Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 2 / b / i

Keep the smooth

β = β_{H}

geometry fixed away from the horizon but identify Euclidean time with arbitrary period

β

. The horizon then has deficit angle

2 π (1 - β / β_{H})

. Its delta-function curvature contributes

\int_{M} g R_{tip} = 4 π (1 - \frac{β}{β _{H}}) A,

and hence

I_{tip} = - \frac{A}{4 G _{N}} (1 - \frac{β}{β _{H}}) .

Using

lo g Z = - I_{grav}

S_{BH} = (1 - β \partial_{β}) lo g Z ∣_{β_{H}} = \frac{A}{4 G _{N}} = \frac{2 π ^{2} r _{H}^{3}}{4 G _{N}} .

All smooth bulk terms, including the cosmological-constant volume term, are proportional to the Euclidean time period and are annihilated by

1 - β \partial_{β}

. The asymptotic Gibbons–Hawking–York boundary term and holographic counterterms are likewise smooth and linear in

β

. Only the curvature singularity at the fixed point of the Euclidean time circle survives.

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