Abreviations: Synchronous reference frame (SynRF), Schwarzschild's coordinate system (SchCS)

Abstract

Analytical spherically symmetric static solution to the set of Einstein and Klein-Gordon equations in a synchronous reference frame (SynRF) is considered. In SynRF, a static solution exists in the ultrarelativistic limit p = - e /3. Pressure p is negative when matter tends to contract. The solution describes a collapsed black hole. At zero temperature, the state of matter with minimum energy is a Bose-Einstein condensate. Its wave function is a sc alar field, satisfying the Klein-Gordon equation. The balance at the boundary with dark matter ensures the static solution for a black hole. With no support by dark matter there is no static solution for a black hole in SynRF. Recently, a static solution for a black hole supported by dark matter has been found in the Schwarzschild's coordinate system (SchCS). In both SchCS and SynRF cases, inside a black hole there is a spherical layer between two "gravitational" radii r_g and r_h > r_g, where the solution exists, but it is not unique. Unlike SchCS, in SynRF detg_ik and g^rr do not change sign, and the signature of the metric is not violated. In SynRF, the solution contains an arbitrary function. It makes easy to transfer from SynRF to SchCS and see that it is the same solution. The non-uniqueness of solutions with boundary conditions at r = r_g and r = r_h makes it possible to find the gravitational field both inside and outside a black hole. In SynRF, time is the same all over the space. In statics, SynRF is a comoving frame with respect to matter. This allows one to find the rest mass of the condensate. In the model "λ|ψ|⁴" total mass is M = ( 3c²/2k ) r_h . It is three times the Schwarzschild's mass M = ( c²/2k ) r_h seen by a distant observer. This gravitational mass defect is the result of bosons being in the ground state, and of the existence of balance between elasticity and density of the condensate.