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Black Holes - Mathematical Tripos - Past Exam, Exams of Mathematics

This is the Past Exam of Mathematical Tripos which includes Class Field Theory, Artin Map, Abelian Extension of Number Fields, Decomposition Group, Inertia Group, Factorisation of Prime Ideals, Version of Hensel’s Lemma, Hilbert Norm Residue Symbol etc. Key important points are: Black Holes, Schwarzschild Metric, Black Hole Horizon, Euclidean Methods, Surface Gravity, Time Translation Killing Vector, Black Hole Area Theorem, Hawking Radiation, Physical Consequences, Collapsing Matter

Typology: Exams

2012/2013

Uploaded on 02/27/2013

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MATHEMATICAL TRIPOS Part III
Monday 13 June, 2005 9 to 12
PAPER 62
BLACK HOLES
Attempt 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.
pf3

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MATHEMATICAL TRIPOS Part III

Monday 13 June, 2005 9 to 12

PAPER 62

BLACK HOLES

Attempt 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.

1 Write an essay describing in detail how one constructs the entirety of the spacetime associated with a single uncharged non-rotating black hole, given as your starting point the Schwarzschild metric. Describe qualitatively the consequences of allowing the hole to be electrically charged.

2 A static asymptotically flat black hole spacetime is given by the metric

ds^2 = −V (r)dt^2 +

dr^2 V (r)

  • r^2 (dθ^2 + sin^2 θdφ^2 ).

As r → ∞, V (r) → 1 and V (r) > 0 for r > r 0. At r = r 0 , V (r) has a simple zero corresponding to a non-degenerate black hole horizon. By using Euclidean methods, derive the temperature T of the horizon.

The surface gravity κ of a black hole is defined by

ka∇akb = κkb

evaluated on the horizon, where ka^ is the time translation Killing vector. Show that

|κ| = 2πT.

3 Sketch a proof of the black hole area theorem.

Describe the effect, over a long period of time, of Hawking radiation on the geometry of a uncharged non-rotating black hole and the physical consequences of this radiation.

How does the mass of an uncharged, non-rotating black hole evolve as a function of time?

Why does your result not contradict the area theorem?

Paper 62