Table Of ContentGravity 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 ORT´IN completed his graduate studies and obtained his Ph.D. at the Universidad Auto´noma 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 Auto´noma 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´SORT´IN
SpanishResearchCouncilandUniversidadAuto´nomadeMadrid
(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
