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

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.
2
1
A spacetime containing a static, spherically symmetric, star has line element
( )−1
ds2 = −e2Φ(r)dt2 +
1 − 2m(r) dr2 + r2dΩ2
r
The matter inside the star is described by a perfect fluid with energy momentum
tensor
Tab = (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
= 4πr2ρ
dr
dΦ
r(r − 2m)
dp
r(r − 2m)
(a)(i) Let R denote the radius of the star, so p, ρ vanish for r > R. Show that
the metric
outside 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 . Explain
why there is a maximum possible mass for the star that is independent of the
equation of
state for ρ > ρ0 .
.
form a 1-