Schwarzschild Massive-Point-Particle Problem in Arbitrary Radial Gauge

TL;DR

This paper provides the first correct solution for the Schwarzschild Massive Point problem in any radial gauge, revealing a two-parameter family of solutions with physical implications for compact objects and gravitational wave echoes.

Contribution

It introduces a mathematically rigorous solution to the SMP problem in arbitrary radial gauges and develops the theory of distributions with finite jumps for this context.

Findings

Two-parameter family of SMP solutions distinguished from Schwarzschild black holes

Solutions exhibit an unavoidable metric jump at the particle's location

Implications for modeling Extremely Compact Objects and gravitational wave echoes

Abstract

We present, for the first time, a correct solution of the Schwarzschild Massive Point (SMP) problem in {\em arbitrary} radial gauge and formulate the strict mathematical assumptions, which are necessary and sufficient for this. In GR, there exists a two-parameter family of such exact SMP solutions to the Einstein equations, which are physically distinguishable from the well-studied one parameter family of vacuum Schwarzschild solutions related with Schwarzschild Black Holes (SBH). The obtained here SMP family of solutions is defined by positive bare mass $M_0>0$ and positive Kepler mass $M<M_0$, or, alternatively, by the standard gravitational radius $\rho_g$ and mass ratio $\varrho=M/M_0 \in (0,1)$. The metrics of spacetime have an unavoidable jump at the place of a massive point particle. We also present a proper development of the theory of distribution defined by kernels with a finite jump which appear in the solution of SMP. The specific properties of these distributions are used for work with SMP. A series of physical properties of SMP solutions are derived and commented. 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.