Introduction to relativistic astrophysics and cosm

arXiv:gr-qc/0103023v18Mar2001IntroductiontorelativisticastrophysicsandcosmologythroughMapleVladimirL.Kalash[e-mail]field,blackholesandcosmologicalmodelsareillus-tratedandanalyzedbymeansofMaple6ApplicationAreas/Subjects:Science,Astrophysics,GeneralRelativity,Ten-sorAnalysis,Differentialgeometry,Differentialequations1Contents1Introduction32Relativisticcelestialmechanicsinweakgravitationalfield32.1Introduction............................32.2Schwarzschildmetric.......................32.3Equationsofmotion.......................62.4Lightraydeflection........................102.5Planet’sperihelionmotion....................132.6Conclusion............................223Relativisticstarsandblackholes223.1Introduction............................223.2Geometricunits..........................233.3Relativisticstar..........................233.4Degeneracystarsandgravitationalcollapse..........403.5Schwarzschildblackhole.....................473.6Reissner-Nordstr¨omblackhole(chargedblackhole)......553.7Kerrblackhole(rotatingblackhole)..............633.8Conclusion............................684Cosmologicalmodels684.1Introduction............................684.2Robertson-Walkermetric....................694.3Standardmodels.........................804.3.1Einsteinstatic......................864.3.2Einstein-deSitter.....................874.3.3deSitterandanti-deSitter...............884.3.4ClosedFriedmann-Lemaitre...............904.3.5OpenFriedmann-Lemaitre................934.3.6Expandingsphericalandrecollapsinghyperbolicaluni-verses...........................964.3.7Bouncingmodel.....................984.3.8Ouruniverse(?).....................994.4Beginning.............................1004.4.1BianchimodelsandMixmasteruniverse........1004.4.2Inflation..........................1175Conclusion12321IntroductionArapidprogressoftheobservationalastrophysics,whichresultedfromtheactiveuseoforbitaltelescopes,essentiallyintensifiestheastrophysicalre-searchesatthelastdecadeandallowstochoosethemoredefinitedirec-tionsoffurtherinvestigations.Atthesametime,thedevelopmentofhigh-performancecomputersadvancesinthenumericalastrophysicsandcosmol-ogy.Againstabackgroundoftheseachievements,thereistherenascenceofanalyticalandsemi-analyticalapproaches,whichisinducedbynewgenera-tionofhigh-efficientcomputeralgebrasystems.Herewepresentthepedagogicalintroductiontorelativisticastrophysicsandcosmology,whichisbasedoncomputationalandgraphicalresourcesofMaple6.ThepedagogicalaimsdefinetheuseonlystandardfunctionsdespitethefactthattherearethepowerfulGeneralRelativity(GR)orientedextensionslikeGRTensor[1].TheknowledgeofbasicsofGRanddifferentialgeometryissupposed.Itshouldbenoted,thatourchoiceofmetricsignature(+2)governsthedefinitionsofLagrangiansandenergy-momentumtensors.Thecomputationsinthisworksheettakeaboutof6minofCPUtime(PIII-500)and9Mbofmemory.2Relativisticcelestialmechanicsinweakgravita-tionalfield2.1IntroductionThefirstresultsintheGR-theorywereobtainedwithoutexactknowledgeofthefieldequations(Einstein’sequationsforspace-timegeometry).Theleadingideawastheequivalenceprinciplebasedontheequalityofiner-tialandgravitationalmasses.WewilldemonstrateherethatthenaturalconsequencesofthisprinciplearetheSchwarzschildmetricandthebasicexperimentaleffectsofGR-theoryinweakgravitationalfield,i.e.planet’sorbitprecessionandlightraydeflection(see[2]).2.2SchwarzschildmetricLetusconsiderthecentrallysymmetricgravitationalfield,whichisproducedbymassM.ThesmallcellK∞fallsalongx-axisonthecentralmass.Intheagreementwiththeequivalenceprinciple,theuniformlyacceleratedmotion3locallycompensatesthegravitationalforcehencethereisnoagravitationalfieldinthefreefallingsystemK∞.ThisresultsinthelocallyLorenzianmetricwithlinearelement(cisthevelocityoflight):ds2=dx∞2+dy∞2+dz∞2-c2dt∞2ThevelocityvandradialcoordinateraremeasuredinthesphericalsystemK,whichareconnectedwithcentralmass.Itisnatural,theobserverinthismotionlesssystem”feels”thegravitationalfield.Sincethefirstsys-temmovesrelativelysecondonetherearethefollowingrelationsbetweencoordinates:dx∞=dr√1−β2(β=vc)dt∞=p1−β2dtdy∞=rdθdz∞=rsin(θ)dφThefirstandsecondrelationsaretheLorentzianlengthshorteningandtimeslowingdowninthemovingsystem.Asresult,K∞fromKlooksas:ds2=(1−β2)(−1)dr2+r2(dθ2+sin(θ)2dφ2)-c2(1-β2)dt2Thesenseoftheadditionaltermsinmetrichastoconnectwiththechar-acteristicsofgravitationalfield.WhatistheenergyofK∞inK?IfthemassofK∞ism,andm0istherestmass,thesumofkineticandpotentialenergiesis:4restart:with(plots):(m-m0)*c^2-G*M*m/r=0;#energyconservationlaw\(wesupposethattheNewtonianlawofgravitation\iscorrectinthefirstapproximation),Gis\thegravitationalconstant%/(m*c^2):subs(m=m0/sqrt(1-beta^2),%):#relativisticmassexpand(%);solve(%,sqrt(1-beta^2)):sqrt(1-beta^2)=expand(%);1-beta^2=taylor((1-subs(op(2,%)=alpha/r,rhs(%)))^2,alpha=0,2);\#alpha=G*M/c^2,weusethefirst-orderapproximation\onalpha(m−m0)c2−GMmr=01−q1