Black Hole Evaporation as a Topological Tunneling

TL;DR

This work implements a Euclidean path integral treatment of the electromagnetic field in a Schwarzschild background, showing that the vacuum near the horizon forms a finite-volume photon atmosphere at $T_H=\frac{1}{8\pi GM}$ and that the Gibbons–Hawking–York boundary term drives the black hole entropy and evaporation. It reveals a topological tunneling interpretation where evaporation corresponds to a transition between spacetimes with different Euler characteristics, connecting the RVB formula to gravitational instantons and generalizing to $D$ dimensions. The atmosphere contributes a finite entanglement entropy correction $S_{out}=\tfrac{16}{3}\sigma_{SB}T^{3}V_A$ and can stabilize the black hole thermodynamically above a critical temperature, offering a coherent picture that aligns with Parikh–Wilczek tunneling. The results establish a robust topological and thermodynamic framework for quantum corrections to black hole spacetimes and suggest avenues for extending the analysis to more general black holes and potential observational probes.

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 $T_{H} = \frac{1}{8πG M}$, 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 ($χ= 2)$ transitions to the flat spacetime ($χ= 1$) via Hawking radiation, where $χ$ 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.

Figures (4)

• Figure 1: This is one of the diagrams of the gravitational contribution to the electromagnetic vacuum polarization. As Maxwell-Einstein theory is not renormalizable, thus, it is not possible to obtain finite results to such processses. For this reason, here we will use quantum field theory in curved backgrounds, an effective approach to quantum gravity.
• Figure 2: This is the curve for the heat capacity of a black hole with the inclusion of the electromagnetic vacuum fluctuations. At $T_{C}$ the system reaches stability as $C_{V}>0$.
• Figure 3: In the particle-antiparticle picture of Hawking radiation, a pair is created due to the intense gravitational field; one of the particles goes into the black hole while the other is scattered to infinity. As electrodynamics does not possess self-interactions, the particle creation is only possible if gravitons are taken into account. In our approach, the gravitons are hidden in the background metric, but their effects appear in the observables. In this diagram, the blob represents loop corrections to the vertex.
• Figure 4: This diagram illustrate the deep connections between spacetime topology and black hole thermodynamics. The relation between the Gibbons-Hawking-York action was found by S. Liberatti and others Liberati:1995jjLiberati:1996ktLiberati:1997sp , and the connection between $\chi$ and $T_{H}$ was discovered by RVB Robson:2018con.