The Era of Gravitational Astronomy and Gravitational Field of Non-Rotating Single Point Particle in General Relativity

Abstract

Utilizing various gauges of the radial coordinate, we give a General Relativistic (GR) description of static spherically symmetric spacetimes with a massive point source and vacuum outside this singularity. We show that in GR there exists a two-parameter family of such solutions to the Einstein equations which are physically distinguishable and describe the gravitational field of a single massive point particle with positive proper mass M0 and positive Keplerian mass M < M0. In particular, we show that the widespread Hilbert form of the Schwarzschild solution, which depends only on the Keplerian mass M and describes Black Holes (BH), does not solve the Einstein equations with a massive point particle stress-energy tensor. Novel normal coordinates for the gravitational field and a new physical class of gauges are proposed, thus 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 φN = –GNM/r to a modified one φG = –GNM/$$\left( {r + {{G}_{{\text{N}}}}{M \mathord{\left/ {\vphantom {M {{{c}^{2}}}}} \right. \kern-0em} {{{c}^{2}}}}\ln \frac{{{{M}_{0}}}}{M}} \right)$$ and the corresponding modification of the four-interval. We show that the proper 3D flat space, where these two potentials can be compared, is the tangent space above the position of the massive point source. In addition, we present invariant characteristics of the physically and geometrically different classes of spherically symmetric static spacetimes created by a point mass. Our findings are important for description of Extremely Compact Objects (ECOs) studied in relation with possible echoes in Gravitational Waves (GW) recently discovered by the LIGO/VIRGO collaboration.