Gravity and Strings

1042 Pages·2015·5.167 MB·English

by  Tomás Ortín

GRAVITY AND STRINGS
Self-contained and comprehensive, this definitive new edition of Gravity and Strings is a
unique resource for graduate students and researchers in theoretical physics.
From basic differential geometry through to the construction and study of black-hole
and black-brane solutions in quantum gravity – via all the intermediate stages – this book
provides a complete overview of the intersection of gravity, supergravity, and superstrings.
Now fully revised, this second edition covers an extensive array of topics, including
new material on non-linear electric-magnetic duality, the embedding-tensor formalism,
matter-coupled supergravity, supersymmetric solutions, the geometries of scalar manifolds
appearing in four- and five-dimensional supergravities, and much more. Covering reviews
of important solutions and numerous solution-generating techniques, and accompanied by
an exhaustive index and bibliography, this is an exceptional reference work.
Toma´s Ort´ın is a Research Professor at the Institute for Theoretical Physics (IFT),
a joint institute of the Autonomous University of Madrid and the Spanish National
Research Council (UAM-CSIC). He has previously worked at the European Laboratory
forParticlePhysics(CERN),heldpostdoctoralpositionsatStanfordUniversityandQueen
Mary University of London, and has taught several graduate courses on advanced general
relativity. His research interests include string theory, gravity, quantum gravity, and black-
hole physics.

CAMBRIDGEMONOGRAPHSONMATHEMATICALPHYSICS
General Editors: P. V. Landshoff, D. R. Nelson, S. Weinberg
S.J.AarsethGravitational N-BodySimulations:ToolsandAlgorithms†
J.Ambjørn,B.DurhuusandT.JonssonQuantumGeometry:AStatistical FieldTheoryApproach†
A.M.AnileRelativistic FluidsandMagneto-fluids:WithApplicationsinAstrophysics andPlasmaPhysics
J.A.deAzc´arragaandJ.M.IzquierdoLieGroups,LieAlgebras,CohomologyandSomeApplicationsinPhysics†
O.Babelon,D.BernardandM.TalonIntroduction toClassical IntegrableSystems†
F.BastianelliandP.vanNieuwenhuizenPathIntegralsandAnomaliesinCurvedSpace†
V.BelinskiandE.VerdaguerGravitational Solitons†
J.BernsteinKineticTheoryintheExpandingUniverse†
G.F.BertschandR.A.BrogliaOscillationsinFiniteQuantumSystems†
N.D.BirrellandP.C.W.DaviesQuantumFieldsinCurvedSpace†
K.Bolejko,A.Krasin´ski,C.HellabyandM.-N.C´el´erierStructuresintheUniversebyExactMethods:Formation,
Evolution,Interactions
D.M.BrinkSemi-Classical MethodsforNucleus-NucleusScattering†
M.BurgessClassical Covariant Fields†
E.A.CalzettaandB.-L.B.HuNonequilibriumQuantumFieldTheory
S.CarlipQuantumGravityin2+1Dimensions†
P.CartierandC.DeWitt-MoretteFunctionalIntegration: ActionandSymmetries†
J. C. Collins Renormalization: An Introduction to Renormalization, the Renormalization Group and the
Operator-Product Expansion†
P.D.B.CollinsAnIntroduction toReggeTheoryandHighEnergyPhysics†
M.CreutzQuarks,GluonsandLattices†
P.D.D’EathSupersymmetricQuantumCosmology†
J.Derezin´skiandC.G´erardMathematicsofQuantization andQuantumFields
F.deFeliceandD.BiniClassical MeasurementsinCurvedSpace-Times
F.deFeliceandC.J.S.ClarkeRelativityonCurvedManifolds†
B.DeWittSupermanifolds, 2nd edition†
P.G.O.FreundIntroduction toSupersymmetry†
F.G.FriedlanderTheWaveEquationonaCurvedSpace-Time†
J.L.FriedmanandN.StergioulasRotating Relativistic Stars
Y. Frishman and J. Sonnenschein Non-Perturbative Field Theory: From Two Dimensional Conformal Field
TheorytoQCDinFourDimensions
J.A.FuchsAffineLieAlgebrasandQuantumGroups:AnIntroduction,withApplicationsinConformalField
Theory†
J.FuchsandC.SchweigertSymmetries,LieAlgebrasandRepresentations:AGraduateCourseforPhysicists†
Y.FujiiandK.MaedaTheScalar-TensorTheoryofGravitation†
J.A.H.Futterman,F.A.HandlerandR.A.MatznerScatteringfromBlackHoles†
A.S.Galperin,E.A.Ivanov,V.I.OgievetskyandE.S.SokatchevHarmonicSuperspace†
R.GambiniandJ.PullinLoops, Knots,GaugeTheoriesandQuantumGravity†
T.GannonMoonshine beyondtheMonster:TheBridgeConnectingAlgebra,ModularFormsandPhysics†
M.G¨ockelerandT.Schu¨ckerDifferentialGeometry,GaugeTheories,andGravity†
C.G´omez,M.Ruiz-AltabaandG.SierraQuantumGroupsinTwo-Dimensional Physics†
M.B.Green,J.H.SchwarzandE.WittenSuperstringTheoryVolume1:Introduction
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 2: Loop Amplitudes, Anomalies and
Phenomenology
V.N.GribovTheTheoryofComplexAngularMomenta:GribovLecturesonTheoretical Physics†
J.B.GriffithsandJ.Podolsky´ExactSpace-TimesinEinstein’sGeneralRelativity†
S.W.HawkingandG.F.R.EllisTheLargeScaleStructureofSpace-Time†
F.IachelloandA.ArimaTheInteractingBoson Model†
F.IachelloandP.vanIsackerTheInteractingBoson-Fermion Model†
C.ItzyksonandJ.M.DrouffeStatistical Field Theory Volume 1: From Brownian Motion to Renormalization
andLattice GaugeTheory†
C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 2: Strong Coupling, Monte Carlo Methods,
ConformalFieldTheoryandRandomSystems†
C.V.JohnsonD-Branes†
P.S.JoshiGravitational CollapseandSpacetime Singularities†
J.I.KapustaandC.GaleFinite-Temperature FieldTheory:PrinciplesandApplications, 2nd edition†
V. E. Korepin, N. M. Bogoliubov and A. G. Izergin Quantum Inverse Scattering Method and Correlation
Functions†
M.LeBellacThermalFieldTheory†
Y.MakeenkoMethodsofContemporary GaugeTheory†
N.MantonandP.SutcliffeTopological Solitons†
N.H.MarchLiquidMetals:ConceptsandTheory†
I.MontvayandG.Mu¨nsterQuantumFieldsonaLattice†
L.O’RaifeartaighGroupStructureofGaugeTheories†
T.Ort´ınGravityandStrings,2nd edition
A.M.OzoriodeAlmeidaHamiltonian Systems:ChaosandQuantization†
L.ParkerandD.TomsQuantumFieldTheoryinCurvedSpacetime:QuantizedFieldsandGravity
R.PenroseandW.RindlerSpinorsandSpace-Time Volume1:Two-SpinorCalculusandRelativistic Fields†
R. Penrose and W. Rindler Spinors and Space-Time Volume 2: Spinor and Twistor Methods in Space-Time
Geometry†
S.PokorskiGaugeFieldTheories,2nd edition†
J.PolchinskiStringTheoryVolume1:AnIntroduction totheBosonic String†
J.PolchinskiStringTheoryVolume2:SuperstringTheoryandBeyond†
J.C.PolkinghorneModelsofHighEnergyProcesses†
V.N.PopovFunctionalIntegralsandCollectiveExcitations†
L.V.ProkhorovandS.V.ShabanovHamiltonian MechanicsofGaugeSystems
A.RecknagelandV.SchiomerusBoundaryConformalFieldTheoryandtheWorldsheetApproachtoD-Branes
R.J.RiversPathIntegralMethodsinQuantumFieldTheory†
R.G.RobertsTheStructureoftheProton:DeepInelasticScattering†
C.RovelliQuantumGravity†
W.C.SaslawGravitational PhysicsofStellarandGalacticSystems†
R.N.SenCausality,Measurement TheoryandtheDifferentiableStructureofSpace-Time
M.ShifmanandA.YungSupersymmetricSolitons
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field
Equations,2nd edition†
J.StewartAdvancedGeneralRelativity†
J.C.TaylorGaugeTheoriesofWeakInteractions†
T.ThiemannModernCanonical QuantumGeneralRelativity†
D.J.TomsTheSchwingerActionPrincipleandEffectiveAction†
A.VilenkinandE.P.S.ShellardCosmic StringsandOtherTopological Defects†
R.S.WardandR.O.Wells,JrTwistor GeometryandFieldTheory†
E.J.WeinbergClassicalSolutionsinQuantumFieldTheory:SolitonsandInstantonsinHighEnergyPhysics
J.R.WilsonandG.J.MathewsRelativistic NumericalHydrodynamics†
†Availableinpaperback
Gravity and Strings
TOMA´S ORT´IN
Spanish National Research Council
(CSIC)
University Printing House, Cambridge CB28BS, United Kingdom
CambridgeUniversity Press ispart of theUniversity of Cambridge.
It furthers theUniversity’smission by disseminating knowledge inthe pursuit of
education, learning and research at the highest international levels ofexcellence.
www.cambridge.org
Informationon thistitle: www.cambridge.org/9780521768139
Second edition© T. Ort´ın2015
This publication isincopyright. Subject to statutoryexception
and to the provisionsof relevant collectivelicensing agreements,
no reproduction of any part may takeplace without the written
permissionof Cambridge University Press.
First published 2004
Second edition2015
Printedin theUnited Kingdomby CPI Group Ltd, Croydon CR0 4YY
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
ISBN 978-0-521-76813-9 Hardback
Cambridge UniversityPress has no responsibility for the persistence oraccuracy of
URLs for external orthird-party internet websites referred to inthis publication,
anddoes not guarantee that any content onsuch websites is, or willremain,
accurateor appropriate.
To Marimar, Diego, and Toma´s, the
sweet strings that tie me to the real
world

Contents
Prefacetothesecondedition pagexxi
Prefacetothefirstedition xxv
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 Einstein–WeylspacetimeEW 14
d
1.3.3 RiemannspacetimeV 14
d
1.4 Tangentspace 16
1.4.1 Weitzenbo¨ckspacetimeA 20
d
1.5 Killingvectors 22
1.6 Dualityoperations 23
1.7 Differentialformsandintegration 25
1.8 Extrinsicgeometry 27
2 SymmetriesandNoether’stheorems 29
2.1 Equationsofmotion 29
2.2 Noether’stheorems 30
2.3 Conservedcharges 34
2.4 Thespecial-relativisticenergy–momentumtensor 35
2.4.1 Conservationofangularmomentum 36
2.4.2 Dilatations 40
2.4.3 Rosenfeld’senergy–momentumtensor 42
ix