Black Hole Entropy and the Dimensional Continuation of the Gauss-Bonnet Theorem

Abstract

The Euclidean black hole has topology ${\Re }^2\times {\mathcal{S}}^{d-2}$. It is shown that -in Einstein's theory- the deficit angle of a cusp at any point in ${\Re }^2$ and the area of the ${\mathcal{S}}^{d-2}$ are canonical conjugates. The black hole entropy emerges as the Euler class of a small disk centered at the horizon multiplied by the area of the ${\mathcal{S}}^{d-2}$ there.These results are obtained through dimensional continuation of the Gauss-Bonnet theorem. The extension to the most general action yielding second order field equations for the metric in any spacetime dimension is given.

7 pages, RevTex

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