Black Hole Evaporation as a Topological Tunneling

TL;DR

This paper models black hole evaporation as a topological tunneling process involving a quantum atmosphere of photons, using Euclidean path integrals and topological invariants, providing new insights into black hole thermodynamics and stability.

Contribution

It introduces a topological perspective on black hole evaporation, showing that the process is a tunneling between spacetimes with different topologies via Euclidean instantons.

Findings

Black hole surrounded by a finite photon atmosphere at Hawking temperature.

Quantum atmosphere contributes positively to black hole specific heat, suggesting possible thermodynamic stability.

Evaporation modeled as a topological tunneling between distinct spacetime topologies.

Abstract

We present the quantization of the electromagnetic field near the event horizon of a Schwarzschild black hole using Euclidean path integrals. Our result for the vacuum energy describes a black hole surrounded by a finite volume of photons at $T_{H} = \frac{1}{8\pi G M}$, the black hole quantum atmosphere. The total entropy includes contributions from this atmosphere, and the Bekenstein entropy, which arises from the Gibbons--Hawking--York boundary term, which encodes topological information. We show that the contribution of the quantum atmosphere to the black hole specific heat is positive, indicating that the system may become thermodynamically stable. By analyzing homology groups, we show that the black hole evaporation is a tunneling between topologically distinct spacetimes: Schwarzschild ($\chi = 2)$ transitions to the flat spacetime ($\chi = 1$) via Hawking radiation, where $\chi$ is the Euler characteristic, a topological invariant. This process resembles instanton-driven tunneling in Yang-Mills theories, where topologically non-trivial solutions dominate the vacuum amplitude. In our case, the Gibbons--Hawking--York term dominates the transition amplitude, which induces the evaporation process. These results corroborate the Parikh-Wilczek picture of Hawking radiation and the interpretation of Euclidean black holes as gravitational instantons.