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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-54/3/e/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 3 / e / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-18

In the ingoing Eddington-Finkelstein coordinates, the stationary Killing vector field is

k = \partial_{v}

with norm

k^{2} = - f

. The normal vector to a level set of

r

is

\nabla^{a} r = (\partial_{v})^{a} + f (\partial_{r})^{a} .

At

r = r_{+}

it equals

k^{a}

, which is null there. Thus this null hypersurface is a Killing horizon of

k

. Its surface gravity is fixed by

\nabla_{a} (k^{2}) = - 2 κ k_{a}

on the horizon. Since

k_{a} = (d r)_{a}

there and

\nabla_{a} (k^{2}) = - f^{'} (r_{+}) (d r)_{a}

,

κ = \frac{1}{2} f^{'} (r_{+}) = \frac{r _{+}^{2} - r _{-}^{2}}{r _{+}^{3}} .

The positive sign follows from the future outer-horizon generator, normalized by the static time at infinity. It vanishes as the two radii approach the extremal limit.

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