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

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.