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

PaperIII_54 - MATHEMATICAL TRIPOS Thursday 4 June 2015 Part...

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MATHEMATICAL TRIPOS Part III Thursday, 4 June, 2015 9:00 am to 12:00 pm PAPER 54 BLACKHOLES 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.
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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 byρcρ(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.
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