Symmetry in Nonlinear Mathematical Physics - 2009

Oleg Zaslavskii (Astronomical Institute of Kharkov V.N. Karazin National University, Ukraine)

Kirill Bronnikov (Center of Gravitation and Fundamental Metrology, Russian Research Institute of Metrological Service & Peoples' Friendship University of Russia, Moscow, Russia)

Static black holes in matter

We study equilibrium conditions between a static spherically symmetric black hole and classical matter in terms of the radial pressure to density ratio w. It is shown that such an equilibrium is possible in two cases: (i) the well-known case w = - 1 at the horizon ("vacuum" matter with non-zero density) and (ii) w = - 1/1+2k where k > 0 is an integer. The whole reasoning is local, hence the results do not depend on any global or asymptotic conditions. They mean, in particular, that a static black hole cannot live inside a star with nonnegative pressure and density. As an example, an exact solution for an isotropic fluid with w = - 1/3 (that is, a fluid of disordered cosmic strings), with or without vacuum matter, is presented. The results are also generalized to an arbitrarily distorted horizon. This became possible due to using Gaussian coordinates in the vicinity of the horizon and the formalism of embeddings that significantly simplifies the problem.