MATHEMATICAL TRIPOS
Part III
Friday 30 May 2008 1.30 to 4.30
PAPER 66
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)
State (without explanation) which of the space-times listed below is (i) extendible;
(ii) geodesically complete; (iii) globally hyperbolic; (iv) asymptotically simple. (Present
your answers in the form of a table.)
The region
r >
2
M
of the Schwarzschild solution (with
M >
0 ),
Kruskal,
the maximal analytic extension of extremal Reissner-Nordstrom,
Rindler,
the Einstein static universe.
(b)
The Majumdar-Papapetrou (MP) solution has metric
ds
2
=
-
H
-
2
dt
2
+
H
2
(
dx
2
+
dy
2
+
dz
2
)
,
H
(
x,y,z
) = 1 +
N
X
i
=1
M
i
|
x
-
x
i
|
where
M
i
>
0 ,
x
= (
), and
x
i
are constant vectors in
R
3
.
