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Gravity and Strings Oneappealingfeatureofstringtheoryisthatitprovidesatheoryofquantumgravity.GravityandStringsisa self-contained,pedagogicalexpositionofthistheory,itsfoundations,anditsbasicresults. InPartI,thefoundationsaretracedbacktotheveryearlyspecial-relativisticfieldtheoriesofgravity,show- inghowsuchtheories,whichareassociatedwiththeconceptofthegraviton,leadtogeneralrelativity.Gauge theoriesofgravityarethendiscussedandusedtointroducesupergravitytheories. Part II covers some of the most interesting solutions of general relativity and its generalizations. These includeSchwarzschildandReissner–Nordstro¨mblackholes,theTaub–NUTsolution,gravitationalinstantons, and gravitational waves. Kaluza–Klein theories and the uses of residual supersymmetries are discussed in detail. Part III presents string theory from the effective-action point of view, using the results found earlier in thebookasbackground.ThesupergravitytheoriesassociatedwithsuperstringsandMtheoryarethoroughly studied,andusedtodescribedualitiesandclassicalsolutionsrelatedtonon-pertubativestatesofthesetheories. Abriefaccountofextremeblack-holeentropycalculationsisalsogiven. Thisuniquebookwillbeusefulasareferenceforgraduatestudentsandresearchers,aswellasacomple- mentarytextbookforcoursesongravity,supergravity,andstringtheory. TOMA´S ORTÍN completed his graduate studies and obtained his Ph.D. at the Universidad Autońoma de Madrid.HethenworkedasapostdoctoralstudentinthePhysicsDepartmentofStanfordUniversitysupported byaSpanishGovernmentgrant.Between1993and1995hewasEUMarieCuriepostdoctoralfellowinthe StringTheoryGroupofthePhysicsDepartmentofQueenMaryCollege,UniversityofLondon,andfrom1995 to 1997, he was a Fellow in the Theory Division of CERN. He is currently a Staff Scientist at the Spanish Research Council and a member of the Institute for Theoretical Physics of the Universidad Autońoma de Madrid.DrOrt´ınhastaughtseveralgraduatecoursesonadvancedgeneralrelativity,supergravity,andstrings. Hisresearchinterestslieinstringtheory,gravity,quantumgravity,andblack-holephysics. CAMBRIDGEMONOGRAPHSON MATHEMATICALPHYSICS Generaleditors:P.V.Landshoff,D.R.Nelson,S.Weinberg S.J.AarsethGravitationalN-BodySimulations J.Ambjørn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheoryApproach A.M.AnileRelativisticFluidsandMagneto-Fluids J.A.deAzca´rrageandJ.M.IzquierdoLieGroups,LieAlgebras,CohomologyandSomeApplications inPhysics† O.Babelon,D.BernardandM.TalonIntroductiontoClassicalIntegralSystems V.BelinkskiandE.VerdaguerGravitationalSolitons J.BernsteinKineticTheoryintheEarlyUniverse 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.SClarkeRelativityonCurvedManifolds† P.G.O.FreundIntroductiontoSupersymmetry† J.FuchsAffineLieAlgebrasandQuantumGroups† J.FuchsandC.SchweigertSymmetries,LieAlgebrasandRepresentations:AGraduateCourse forPhysicists† Y.FujiiandK.MaedaTheScalar–TensorTheoryofGravitation A.S.Galperin,E.A.Ivanov,V.I.OrievetskyandE.S.SokatchevHarmonicSuperspace R.GambiniandJ.PullinLoops,Knots,GaugeTheoriesandQuantumGravity† M.Go¨ckelerandT.Schu¨ckerDifferentialGeometry,GaugeTheoriesandGravity† C.Go´mez,M.RuizAltabaandG.SierraQuantumGroupsinTwo-dimensionalPhysics M.B.Green,J.H.SchwarzandE.WittenSuperstringTheory,volume1:Introduction† M.B.Green,J.H.SchwarzandE.WittenSuperstringTheory,volume2:LoopAmplitudes,Anomalies andPhenomenology† V.N.GribovTheTheoryofComplexAngularMomenta S.W.HawkingandG.F.R.EllisTheLarge-ScaleStructureofSpace-Time† F.IachelloandA.ArunaTheInteractingBosonModel F.IachelloandP.vanIsackerTheInteractingBoson–FermionModel C.ItzyksonandJ.-M.DrouffeStatisticalFieldTheory,volume1:FromBrownianMotionto RenormalizationandLatticeGaugeTheory† C.ItzyksonandJ.-M.DrouffeStatisticalFieldTheory,volume2:StrongCoupling,MonteCarlo Methods,ConformalFieldTheory,andRandomSystems† C.JohnsonD-Branes J.I.KapustaFinite-TemperatureFieldTheory† V.E.Korepin,A.G.IzerginandN.M.BoguliubovTheQuantumInverseScatteringMethodand CorrelationFunctions† M.LeBellacThermalFieldTheory† Y.MakeenkoMethodsofContemporaryGaugeTheory N.H.MarchLiquidMetals:ConceptsandTheory I.M.MontvayandG.Mu¨nsterQuantumFieldsonaLattice† T.Ort´ınGravityandStrings A.OzoriodeAlmeidaHamiltonianSystems:ChaosandQuantization† R.PenroseandW.RindlerSpinorsandSpace-Time,volume1:Two-SpinorCalculusandRelativistic Fields† R.PenroseandW.RindlerSpinorsandSpace-Time,volume2:SpinorandTwistorMethodsin Space-TimeGeometry† S.PokorskiGaugeFieldTheories,2ndedition J.PolchinskiStringTheory,volume1:AnIntroductiontotheBosonicString J.PolchinskiStringTheory,volume2:SuperstringTheoryandBeyond V.N.PopovFunctionalIntegralsandCollectiveExcitations† R.G.RobertsTheStructureoftheProton† H.Stephani,D.Kramer,M.A.H.MacCallum,C.HoenselaersandE.HerltExactSolutionsof Einstein’sFieldEquations,2ndedition J.M.StewartAdvancedGeneralRelativity† A.VilenkinandE.P.S.ShellardCosmicStringsandOtherTopologicalDefects† R.S.WardandR.O.WellsJrTwistorGeometryandFieldTheories† J.R.WilsonandG.J.MathewsRelativisticNumericalHydrodynamics †Issuedasapaperback Gravity and Strings TOMA´SORTÍN SpanishResearchCouncilandUniversidadAutońomadeMadrid (CSIC) cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521824750 © Tomas Ortin 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2004 isbn-13 978-0-511-18476-5 eBook (NetLibrary) isbn-10 0-511-18476-x eBook (NetLibrary) isbn-13 978-0-521-82475-0 hardback isbn-10 0-521-82475-3 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To Marimar, Diego, and Toma´s, the sweet strings that tie me to the real world Contents Preface pagexix PartI Introductiontogravityandsupergravity 1 1 Differentialgeometry 3 1.1 Worldtensors 3 1.2 Affinelyconnectedspacetimes 5 1.3 Metricspaces 9 1.3.1 Riemann–CartanspacetimeU 11 d 1.3.2 RiemannspacetimeV 13 d 1.4 Tangentspace 14 1.4.1 Weitzenbo¨ckspacetimeA 19 d 1.5 Killingvectors 20 1.6 Dualityoperations 21 1.7 Differentialformsandintegration 23 1.8 Extrinsicgeometry 25 2 Noether’stheorems 26 2.1 Equationsofmotion 26 2.2 Noether’stheorems 27 2.3 Conservedcharges 31 2.4 Thespecial-relativisticenergy–momentumtensor 32 2.4.1 Conservationofangularmomentum 33 2.4.2 Dilatations 37 2.4.3 Rosenfeld’senergy–momentumtensor 39 2.5 TheNoethermethod 41 3 Aperturbativeintroductiontogeneralrelativity 45 3.1 ScalarSRFTsofgravity 46 ix