General static black holes in matter

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 $w=p_1\mathrm{/}\rho$, 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 $w=-1\mathrm{/}3$, 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.