General static black holes in matter

TL;DR

This paper investigates conditions under which static space-times can have matter in equilibrium with black hole horizons, revealing discrete parameter values and extending previous spherical symmetry results to more general geometries.

Contribution

It generalizes the conditions for matter-horizon equilibrium beyond spherical symmetry, identifying specific parameter values and the possibility of higher-order horizons without vacuum matter.

Findings

Equilibrium with matter occurs only at discrete parameter values of w.

For simple horizons, w = -1/3 corresponds to cosmic string gas.

Higher-order horizons can exist without vacuum matter if the horizon surface has zero curvature.

Abstract

For arbitrary static space-times, it is shown that an equilibrium between a Killing horizon and matter is only possible for some discrete values of the parameter $w = p_1/\rho$, where $\rho$ is the density and $p_1$ is pressure in the direction normal to the horizon. In the generic situation of a simple (non-extremal) horizon and the slowest possible density decrease near the horizon, this corresponds to $w = -1/3$, the value known for a gas of disordered cosmic strings. An admixture of "vacuum matter", characterized by $w=-1$ and nonzero density at the horizon, is also admitted. This extends the results obtained previously for static, spherically symmetric space-times. A new feature as compared to spherical symmetry is that higher-order horizons can exist in the absence of vacuum matter if the horizon is a surface of zero curvature, which can occur, e.g., in cylindrically symmetric space-times.