An Extension of Schwarzschild Space to r=0

TL;DR

This paper extends the Schwarzschild solution to include the region at r=0 by incorporating the energy-momentum tensor of a point particle, revealing a discontinuity that causes a repulsive force and alters the fate of falling objects.

Contribution

It provides a more rigorous Schwarzschild metric extension to r=0 using the energy-momentum tensor, showing a discontinuity and resulting repulsive force.

Findings

Discontinuity at r=0 causes infinite repulsive force.

Modified metric includes additional terms involving theta(r).

Falling particles bounce instead of collapsing into a singularity.

Abstract

A more rigorous treatment of the Schwarzschild metric by making use of the energy-momentum tensor of a single point particle as source term shows that $g_{00}=-\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(\theta (r)-1)\}\exp [2(\theta (r)-1)]$ $g_{rr}=\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(\theta (r)-1)\}^{-1}$ The existence of a discontinuity at r=0 leads to an infinite repulsive force that will change the ultimate fate of a free fall test particle to a bouncing state.