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

TL;DR

The Gauss-Bonnet theorem is extended to the most general action yielding second order field equations for the metric in any spacetime dimension and the black hole entropy emerges as the Euler class of a small disk centered at the horizon.

Abstract

The Euclidean black hole has topology $\Re^2 \times {\cal S}^{d-2}$. 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 ${\cal S}^{d-2}$ 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 ${\cal S}^{d-2}$ 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.