Eur. Phys. J. C (2017) 77: 244
https://doi.org/10.1140/epjc/s10052-017-4812-y

Regular Article - Theoretical Physics

Effective Tolman temperature induced by trace anomaly

¹ Department of Civil Engineering, Sangmyung University, Cheonan, 31066, Republic of Korea
² Department of Physics, Sogang University, Seoul, 04107, Republic of Korea
³ Research Institute for Basic Science, Sogang University, Seoul, 04107, Republic of Korea

^* e-mail: wtkim@sogang.ac.kr

Received: 20 January 2017
Accepted: 7 April 2017
Published online: 17 April 2017

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 blue-shift 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 an infinite blue-shift of the Hawking temperature at the event horizon does not appear any more. In particular, it is vanishing on the horizon, so that the equivalence principle is exactly recovered at the horizon.