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.
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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= 4πr2ρdrdΦr(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 .
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