Gravitational Field of Massive Point Particle in General Relativity
Abstract
Description
Utilizing various gauges of the radial coordinate we give a description of static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist a two-parameters family of such solutions to the Einstein equations which are physically distinguishable but only some of them describe the gravitational field of a single massive point particle with nonzero bare mass $M_0$. In particular, we show that the widespread Hilbert's form of Schwarzschild solution, which depends only on the Keplerian mass $M<M_0$, does not solve the Einstein equations with a massive point particle's stress-energy tensor as a source. Novel normal coordinates for the field and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravitational mass defect of a point particle and determine the dependence of the solutions on this mass defect. The result can be described as a change of the Newton potential $ϕ_{{}_N}=-G_{{}_N}M/r$ to a modified one: $ϕ_{{}_G}=-G_{{}_N}M/ (r+G_{{}_N} M/c^2\ln{{M_0}\over M})$ and a corresponding modification of the four-interval. In addition we give invariant characteristics of the physically and geometrically different classes of spherically symmetric static space-times created by one point mass. These space-times are analytic manifolds with a definite singularity at the place of the matter particle.
16 pages, no figures, latex file, changes in the Abstract,typos corrected
16 pages, no figures, latex file, changes in the Abstract,typos corrected