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

TL;DR

This paper explores the geometric origin of black hole entropy using dimensional continuation of the Gauss-Bonnet theorem, revealing its relation to topological invariants and extending the analysis to general second-order gravity actions.

Contribution

It introduces a novel approach linking black hole entropy to topological invariants via dimensional continuation of the Gauss-Bonnet theorem, applicable to general second-order gravity theories.

Findings

Black hole entropy is related to the Euler class of a small disk at the horizon.

Deficit angle and horizon area are canonical conjugates in Einstein's theory.

Extension of the topological approach to general second-order gravity actions.

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.