultypositionsinseveralcountries, lecturinganddoingresearchonthese topics. Professor Joshi has an excellent international reputation for his work in the field of gravitation theory. His extensive analysis of general relativistic gravita- tional collapse has been widely recognized as providing significant insights into thefinalendstatesofacontinualcollapse, formationofvisiblesingularities, and nature of cosmic censorship and blackholes. JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE i — #1 CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS General editors: P. V. Landshoff, D. R. Nelson, S. Weinberg S.J.AarsethGravitationalN-BodySimulations J.Ambjørn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheoryApproach A.M.AnileRelativisticFluidsandMagneto-Fluids J. A. de Azc´arrage and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics† O.Babelon,D.BernardandM.TalonIntroductiontoClassicalIntegrableSystems F.BastianelliandP.vanNieuwenhuizenPathintegralsandAnomaliesinCurvedSpace V.BelinkskiandE.VerdaguerGravitationalSolutions J.BernsteinKineticTheoryintheExpandingUniverse G.F.BertschandR.A.BrogliaOscillationsinFiniteQuantumSystems N.D.BirrellandP.C.W.DaviesQuantumFieldsinCurvedSpace† M.BurgessClassicalCovariantFields S.CarlipQuantumGravityin2+1Dimensions J.C.CollinsRenormalization† M.CreutzQuarks,GluonsandLattices† P.D.D’EathSupersymmetricQuantumCosmology F.deFeliceandC.J.S.ClarkeRelativityonCurvedManifolds† B.S.DeWittSupermanifolds,2ndedition† P.G.O.FreundIntroductiontoSupersymmetry† J.FuchsAffineLieAlgebrasandQuantumGroups† J.FuchsandC.SchweigertSymmetries,LieAlgebrasandRepresentations:AGraduateCourseforPhysicists† Y.FujiiandK.MaedaTheScalarTensorTheoryofGravitation A.S.Galperin,E.A.Ivanov,V.I.OrievetskyandE.S.SokatchevHarmonicSuperspace R.GambiniandJ.PullinLoops,Knots,GaugeTheoriesandQuantumGravity† M.Go¨ckelerandT.Schu¨ckerDifferentialGeometry.GaugeTheoriesandGravity† C.G´omez,M.RuizAltabaandG.SierraQuantumGroupsinTwo-dimensionalPhysics M.B.Green,J.H.SchwarzandE.WittenSuperstringTheory,volume1:Introduction† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 2: Loop Amplitudes, Anomalies and Phenomenology† V.N.GribovTheTheoryofComplexAngularMomenta S.W.HawkingandG.F.R.EllisTheLarge-ScaleStructureofSpace-Time† F.IachelloandA.ArimaTheInteractingBosonModel F.IachelloandP.vanIsackerTheInteractingBoson–FermionModel C.ItzyksonandJ.-M.DrouffeStatisticalFieldTheory,volume1:FromBrownianMotiontoRenormalization andLatticeGaugeTheory† C. Itzykson and J. -M. Drouffe Statistical Field Theory, volume 2: Strong Coupling, Monte Carlo Methods, ConformalFieldTheory,andRandomSystems† C.JohnsonD-Branes P.S.JoshiGravitationalCollapseandSpacetimeSingularities J.I.KapustaFinite-TemperatureFieldTheory† V.E.Korepin,A.G.IzerginandN.M.BoguliubovTheQuantumInverseScatteringMethodandCorrelation Functions† M.LeBellacThermalFieldTheory† Y.MakeenkoMethodsofContemporaryGaugeTheory N.MantonandP.SutcliffeTopologicalSolitons N.H.MarchLiquidMetals:ConceptsandTheory I.M.MontvayandG.Mu¨nsterQuantumFieldsonaLattice† L.O’RaifeartaighGroupStructureofGaugeTheories† T.Ort´ınGravityandStrings A.OzoriodeAlmeidaHamiltonianSystems:ChaosandQuantization† R.PenroseandW.RindlerSpinorsandSpace-Time,volume1:Two-SpinorCalculusandRelativisticFields† R. Penrose and W. Rindler Spinors and Space-Time, volume 2: Spinor and Twistor Methods in Space-Time Geometry† S.PokorskiGaugeFieldTheories,2ndedition J.PolchinskiStringTheory,volume1:AnIntroductiontotheBosonicString† J.PolchinskiStringTheory,volume2:SuperstringTheoryandBeyond† V.N.PopovFunctionalIntegralsandCollectiveExcitations† R.J.RiversPathIntegralMethodsinQuantumFieldTheory† R.G.RobertsTheStructureoftheProton† C.RovelliQuantumGravity W.C.SaslawGravitationalPhysicsofStellarandGalacticSystems† H.Stephani,D.Kramer,M.A.H.MacCallum,C.HoenselaersandE.HerltExactSolutionsofEinstein’sField Equations,2ndedition J.M.StewartAdvancedGeneralRelativity† A.VilenkinandE.P.S.ShellardCosmicStringsandOtherTopologicalDefects† R.S.WardandR.O.WellsJrTwistorGeometryandFieldTheories† J.R.WilsonandG.J.MathewsRelativisticNumericalHydrodynamics †Issuedasapaperback JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE ii — #2 Gravitational Collapse and Spacetime Singularities PANKAJ S. JOSHI Tata Institute of Fundamental Research, Mumbai, India JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE iii — #3 cambridgeuniversitypress Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,S˜aoPaulo CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521871044 (cid:1)c PankajS.Joshi2007 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2007 PrintedintheUnitedKingdomattheUniversityPress,Cambridge A catalogue record for this publication is available from the British Library ISBN978-0-521-87104-4hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurlsfor externalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE iv — #4 To my parents, Arunadevi Shantilal Joshi and Shantilal Ramshankar Joshi JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE v — #5 JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE vi — #6 Contents Preface pageix 1 Introduction 1 2 The spacetime manifold 10 2.1 The manifold model 10 2.2 The metric tensor 21 2.3 Connection 24 2.4 Non-spacelike geodesics 29 2.5 Spacetime curvature 32 2.6 The Einstein equations 38 2.7 Exact solutions 43 3 Spherical collapse 60 3.1 Basic framework 62 3.2 Regularity conditions 69 3.3 Collapsing matter clouds 71 3.4 Nature of singularities 79 3.5 Exterior geometry 87 3.6 Dust collapse 90 3.7 Equation of state 129 4 Cosmic censorship 135 4.1 Causal structure 136 4.2 Spacetime singularities 149 4.3 Blackholes 161 4.4 Higher spacetime dimensions 169 4.5 Formulating the censorship 175 4.6 Genericity and stability 190 5 Final fate of a massive star 210 5.1 Life cycle of massive stars 213 5.2 Evolution of a physically realistic collapse 215 vii JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE vii — #7 viii Contents 5.3 Non-spherical models 225 5.4 Blackhole paradoxes 235 5.5 Resolution of a naked singularity 238 References 255 Index 269 JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE viii — #8 Preface Thephysicalphenomenainastrophysicsandcosmologyinvolvegravitational collapse in a fundamental way. The final fate of a massive star, when it collapses under its own gravity at the end of its life cycle, is one of the most important questions in gravitation theory and relativistic astrophysics today. The applications and basic theory of blackholes vigorously developed over the past decades crucially depend on this outcome. A sufficiently massive star many times the size of the Sun would undergo a continual gravitational collapse on exhausting its nuclear fuel, without achieving an equilibrium state such as a neutron star or white dwarf. The singularity theorems in general relativity then predict that the collapse gives risetoaspacetimesingularity, eitherhiddenwithinaneventhorizonofgrav- ityorvisibletotheexternaluniverse.Thedensitiesandspacetimecurvatures get arbitrarily high and diverge at these ultra-strong gravity regions. Their visibility to outside observers is determined by the causal structure within the dynamically developing collapsing cloud, as governed by the Einstein field equations. When the internal dynamics of the collapse delays the hori- zon formation, these become visible, and may communicate physical effects to the external universe. These issues are investigated here, and the treat- mentisaimedatshowinghowsuchvisibleultra-denseregionsarisenaturally and generically as the outcome of a dynamical gravitational collapse in Ein- stein gravity. While it predicts the existence of visible singularities; classical general relativity may no longer hold in these very late stages of the col- lapse, and quantum gravity may take over to resolve the classical spacetime singularity. The quantum effects from a visible, the extreme gravity region could then propagate to outside observers to provide a useful laboratory for quantum gravity. Blackholes need not form in such a scenario and there may be interesting consequences for ultra-high energy astrophysical phenomena in the universe. Thegeneraltheoryofrelativity, whichhasstrongexperimentalsupport, is used here, and its basics and useful features of spacetimes are reviewed. The necessary tools are developed as needed, but a prior familiarity with general relativity would help. It is a pleasure to thank many friends and colleagues ix JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE ix — #9 x Preface for numerous discussions and work as cited, on the themes described here. Special thanks are due to R. Goswami and I. H. Dwivedi for their ideas and help and for our studies together. A. Mahajan and S. Khedekar helped with the manuscript. JOSHI: “FM” — 2007/8/18 — 18:41 — PAGE x — #10