Black Hole and Dark Matter in the Synchronous Coordinate System

Abstract

The static state of a black hole in interaction with dark matter is considered in the synchronous coordinate system. Just as in Schwarzschild coordinates, in synchronous coordinates there exists a regular static spherically symmetric solution of the system of Einstein and Klein–Gordon equations that describes the state of matter extremely compressed by its own gravitational field. There is also no constraint on the mass. There also exist two gravitational radii with the boundary conditions at which the solutions are not unique. In contrast to Schwarzschild coordinates, in synchronous coordinates the determinant of the metric tensor and the component g¹¹(r) do not become zero at the gravitational radii. In synchronous coordinates, in contrast to Schwarzschild coordinates, in the spherical layer between the gravitational radii the signature of the metric tensor is not violated. In synchronous coordinates the Einstein and Klein–Gordon equations are reduced to a system of the second (rather than fourth) order. The solutions were obtained analytically, so that no numerical calculations were required. The gravitational mass defect in the λψ⁴ model was determined. The total mass of matter turns out to be thrice the Schwarzschild mass determined by a remote observer when compared with Newtonian gravity.