The Era of Gravitational Astronomy and Gravitational Field of Non-Rotating Single Point Particle in General Relativity
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he Era of Gravitational Astronomy and Gravitational Field of Non-Rotating Single Point Particle in General Relativity P. P. Fizieva, b, *, ** a
Sofa University Foundation for Theoretical and Computational Physics and Astrophysics, Sofia, 1164 Bulgaria b BLTF, JINR, Dubna, Moscow oblast, 141980 Russia *e-mail: [email protected] **e-mail: [email protected] Received July 15, 2019; revised August 12, 2019; accepted August 12, 2019
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 M Newton potential ϕN = –GNM/r to a modified one ϕG = –GNM/ r + GN M c 2 ln 0 and the correspondM ing 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.
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Keywords: General Relativity, gauge conditions, the Schwarzschild problem, massive point source, proper mass, the Keplerian mass, mass defect, gauge classes of solutions, modified gravitational potential, invariants, extremely compact objects, black holes DOI: 10.1134/S1063779620010037
1. INTRODUCTION LIGO detection of Gravitational Waves (GW) is the most important finding in gravity after Sir Isaac Newton’s discovery of the gravitational field. Indeed, Sir Isaac Newton discovered the gravitational field attached to bodies. LIGO discovered the gravitational field detached from bodies and freely spreading in space. The last being a qualitatively different novel gravitational phenomenon. Without any doubts, with the LIGO/VIRGO observations of more than 11 GW events (see the references at the WEB address [1]), a new era in fundamental physics started [2]. Fina
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