Comment on ''The space-time line element for static ellipsoidal objects''

TL;DR

This paper critically examines proposed static ellipsoidal spacetime metrics, revealing they do not satisfy vacuum Einstein equations and thus do not represent vacuum solutions, contrary to prior claims.

Contribution

It provides a detailed analysis showing that previously proposed ellipsoidal metrics are not valid vacuum solutions of Einstein's equations.

Findings

The corrected metrics have non-zero Einstein tensor and Ricci scalar.

The metrics imply non-zero pressures, indicating non-vacuum solutions.

Previously cited metrics also fail to satisfy vacuum Einstein equations.

Abstract

A recent paper proposes a static ellipsoidal space-time metric that reduces to Schwarzschild. After examining the corrected line element using Maple's Differential geometry package, we find that the Einstein tendor and Ricci scalar are non-zero, and the metric yields non-zero pressures when interpreted as an anisotropic fluid. Thus, it does not represent a vacuum solution. We also checked a cited ellipsoidal metric from the literature and found it likewise fails to satisfy the vacuum Einstein equations.