Schwarzschild Black Hole Lives to Fight Another Day - Comment on the paper J. Math. Phys. 50, 042502 (2009) by A. Mitra

TL;DR

This paper critiques a previous claim that the Schwarzschild black hole's mass must be zero, clarifying that the earlier proof was based on a misunderstanding of coordinate volume elements in differential geometry.

Contribution

It refutes Mitra's incorrect proof regarding the mass of Schwarzschild black holes, emphasizing proper mathematical treatment of coordinate volume elements.

Findings

Mitra's proof is incorrect due to a misunderstanding of coordinate volume elements.

The Schwarzschild black hole's mass parameter does not necessarily vanish.

Proper mathematical analysis invalidates the previous claim.

Abstract

In a comment published several years ago in this Journal [J. Math. Phys. 50, 042502 (2009)] Mitra has claimed to prove that a neutral point particle in general relativity as described by the Schwarzschild metric must have zero gravitational mass, i.e. the mass parameter $M_{0}$ of a Schwarzschild black hole necessarily vanishes. It is shown that the purported proof is incorrect. The error stems from a basic misunderstanding of the mathematical description of coordinate volume element in a differentiable manifold.