A Quantal Tolman Temperature

TL;DR

This paper introduces a quantum-mechanical generalization of the Tolman temperature accounting for trace anomalies, resulting in a finite temperature at the horizon and resolving related puzzles.

Contribution

It derives a trace anomaly-induced Stefan-Boltzmann law and generalizes the Tolman temperature to include quantum effects, ensuring finiteness at the horizon.

Findings

Generalized Tolman temperature is finite everywhere

The temperature vanishes at the horizon

The equivalence principle holds at the horizon

Abstract

The conventional Tolman temperature based on the assumption of the traceless condition of energy-momentum tensor for matter fields is infinite at the horizon if Hawking radiation is involved. However, we note that the temperature associated with Hawking radiation is of relevance to the trace anomaly, which means that the traceless condition should be released. So, a trace anomaly-induced Stefan-Boltzmann law is newly derived by employing the first law of thermodynamics and the property of the temperature independence of the trace anomaly. Then, the Tolman temperature is quantum-mechanically generalized according to the anomaly-induced Stefan-Boltzmann law. In an exactly soluble model, we show that the Tolman factor does not appear in the generalized Tolman temperature which is eventually finite everywhere, in particular, vanishing at the horizon. It turns out that the equivalence principle survives at the horizon with the help of the quantum principle, and some puzzles related to the Tolman temperature are also resolved.