Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 2 / a / ii

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- Solution ii

Solution

After Wick rotation

t = - i t_{E}

, the near-horizon metric has

d s_{E}^{2} ≃ f^{'} (r_{H}) (r - r_{H}) d t_{E}^{2} + \frac{d r ^{2}}{f ^{'} ( r _{H} ) ( r - r _{H} )} .

Regularity at the origin of this polar plane requires the Euclidean black-hole regularity condition

β_{H} = 4 π / f^{'} (r_{H})

. Since

μ = r_{H}^{2} (1 + \frac{r _{H}^{2}}{L ^{2}}), f^{'} (r_{H}) = \frac{2}{r _{H}} + \frac{4 r _{H}}{L ^{2}},

we obtain

β_{H} = \frac{2 π L ^{2} r _{H}}{2 r _{H}^{2} + L ^{2}} .

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