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 = p1/ρ , where ρ is the density and p1 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.