Paper56 (2) - MATHEMATICAL TRIPOS Part III Thursday 4 June...

MATHEMATICAL TRIPOS
Part III
Thursday, 4 June, 2009 1:30 pm to 4:30 pm
PAPER 56
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 variant of general relativity admits an electrically charged black hole
solution with
metric
,
ds
2
=
H(r)
1 / 2
F (r)dt
2
+ H(r)
1 / 2
F (r)
1
dr
2
+ r
2
dΩ
2
−
−
2m sinh
2
α
−
2m
H(r) = 1 +
,
F (r) = 1 −
,

r
r
where m and α are real constants with m > 0.
(a) Calculate the Komar mass of this solution. [4]
(b) Show that one can define a quantity r
∗
such that u = t − r
∗
and v = t + r
∗
are constant on outgoing
and ingoing radial null geodesics respectively. (You may express r
∗
as an
integral.)
[2]
(c) Obtain the above metric in ingoing Eddington-Finkelstein
coordinates (v, r, θ, φ).
Hence show that it can be analytically extended through r = 2m.
[2]
(d) Define the black hole region of an asymptotically flat spacetime. Prove that
the region r < 2m (in ingoing Eddington-Finkelstein coordinates) is within the
black hole region, and that the region r > 2m does not intersect the black hole
region. [5]