Dirty black holes: Thermodynamics and horizon structure

Abstract

Considerable interest has recently been expressed in (static spherically symmetric) black holes in interaction with various classical matter fields (such as electromagnetic fields, dilaton fields, axion fields, Abelian Higgs fields, non-Abelian gauge fields, 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 black hole of equal horizon area. That is, ${\mathit{kT}}_{\mathit{H}}$\ensuremath{\le}\ensuremath{\Elzxh}/(4\ensuremath{\pi}${\mathit{r}}_{\mathit{H}}$)==\ensuremath{\Elzxh}/ \ensuremath{\surd}4\ensuremath{\pi}${\mathit{A}}_{\mathit{H}}$. This paper will argue that this suppression is generic. Specifically, it will be shown that ${\mathit{kT}}_{\mathit{H}}$=(\ensuremath{\Elzxh}/4\ensuremath{\pi}${\mathit{r}}_{\mathit{H}}$)${\mathit{e}}_{\mathit{H}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varphi}}(\mathit{r}}$) (1-8\ensuremath{\pi}G${\mathrm{\ensuremath{\rho}}}_{\mathit{H}}$${\mathit{r}}_{\mathit{H}}^{2}$). Here \ensuremath{\varphi}(${\mathit{r}}_{\mathit{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 behavior will be discussed. Generalizations of this behavior to nonsymmetric nonstatic black holes are conjectured.