Dirty blackholes: Thermodynamics and horizon structure

TL;DR

This paper demonstrates that the Hawking temperature of static spherically symmetric black holes interacting with matter fields is generally suppressed compared to vacuum black holes, due to matter distribution effects and energy conditions.

Contribution

It shows that Hawking temperature suppression is a generic feature for such black holes, extending previous observations and providing a formula involving matter distribution and energy conditions.

Findings

Hawking temperature is suppressed relative to vacuum blackholes.

The suppression factor depends on matter distribution and energy conditions.

Examples illustrate the temperature suppression behavior.

Abstract

Considerable interest has recently been expressed in (static spherically symmetric) blackholes in interaction with various classical matter fields (such as electromagnetic fields, dilaton fields, axion fields, Abelian Higgs fields, non--Abelian gauge fields, {\sl etc}). A common feature of these investigations that has not previously been remarked upon is that the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum blackhole of equal horizon area. That is: $k T_H \leq \hbar/(4\pi r_H) \equiv \hbar/\sqrt{4\pi A_H}$. This paper will argue that this suppression is generic. Specifically, it will be shown that \[ k T_H = {\hbar\over4\pi r_H} \; e^{-\phi(r_H)} \; \left( 1 - 8\pi G \; \rho_H \; r_H^2 \right). \] Here $\phi(r_H)$ is an integral quantity, depending on the distribution of matter, that is guaranteed to be positive if the Weak Energy Condition is satisfied. Several examples of this behaviour will be discussed. Generalizations of this behaviour to non--symmetric non--static blackholes are conjectured.