Effective Tolman temperature induced by trace anomaly

TL;DR

This paper introduces an effective Tolman temperature that accounts for trace anomaly effects, resolving the divergence issue at the black hole horizon and restoring the equivalence principle.

Contribution

It proposes a modified Tolman temperature incorporating trace anomaly, ensuring finiteness and physical consistency at the horizon.

Findings

Effective Tolman temperature is finite everywhere outside the horizon.

The temperature vanishes at the horizon, restoring the equivalence principle.

The approach aligns with the presence of trace anomaly in black hole thermodynamics.

Abstract

Despite the finiteness of stress tensor for a scalar field on the four-dimensional Schwarzschild black hole in the Israel-Hartle-Hawking vacuum, the Tolman temperature in thermal equilibrium is certainly divergent on the horizon due to the infinite blueshift of the Hawking temperature. The origin of this conflict is due to the fact that the conventional Tolman temperature was based on the assumption of a traceless stress tensor, which is, however, incompatible with the presence of the trace anomaly responsible for the Hawking radiation. Here, we present an effective Tolman temperature which is compatible with the presence of the trace anomaly by using the modified Stefan-Boltzmann law. Eventually, the effective Tolman temperature turns out to be finite everywhere outside the horizon, and so there does not appear infinite blueshift of the Hawking temperature at the event horizon any more. In particular, it is vanishing on the horizon, so that the equivalence principle is exactly recovered at the horizon.