Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem

Abstract

The Euclidean black hole has topology ${\mathit{gerR}}^{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{scrS}}^{\mathit{d}\mathrm{\ensuremath{-}}2}$. It is shown that, in Einstein's theory the deficit angle of a cusp at any point in ${\mathit{gerR}}^{2}$ and the area of the ${\mathit{scrS}}^{\mathit{d}\mathrm{\ensuremath{-}}2}$ 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 ${\mathit{scrS}}^{\mathit{d}\mathrm{\ensuremath{-}}2}$ 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.