Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 4 / d / i

Table of contents
- Solution i

Solution

d s^{2} = - f (r) d v^{2} + 2 d v d r + r^{2} d Ω_{2}^{2}

with a simple Killing horizon, the surface gravity of

k = \partial_{v}

is

κ = f^{'} (r_{+}) /2

. Here

f^{'} (r) = \frac{2 r}{L ^{2}} + \frac{r _{+}}{r ^{2}} (1 + \frac{r _{+}^{2}}{L ^{2}}),

so

κ = \frac{1}{2 r _{+}} + \frac{3 r _{+}}{2 L ^{2}} .

The Hawking temperature of the Schwarzschild--anti-de Sitter black hole is therefore

T_{H} = \frac{κ}{2 π} = \frac{1 + 3 r _{+}^{2} / L ^{2}}{4 π r _{+}} .

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