SPHERICALLY SYMMETRIC SOLUTIONS OF GENERAL RELATIVITY WITH RADIAL MOTION

In this paper, the model of radial motion of dusty matter for the spherically symmetric case in General Relativity (GR) was developed. The model can be used, in particular, to describe galactic halos of dark matter. The question of the qualitative and quantitative composition of dark matter is of great importance both for understanding the current structure of the Universe and for choosing the most realistic scenario of its evolution. Since dark matter effectively manifests itself only gravitationally, its pressure can be neglected, and the equation of state, despite its physical nature, can be considered dusty. The presence of supermassive black holes in the centers of galaxies and the scale of the phenomena necessitate the use of GR equations. The spherical shapes of galactic halos require the use of spherical symmetry. The aim of this work is to model the radial motion of dark matter for a spherically symmetric GR interval, taking into account a possible central mass. Dark matter is assumed to be dusty and moving both toward and away from the center. It is proved that such a stationary case corresponds to a static interval of spacetime in the coordinates of curvatures. The system of Einstein's equations for this case is significantly simplified and solved numerically. The Lichnerowicz-Darmois conditions of crosslinking the spacetime of the proposed model with the external spherically symmetric Schwarzschild spacetime are chosen as boundary conditions. In the proposed model, there is an event horizon, approaching which the motion of particles seems to “freeze”, similar to the motion near the event horizon in the Schwarzschild field. Thus, the model also takes into account the presence of a black hole in the center, which should have been formed as a result of the considered motion of matter. The proposed model can be applied to spherical galaxy clusters and even to stellar systems in space beyond the star, assuming the existence of a halo of cold dark matter particles around them.