Gravitational singularities via acceleration: the case of the Schwarzschild solution and Bach's gamma metric

TL;DR

This paper investigates the nature of singularities in the gamma metric, a family of solutions to Einstein's equations, revealing invariant singular behavior related to acceleration beyond traditional curvature measures.

Contribution

It demonstrates that acceleration norms exhibit singularities in gamma metrics, including Schwarzschild's horizon, providing new insights into gravitational singularities beyond curvature scalars.

Findings

Acceleration norm is singular beyond Kretschmann scalar behavior.

Singular acceleration occurs on the Schwarzschild horizon.

Invariant singular behavior is identified in gamma metrics.

Abstract

The so called gamma metric corresponds to a two-parameter family of axially symmetric, static solutions of Einstein's equations found by Bach. It contains the Schwarzschild solution for a particular value of one of the parameters, that rules a deviation from spherical symmetry. It is shown that there is invariantly definable singular behaviour beyond the one displayed by the Kretschmann scalar when a unique, hypersurface orthogonal, timelike Killing vector exists. In this case, a particle can be defined to be at rest when its world-line is a corresponding Killing orbit. The norm of the acceleration on such an orbit proves to be singular not only for metrics that deviate from Schwarzschild's metric, but also on approaching the horizon of Schwarzschild metric itself, in contrast to the discontinuous behaviour of the curvature scalar.