Deformed Spheres in General Relativity and Beyond

Abstract

Currently,Einstein'sgeneraltheoryofrelativity(GR)providesthebestdescriptionfor the phenomenacalledgravity.Butitisnottheonlytheorythatdoesthejob.Thereis the versiongivenbyNewtonalso.Thisversiondescribesgravityastheforcebetweenthe objects.Suchaforcedependsonmassesoftheobjectsinvolvedandalsoonthedistance from eachother.InGR,gravityisnotaforce.Itisthecurvatureofthespacetime resulting duetothepresenceofthematter.Thegravitational eldasdescribedby GR isamanifestationthatspaceisthecurvedRiemannianoneinsteadofthe at Minkowski.Thegravitational eldgetsgeometrizedinGR,whichisatensortheoryof the gravitational eldinsteadofscalarone(Newtoniantheoryisascalartheory).The gravitational eldisrepresentedintermsofthemetrictensoroftheRiemannianspace, its sourcebeingthemattertensor.Thecomponentsofthemattertensorsourcethe gravitational eldinanelegantwaydeterminedbytheEinstein ledequations(EFEs). Continuousprogressisbeingmadein ndingthesolutionsoftheEFEs.Schwarzschild, Reissner-Nordstr om,KerrandKerr-Newmanspacetimesarethesimplestvacuumsolu- tions ofEFEsthatdescribetheblackholes.Reissner-Nordstr omisthechargedgener- alization oftheSchwarzschildsolution,bothbeingsphericallysymmetric.Kerrmetric is therotatinggeneralizationoftheSchwarzschildmetric.Introductionofthecharge in theKerrmetricgivestheKerr-Newmanmetric.KerrandKerr-Newmanspacetimes are axiallysymmetric.Inthelimitofmassbeingvanished,theyreducetoMinkowski metric inspheroidalcoordinates.Spheroidsarethegeometricobjectswhichwecantake as deformedspheres. Apart fromtheresearchandinterestinGR,therehasbeenagrowinginterestinalternate theories ofgravity.OnesuchtheoryistheChern-Simons(CS)theory.Theactionof this theoryconsistsoftheusualEinstein-Hilberttermandanewparityviolatingfour- dimensional correction.TwokindsofformulationsexistinCStheory,namelydynamical and non-dynamical.Blackholesolutionshavebeendevelopedinboththecases.Our interestasregardstothisthesisisthespacetimewhichhasbeendevelopedintheformer formulation. The solutionsbeyondGRcanalsobeformulatedbyanothermethod.Suchmethodin- volvesthemodelindependentparameterizationofthemetric.Themetricthusobtained mustdescribetheblackholesolutioninanytheoryofgravity.Thepossibledeviations from theKerrspacetimearemeasuredbythedeviationparameters. The detailedoutlineofthethesisisasfollows:Chapter1isaboutthepreliminaries.In Chapter 2,theMisner-Sharpmassisgeneralizedforthespheroidalgeometry.Misner- Sharp massisatypeofquasilocalmassthatpreviouslyworkedonlyinthespherically symmetric spacetimes.Italsogivesthelocationofthemarginallyoutertrappedsurface in suchspacetimes.TheMisner-SharpmassisextendedforspheroidswithinGRand iii the locationofmarginallyoutertrappedsurfaceisdeterminedinthisnewsetting.The parameter whichgivesdeviationfromsphericalgeometryiskeptsmallthroughoutthe analysis. Inquantumphysics,theenergydensitywhichde nestheMisner-Sharpmass (and ADMmass,namedafterRichardArnowitt,StanleyDeserandCharlesMisner) becomesaquantumobservableandonecouldconjecturethatthegravitationalradius admits asimilardescription.Thegravitationalradiusismadeaquantummechanical operatorwhichactsonthe\horizonwavefunction".Thehorizonwavefunctionisgiven bythequantumstateofthesource.Thehorizonquantummechanicshasbeenextended to thecaseofspheroidalsourcesattheendofthechapter. The nexttwochaptersdealwiththespacetimesinthealternatetheoriesofgravity. Chapter 3involvesspacetimeindynamicalCStheory.Thisspacetimeisvalidinslow rotation approximationandsmallcouplingconstant.Thee ectsoftheCScoupling constantonsomephysicalphenomenae.g.quasilocalmass,particlemotionandenergy extraction processarestudied. Johannsen andPsaltisdevelopedarotatingdeformedKerr-likemetricinanalternate theory ofgravityotherthanGR.ItisobtainedbyapplyingNewman-Janisalgorithmto a deformedSchwarzschildmetric.Motivatedbythisspacetime,achargedanalogueof the Johannsen-PsaltismetricisdevelopedinChapter4.Heretheseedmetricistaken as theReissner-Nordstr omspacetime.Thenewmetricisstudiedfortheeventand Killing horizons,thelatterarealsorepresentedgraphically.Lorentzviolatingregions are analyzedbythedeterminantofthechargedversionoftheJohannsen-Psaltismetric. Analysis oftheclosedtime-likecurvesarealsoincludedinthischapter.Considering the motionofaparticleontheequatorialplane,weobtainitsenergyandangular momentum.Locationofthecircularphotonorbitsandinnermoststablecircularorbits are alsodetermined. The Chapter5containsthesummaryandconclusionofthethesis.