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

TL;DR

This paper links the temperature of spacetime horizons to topological invariants using the Chern-Gauss-Bonnet theorem, revealing a geometric origin of thermodynamic properties in curved spacetimes.

Contribution

It demonstrates a novel topological approach to understanding horizon temperature, connecting global spacetime topology with thermodynamic behavior.

Findings

Horizon temperature relates to the Euler characteristic of Euclidean spacetime

Topological invariants determine thermal properties of de Sitter and Schwarzschild horizons

Spacetime thermodynamics emerge from geometrical and topological structures

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.