Topological Origin of Horizon Temperature via the Chern-Gauss-Bonnet Theorem

TL;DR

This work proposes a topological origin for horizon thermodynamics by applying the Chern-Gauss-Bonnet theorem to the Wick-rotated near-horizon Euclidean geometry of four-dimensional vacuum spacetimes. By enforcing compactness through the Euclidean time periodicity $\beta = 1/T_H$ and analyzing the near-horizon topology (e.g., $D^2\times\mathbb{S}^2$ for de Sitter, a Rindler-like region for Schwarzschild), the Hawking temperature is tied to the Euler characteristic $\chi(\mathcal{M}_E)$ via $T_H = \frac{1}{\chi(\mathcal{M}_E)\pi l}$ (with appropriate global/topology refinements). The results reproduce the standard $T_H$ values, explain the factor-of-two differences in de Sitter between local and global temperatures, and offer a unified topological framework for non-extremal horizon thermodynamics, while noting the complexity of extending to Kerr. This indicates a deep link between global spacetime topology and semi-classical horizon thermodynamics with potential implications for holography and quantum gravity.

Abstract

This paper establishes a connection between the Hawking temperature of spacetime horizons and global topological invariants, specifically the Euler characteristic of Wick-rotated Euclidean spacetimes. This is demonstrated for both de Sitter and Schwarzschild, where the compactification of the near-horizon geometry allows for a direct application of the Chern-Gauss-Bonnet theorem. For de Sitter, a simple argument connects the Gibbon-Hawking temperature to the global thermal de Sitter temperature of the Bunch-Davies state. This establishes that spacetime thermodynamics are a consequence of the geometrical structure of spacetime itself, therefore suggesting a deep connection between global topology and semi-classical analysis.