Nonthermal nature of incipient extremal black holes

TL;DR

This paper investigates particle production during the formation of extremal black holes, revealing a nonthermal spectrum and challenging the notion that such black holes can be considered thermodynamic objects.

Contribution

It introduces a generalized coordinate system for extremal black holes and demonstrates that their particle emission spectrum is nonthermal, with implications for black hole thermodynamics.

Findings

Particle spectrum is nonthermal and depends on collapse history.

Stress-energy tensor expectation value is zero, variance decays as a power law.

Extremal black holes do not behave as thermal objects at any stage.

Abstract

We examine particle production from spherical bodies collapsing into extremal Reissner-Nordstr\"om black holes. Kruskal coordinates become ill-defined in the extremal case, but we are able to find a simple generalization of them that is good in this limit. The extension allows us to calculate the late-time worldline of the center of the collapsing star, thus establishing a correspondence with a uniformly accelerated mirror in Minkowski spacetime. The spectrum of created particles associated with such uniform acceleration is nonthermal, indicating that a temperature is not defined. Moreover, the spectrum contains a constant that depends on the history of the collapsing object. At first sight this points to a violation of the no-hair theorems; however, the expectation value of the stress-energy-momentum tensor is zero and its variance vanishes as a power law at late times. Hence, both the no-hair theorems and the cosmic censorship conjecture are preserved. The power-law decay of the variance is in distinction to the exponential fall-off of a nonextremal black hole. Therefore, although the vanishing of the stress tensor's expectation value is consistent with a thermal state at zero temperature, the incipient black hole does not behave as a thermal object at any time and cannot be regarded as the thermodynamic limit of a nonextremal black hole, regardless of the fact that the final product of collapse is quiescent.