PaperIII_54 - MATHEMATICAL TRIPOS Part III Thursday 4 June 2015 9:00 am to 12:00 pm PAPER 54 BLACK HOLES Attempt no more than THREE questions There are

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MATHEMATICAL TRIPOS Part III Thursday, 4 June, 2015 9:00 am to 12:00 pm PAPER 54 BLACK HOLES Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 21A spacetime containing a static, spherically symmetric, star has line element( )−1ds2 = −e2Φ(r)dt2 + 1 − 2m(r) dr2 + r2dΩ2 rThe matter inside the star is described by a perfect fluid with energy momentum tensorTab = (ρ+p)uaub +pgab and barotropic equation of state p = p(ρ) with ρ, p ≥ 0, dp/dρ > 0.The Einstein equation reduces to the Tolman-Oppenheimer-Volkov (TOV) equations:dm= 4πr2ρdrr(r − 2m)dpr(r − 2m)(a)(i) Let R denote the radius of the star, so p, ρ vanish for r > R. Show that the metricoutside the star is the Schwarzschild metric.(ii) Explain why smooth solutions of the TOV equations parameter family,labelled uniquely by ρc ≡ ρ(0).(iii) Assume that the equation of state is known for ρ ≤ ρ0 but not for ρ > ρ0 . Explainwhy there is a maximum possible mass for the star that is independent of the equation ofstate for ρ > ρ0 . . form a 1-