Gravity, a Geometrical Course Pietro Giuseppe Frè Gravity, a Geometrical Course Volume 2: Black Holes, Cosmology and Introduction to Supergravity PietroGiuseppeFrè DipartimentodiFisicaTeorica UniversityofTorino Torino,Italy Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Thisbookisdedicatedtomybeloved daughterLaura andtomydarlingwifeOlga. Preface This book grew out from the Lecture Notes of the course in General Relativity whichIgaveformorethan15yearsattheUniversityofTorino.Thatcoursehasa long tradition since it was attached to the Chair of Relativity created at the begin- ningofthe1960sforprof.TullioRegge.Intheyears1990–1996,whileprof.Regge was Member of the European Parliament the course was given by my long time excellent friend and collaborator prof. Riccardo D’Auria. In 1996 I had the honor tobeappointedonRegge’schair1 andIleftSISSAofTriestetotakethismomen- tousandchallenginglegacy.FeelingtheburdenofthetasklaidonmyshouldersI humblytriedtodomybestandcreateanewcoursewhichmightkeepalivethetra- ditionestablishedbymysomuchdistinguishedpredecessors.Inmyeffortstocope with the expected standards, I obviously introduced my own choices, view-points and opinions that are widely reflected in the present book. The length of the orig- inal course was of about 120 hours (without exercises). In the new 3+2 system introduced by the Bologna agreements it was split in two courses but, apart from minorreadjustments,Icontinuedtoconsiderthemjustaspartoneandparttwoof auniqueentity.Thiswasnotarandomchoicebutitsprangfromtheviewsthatin- spiredmyteachingandthepresentbook.IalwaysheldtheopinionthatUniversity coursesshouldbelong,complexandarticulatedinmanyaspects.Theyshouldnot aimataquicktransmissionofcalculatingabilitiesandofreadytouseinformation, rather they should be as much formative as informative. They should offer a gen- eraloverviewofasubjectasseenbytheprofessor,inthiswaygivingthestudents theopportunityofdevelopingtheirownopinionsthroughthecriticalabsorptionof thoseoftheteacher. One aspect that I always considered essential is the historical one, concerning ononesidethefacts,thelifeandthepersonalitiesofthescientistswhoshapedour presentunderstanding,ontheotherhandconcerningtheusuallyintricatedevelop- mentoffundamentalideas. ThesecondaspecttowhichIpaidalotofattentionistheuseofanupdatedand as much as possible rigorous mathematical formalism. Moreover I always tried to 1AtthattimeReggehadshiftedfromtheUniversitytothePolitecnicoofTorino. vii viii Preface convey the view that Mathematics should not be regarded as a technical tool for the solution of Physical Problems or simply as a language for the formulation of PhysicalLaws,ratherasanessentialintegralpartofthewholegame. The third aspect taken not only into account but also into prominence, is the emphasisonimportantphysicalapplicationsofthetheory:notjustexercises,from whichIcompletelyrefrained,butthefull-fledgedabinitiodevelopmentofrelevant applications in Astrophysics, Cosmology or Particle Theory. The aim was that of showing,fromAtoZ,asonegoesfromthefirstprinciplestotheactualprediction of experimentally verifiable numbers. For the reader’s or student’s convenience I includedthelistingofsomecomputercodes,writteninMATHEMATICA,thatsolve some of the posed problems or parts thereof. The aim was, once again formative. In the course of their theoretical studies the students should develop the ability to implementformalcalculationsonamachine,freeingthemselvesfromtheslaveryto accidentalerrorsandfocusinginsteadalltheirmentalenergiesonconceptualpoints. Furthermoreimplementationofformulaeinacomputercodeistherealtestoftheir comprehensionbythelearners,moreefficientinitstaskthananyad-hocprepared exercise. As for the actual choice of the included and developedmaterial, I was inspired bythefollowingviewontheroleofthecourseIusedtogave,whichIextendedasa missiontothepresentbook.GeneralRelativity,QuantumMechanics,GaugeTheo- riesandStatisticalMechanicsarethefourpillarsofthePhysicalThoughtdeveloped intheXXthcentury.Thatcenturylaidalsothefoundationsfornewtheories,whose actualrelations with the experimentaltruth andwith observations will be clarified onlyinthepresentmillennium,butwhoseprofoundinfluenceonthecurrentthought issoprofoundthatno-oneapproachingtheoreticalstudiescanignorethem:Irefer tosupersymmetry,supergravity,stringsandbranes.TheroleofthecourseinGen- eral Relativity which I assumed as given, was not only that of presenting Einstein Theory, in its formulation, historical development and applications, but also that of comparing the special structure of Gravity in relation with the structure of the Gauge-Theories describing the other fundamental interactions. This was specially aimedatthedevelopmentofcriticalthinkinginthestudentandasatoolofforma- tiveeducation,preparatorytothestudyofunifiedtheories. ThepresentoneisaGraduateTextBookbutitisalsomeanttobeaself-contained accountofGravitationalTheoryattractiveforthepersonwithabasicscientificed- ucationandacuriosityforthetopicwhowouldliketolearnitfromscratch,being his/herowninstructor. Just as the original course given in Torino after the implementation of the Bolognaagreements,thisbookisdividedintwovolumes: 1. Volume1:DevelopmentoftheTheoryandBasicPhysicalApplications. 2. Volume2:BlackHoles,CosmologyandIntroductiontoSupergravity. Volume1,startingfromasummaryofSpecialRelativityandasketchyhistorical introductionofitsbirth,giveninChap.1,developsthegeneralcurrentdescription ofthephysicalworldintermsofgaugeconnectionsandsectionsofthebundleson Preface ix which such connections are constructed. The special role of Gravity as the gauge theory of the tangent bundle to the base manifold of all other bundles is empha- sized.ThemathematicalfoundationsofthetheoryarecontainedinChaps.2and3. Chapter 2 introduces the basic notions of differential geometry, the definition of manifoldsandfibre-bundles,differentialforms,vectorfields,homologyandcoho- mology.Chapter3introducesthetheoryofconnectionsandmetrics.Itincludesan extensivehistoricalaccountofthedevelopmentofmathematicalandphysicalideas which eventually lead to both general relativity and modern gauge theories of the non-gravitational interactions. The notion of geodesics is introduced and exempli- fied with the detailed presentation of a pair of examples in two dimensions, one withLorentziansignature,theotherwithEuclidiansignature.Chapter4isdevoted totheSchwarzschildmetric.ItisshownhowgeodesicsoftheSchwarzschildmetric retrievethewholebuildingofNewtonianPhysicspluscorrectionsthatcanbevery tinyinweakgravitationalfields,likethatoftheSolarSystem,orgiganticinstrong fields, where they lead to qualitatively new physics. The classical tests of General Relativity are hereby discussed, perihelion advance and the bending of light rays, inparticular.Chapter5introducestheCartanapproachtodifferentialgeometry,the vielbeinandthespinconnection,discussesBianchiidentitiesandtheirrelationwith gaugeinvariancesandeventuallyintroducesEinsteinfieldequations.Thedynamical equationsofgravityandtheirderivationfromanactionprinciplearedevelopedina parallelwaytotheiranaloguesforelectrodynamicsandnon-Abeliangaugetheories whose structure and features are constantly compared to those of gravity. The lin- earizationofEinsteinfieldequationsandthespinofthegravitonarethendiscussed. Afterthatthebottom-upapproachtogravityisdiscussed,namely,followingFeyn- man’sideas,itisshownhowaspecialrelativisticlineartheoryofthegravitonfield couldbeuniquelyinferredfromtheconservationofthestress-energytensorandits non-linearupgradingfollows,oncethestress-energytensorofthegravitationalfield itself is taken into account. The last section of Chap. 5 contains the derivation of theSchwarzschildmetricfromEinsteinequations.Chapter6addressestheissueof stellarequilibriuminGeneralRelativity,derivestheTolmanOppenheimerVolkhoff equationandthecorrespondingmasslimits.Next,consideringtheroleofquantum mechanicstheChandrasekharmasslimitsforwhitedwarfsandneutronstarsarede- rived.Chapter7isdevotedtotheemissionofgravitationalwavesandtothetestsof GeneralRelativitybasedontheslowingdownoftheperiodofdoublestarsystems. Volume2,afterashortintroductorychapter,thefollowingtwochaptersarede- voted to Black Holes. In Chap. 2 we begin with a historical account of the notion ofblackholesfromLaplacetothepresentidentificationofstellarmassblackholes inthegalaxyandelsewhere.NexttheKruskalextensionoftheSchwarzschildsolu- tionisconsideredinfulldetailprecededbythepedagogicaltoyexampleofRindler space-time. Basic concepts about Future, Past and Causality are introduced next. Conformal Mappings, the Causal Structure of infinity and Penrose diagrams are discussedandexemplified. Chapter3dealswithrotatingblack-holesandtheKerr-Newmanmetric.Theusu- ally skippedform of the spin connectionand of the Riemanntensor of this metric is calculated and presented in full detail, together with the electric and magnetic x Preface field strengths associated with it in the case of a charged hole. This is followed by a careful discussion of the static limit, of locally non-rotating observers, of the horizon and of the ergosphere. In a subsequent section the geodesics of the Kerr metric are studied by using the Hamilton Jacobi method and the system is shown tobeLiouvilleintegrablewiththederivationofthefourthHamiltonian(theCarter constant)completingtheneededshelloffour,togetherwiththeenergy,theangular momentumandthemass.Thelastsectioncontainsadiscussionoftheanalogybe- tween the Laws of Thermodynamics and those of Black Hole dynamics including theBekenstein-Hawkingentropyinterpretationofthehorizonarea. Chapters4and5aredevotedtocosmology.Chapter4containsahistoricalout- lineofmodernCosmologystartingfromKant’sproposalthatnebulaemightbedif- ferentisland-universes(galaxiesinmodernparlance)tothecurrentspacemissions that have measured the Cosmic Microwave Background anisotropies. The crucial historical steps in building up the modern vision of a huge expanding Universe, which is even accelerating at the present moment, are traced back in some detail. From the Olbers paradox to the discovery of the stellar parallax by Bessel, to the GreatDebateof1920betweenCurtisandShapley,howthehumanestimationofthe Universe’s size enlarged, is historically reported. The discovery of the Cepheides law by Henrietta Leavitt, the first determination of the distance to nearby galax- iesbyHubbleandfinallythefirstmeasuringoftheuniversalcosmicrecessionare the next episodes of this tale. The discovery of the CMB radiation, predicted by Gamow,thehuntforitsanisotropiesandtherecentadventoftheInflationaryUni- verseparadigmarethesubsequentlandmarks,whicharereportedtogetherwithbi- ographical touches upon the life and personalities of the principal actors in this excitingadventureofthehumanthought. Chapter 5, entitled Cosmology and General Relativity: Mathematical Descrip- tionoftheUniverse,providesafull-fledgedintroductiontoRelativisticCosmology. The chapter begins with a long mathematical interlude on the geometry of coset manifolds. These notions are necessary for the mathematical formulation of the CosmologicalPrinciple, stating homogeneityand isotropy, but havea much wider spectrumofapplications.Inparticulartheywillbeveryimportantinthesubsequent chaptersaboutSupergravity.Havingpreparedthestagewiththismathematicalpre- liminaries,thenextsectionsdealwithhomogeneousbutnotisotropiccosmologies. Bianchi classification of three dimensional Lie groups is recalled, Bianchi metrics aredefinedand,withinBianchitypeI,theKasnermetricsarediscussedwithsome glimpsesaboutthecosmicbilliards,realizedinSupergravity.Next,asapedagogi- calexampleofahomogeneousbutnotisotropiccosmology,anexactsolution,with andwithoutmatter,ofBianchitypeIIspace-timeistreatedinfulldetail.Afterthis, weproceedtotheStandardCosmologicalModel,includingbothhomogeneityand isotropy. Freedman equations, all their implications and known solutions are dis- cussedindetailandaspecialattentionisgiventotheembeddingofthethreetype ofstandardcosmologies(open,flatandclosed)intodeSitterspace.Theconceptof particle and event horizons is next discussed together with the derivation of exact formulaeforread-shiftdistances.Theconceptualproblems(horizonandflatness)of theStandardCosmologicalModelarenextdiscussedasanintroductiontothenew Preface xi inflationary paradigm. The basic inflationary model based on one scalar field and the slow rolling regime are addressed in the following sections with fully detailed calculations.Perturbations,thespectrumoffluctuationsuptotheevaluationofthe spectral index and the principles of comparison with the CMB data form the last partofthisverylongchapter. The last four chapters of the book provide a conceptual, mathematical and de- scriptiveintroductiontoSupergravity,namelytotheBeyondGRWorld. Chapter6startswithahistoricaloutlinethatdescribesthebirthofsupersymme- trybothinStringTheoryandinFieldTheory,touchingalsoonthebiographiesand personalitiesofthetheoristswhocontributedtocreatethisentirenewfieldthrough acomplicatedand,asusual,farfromstraight,path.Thechapterproceedsthanwith the conceptual foundations of Supergravity, in particular with the notion of Free DifferentialAlgebrasandwiththeprincipleofrheonomy.Sullivan’sstructuralthe- oremsarediscussedanditisemphasizedhowtheexistenceofp-forms,thatclose the supermultiplets of fundamental fields appearing in higher dimensional super- gravities,isattheendofthedayaconsequenceofthesuperPoincaréLiealgebras throughtheircohomologies.ThestructureofM-theory,theconstructiveprinciples tobuildsupergravityLagrangiansandthefundamentalroleofBianchiidentitiesis emphasized.Thelasttwosectionsofthechaptercontainathoroughaccountoftype IIA and type IIB supergravities in D=10, the structure of their FDAs, the rheo- nomic parameterization of their curvatures and the full-fledged form of their field equations. Chapter7dealswiththebrane/bulkdualism.Thefirstsectioncontainsaconcep- tualoutlinewherethethreesidedviewofbranesas1)classicalsolitonicsolutionsof thebulktheory,2)worldvolumegauge-theoriesdescribedbysuitableworld-volume actionsendowedwithκ-supersymmetryand3)boundarystatesinthesuperconfor- mal field theory description of superstring vacua is spelled out. Next a New First OrderFormalism,inventedbytheauthorofthisbookatthebeginningoftheXXIst centuryandallowingforanelegantandcompactconstructionofκ-supersymmetric Born-Infeldtypeworld-volumeactionsonarbitrarysupergravitybackgroundsisde- scribed.ItissubsequentlyappliedtothecaseoftheD3-brane,bothasanillustration andfortheitsintrinsicrelevanceinthegauge/gravitycorrespondence.Thelastsec- tionsofthechapteraredevotedtothepresentationsofbranesasclassicalsolitonic solutionsofthebulktheory.Generalfeaturesofthesolutionsintermsofharmonic functions are presented including also a short review of domain walls and some sketchydescriptionoftheRandall-Sundrunmechanism. Chapter 8 is a bestiary of Supergravity Special Geometries associated with its scalarsector.Thechapterclarifiesthecodifyingroleofthescalargeometryincon- structingthebosonicpartofasupergravityLagrangian.Thedominantroleamong the scalar manifolds of homogeneous symmetric spaces is emphasized illustrating theprinciplesthatallowthedeterminationofsuchU/Hcosetsforanysupergravity theory.Themechanismofsymplecticembeddingthatallowstoextendtheactionof U-isometriesfromthescalartothevectorfieldsectorareexplainedindetailwithin thegeneraltheoryofelectric/magneticdualityrotations.Nextthechapterprovides a self-contained summary of the most important special geometries appearing in