Gravitational Waves and Black Holes An Introductio

NIKHEF/97-017HD-THEP-97-6GravitationalWavesandBlackHolesAnIntroductiontoGeneralRelativityJ.W.vanHoltenNIKHEF,P.O.Box418821009DBAmsterdamNLAbstractIntheselecturesgeneralrelativityisoutlinedastheclassicaleldtheoryofgravity,emphasizingphysicalphenomenaratherthanformalism.Dynamicalsolutionsrepresentingtravelingwavesaswellasstationaryeldslikethoseofblackholesarediscussed.Theirpropertiesareinvestigatedbystudyingthegeodesicstructureofthecorrespondingspace-times,asrepresentingthemotionofpoint-liketestparticles.Theinteractionbetweengravitational,electro-magneticandscalareldsisalsoconsidered.c1997LecturespresentedattheUniversityofHeidelberg,february1997Contents1GravityandGeometry11.1Thegravitationalforce.............................11.2Fields......................................21.3Geometricalinterpretationofgravity.....................31.4Curvature....................................51.5TheEinsteinequations.............................71.6Theactionprinciple..............................92Geodesics112.1Curvesandgeodesics..............................112.2Canonicalformulation.............................152.3Actionprinciples................................172.4SymmetriesandKillingvectors........................182.5Phase-spacesymmetriesandconservationlaws...............222.6Example:therigidrotor............................243Dynamicsofspace-time273.1Classicalsolutionsofthegravitationaleldequations............273.2Planefrontedwaves..............................303.3Natureofthespace-time............................363.4Scatteringoftestparticles...........................393.5Generalplanarwaves..............................413.6Thegravitationaleldofalightwave.....................443.7Symmetrybreakingasasourceofgravitationaloscillations........504Blackholes574.1Horizons....................................574.2TheSchwarzschildsolution..........................584.3Discussion...................................614.4TheinterioroftheSchwarzschildsphere...................634.5Geodesics....................................664.6ExtendedSchwarzschildgeometry......................704.7Chargedblackholes..............................74iii4.8Spinningblackholes..............................774.9TheKerrsingularity..............................834.10Black-holesandthermodynamics.......................85Chapter1GravityandGeometry1.1ThegravitationalforceGravityisthemostuniversalforceinnature.Asfaraswecantellfromobservationsandexperimentseveryobject,everyparticleintheuniverseattractsanyotheronebyaforceproportionaltoitsmass.Forslowmovingbodiesatlargedistancesthisisacentralforce,inverselyproportionaltothesquareofthedistance.Astheactionisreciprocal,andsinceaccordingtoNewtonactionandreactionforcesareequalinmagnitude,theexpressionforthegravitationalforcebetweentwoobjectsofmassM1andM2atadistanceRisthendeterminedtohavetheuniqueformF=GM1M2R2:(1.1)Theconstantofproportionality,Newton’sconstantofgravity,hasdimensionsofaccelera-tionperunitofmasstimesanarea.Thereforeitsnumericalvalueobviouslydependsonthechoiceofunits.IntheMKSsystemthisisG=6:67259(85)1011m3kg1s2:(1.2)Itisalsopossible,andsometimesconvenient,toxtheunitofmassinsuchawaythatNewtonsconstanthasthenumericalvalueG=1.Inthenaturalsystemofunits,inwhichalsothevelocityoflightandPlanck’sconstantareunity:c=h=1,thisunitofmassisthePlanckmassmP:mP=qhc=G=2:17671108kg=1:2210471019GeV=c2:(1.3)Newton’slawofgravity(1.1)isvalidforanytwomassivebodies,aslongastheyarefarapartanddonotmovetoofastwithrespecttooneanother.Inparticular,itdescribesthemotionsofcelestialbodieslikethemooncirclingtheearth,ortheplanetsorbitingthesun,aswellasthoseofterrestrialobjectslikeapplesfallingfromatree,orcanonballs12infreeight.EversinceNewtonthisunicationofcelestialandterrestialmechanicshascontinuedtoimpresspeopleandhashadatremendousimpactonourviewoftheuniverse.Itistheoriginandbasisforthebeliefinthegeneralvalidityofphysicallawsindependentoftimeandplace.1.2FieldsAlthoughasaforcegravityisuniversal,Newton’slaw(1.1)itselfhasonlylimitedvalidity.LikeCoulomb’slawfortheelectrostaticforcebetweentwoxedcharges,Newton’slawholdsstrictlyspeakingonlyforstaticforcesbetweenbodiesatrest.Moreover,andunliketheelectricforces,therearemodicationsatsmaller,nitedistanceswhichcanbeobservedexperimentally.Forexample,ifthegravitationalforcewouldhaveapure1=R2dependence,theorbitsofparticlesaroundaveryheavycentralbodywouldbeconicsections:ellipses,parabola’sorhyperbola’s,dependingontheenergyandangularmomentum,inaccordancewithKepler’slaws.TheobservationofanexcessintheprecessionoftheperihelionoftheorbitofMercuryaroundthesunbyLeVerrierin1845,andimprovedbyNewcombin1882[1],wasoneoftherstclearindicationsthatthisisactuallynotthecase,andthatthegravitationalforceismorecomplicated.Theexactformofthegravitationalforcesexertedbymovingbodiesisaproblemwithmanysimilaritiestotheanalogousprobleminelectrodynamics.Theunderstandingofelectrodynamicalphenomenagreatlyimprovedwiththeintroductionoftheconceptoflocaleldofforce.Thisconceptreferstothefollowingcharacteristicsofe