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

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.

2
1
A spacetime containing a static, spherically symmetric, star has line element
ds
2
= - e
2Φ( r )
dt
2
+
parenleftbigg
1 -
2 m ( r )
r
parenrightbigg
- 1
dr
2
+ r
2
d Ω
2
.
The matter inside the star is described by a perfect fluid with energy momentum tensor
T
ab
= ( ρ + p ) u
a
u
b
+ pg
ab
and barotropic equation of state p = p ( ρ ) with ρ, p greaterorequalslant 0, dp/dρ > 0.
The Einstein equation reduces to the Tolman-Oppenheimer-Volkov (TOV) equations:
dm
dr
= 4 πr
2
ρ
d Φ
dr
=
m + 4 πr
3
p
r ( r - 2 m )
dp
dr
= - ( p + ρ )
( m + 4 πr
3
p )
r ( r - 2 m )
(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 form a 1-parameter family,
labelled uniquely by ρ
c
≡ ρ (0).
(iii) Assume that the equation of state is known for ρ lessorequalslant ρ
0
but not for ρ > ρ
0
. Explain
why there is a maximum possible mass for the star that is independent of the equation of
state for ρ > ρ
0
.