Thermal divergences on the event horizons of two-dimensional black holes

TL;DR

This paper calculates the stress-energy tensor of a scalar field near two-dimensional black holes, showing divergences unless the field is at the black hole's natural temperature, highlighting the importance of temperature in black hole equilibrium.

Contribution

It demonstrates that in two-dimensional black holes, the stress-energy tensor diverges unless the field is at the natural temperature, revealing the necessity of this temperature for equilibrium.

Findings

Stress-energy tensor diverges at the horizon unless at natural temperature.

Extreme and nonextreme black holes require the natural temperature for equilibrium.

The natural temperature is determined by the surface gravity of the horizon.

Abstract

The expectation value of the stress-energy tensor $\langleT_{\mu\nu}\rangle$ of a free conformally invariant scalar field is computed in a general static two-dimensional black hole spacetime when the field is in either a zero temperature vacuum state or a thermal state at a nonzero temperature. It is found that for every static two-dimensional black hole the stress-energy diverges strongly on the event horizon unless the field is in a state at the natural black hole temperature which is defined by the surface gravity of the event horizon. This implies that both extreme and nonextreme two-dimensional black holes can only be in equilibrium with radiation at the natural black hole temperature.