On singularities, horizons, invariants, and the results of Antoci, Liebscher and Mihich (GRG 38, 15 (2006) and earlier)

TL;DR

This paper critiques the claim that certain horizon-related invariants indicate singularities, clarifies their nature as regular features, and proposes a new invariant criterion for horizons based on Cartan invariants.

Contribution

It introduces a new invariant criterion for horizons and clarifies the distinction between true singularities and regular horizons using invariant analysis.

Findings

Divergent acceleration scalars at horizons are not true singularities.

Cartan invariants remain finite at horizons, characterizing their regularity.

A new invariant criterion for horizons is proposed.

Abstract

Antoci et al. have argued that the horizons of the boost-rotation, Kerr and Schwarzschild solutions are singular, having shown that a certain invariantly-defined acceleration scalar blows up at the horizons. Their examples do not satisfy the usual definition of a singularity. It is argued that using the same term is seriously misleading and it is shown that such divergent functions are natural concomitants of regular horizons. In particular it is noted that the divergence is given by the special relativistic approximation to the overall metric. Earlier work on characterization of horizons by invariants is revisited, a new invariant criterion for them is proposed, and the relation of the acceleration invariant to the Cartan invariants, which are finite at the horizons and completely determine the spacetimes, is examined for the C-metric, Kerr and Schwarzschild cases. An appendix considers coordinate identifications at axes and horizons.