
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
Abstract:
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 wellknown case w =  1 at the horizon ("vacuum" matter with nonzero
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.

