MATHEMATICAL TRIPOSPart IIIThursday, 4 June, 20159:00 am to 12:00 pmPAPER 54BLACKHOLESAttempt no more thanTHREEquestions.There areFOURquestions in total.The questions carry equal weight.STATIONERY REQUIREMENTSSPECIAL REQUIREMENTSCover sheetNoneTreasury TagScript paperYou may not start to read the questionsprinted on the subsequent pages untilinstructed to do so by the Invigilator.
21A spacetime containing a static, spherically symmetric, star has line elementds2=-e2Φ(r)dt2+parenleftbigg1-2m(r)rparenrightbigg-1dr2+r2dΩ2.The matter inside the star is described by a perfect fluid with energy momentum tensorTab= (ρ+p)uaub+pgaband barotropic equation of statep=p(ρ) withρ, pgreaterorequalslant0,dp/dρ >0.The Einstein equation reduces to the Tolman-Oppenheimer-Volkov (TOV) equations:dmdr= 4πr2ρdΦdr=m+ 4πr3pr(r-2m)dpdr=-(p+ρ)(m+ 4πr3p)r(r-2m)(a)(i) LetRdenote the radius of the star, sop, ρvanish forr > R. Show that the metricoutside the star is the Schwarzschild metric.(ii) Explain why smooth solutions of the TOV equations form a 1-parameter family,labelled uniquely byρc≡ρ(0).(iii) Assume that the equation of state is known forρlessorequalslantρ0but not forρ > ρ0. Explainwhy there is a maximum possible mass for the star that is independent of the equation ofstate forρ > ρ0.