A Purely Relativistic Point-Source Boundary Condition for the Schwarzschild Solution

TL;DR

This paper derives a boundary condition for the Schwarzschild solution directly relating the Schwarzschild radius to the source's mass within general relativity, avoiding Newtonian approximations and clarifying the nature of point-like sources.

Contribution

It provides a purely relativistic derivation of the point-source boundary condition, linking the Schwarzschild radius to the mass without relying on asymptotic or Newtonian limits.

Findings

Established a direct relation between Schwarzschild radius and source mass within GR

Clarified why distributional methods often lead to unphysical source distributions

Provided a boundary condition that connects source properties to spacetime geometry

Abstract

We present a simple derivation of a point-source boundary condition for the Schwarzschild solution that relates the Schwarzschild radius to the mass of its source without appealing to the Newtonian limit. Interpretation of the Schwarzschild radius in terms of the mass of a point-like source traditionally means resorting to distant asymptotics and the safety of Newtonian gravity, but here we instead show a direct connection between a point-particle's invariant mass and the length parameter of the Schwarzschild solution it sources, fully within the framework of general relativity. As a corollary, we also explain why attempts to show this by distributional techniques often result in a physically unmotivated spatial distribution for the source stress-energy tensor.