Nakedness and curvature strength of shell-focusing singularity in the spherically symmetric space-time with vanishing radial pressure

TL;DR

This paper analyzes the nature and strength of shell-focusing singularities in spherically symmetric spacetimes with zero radial pressure, establishing relations between geometric parameters and curvature conditions, and exploring the implications of the gravity dominance condition.

Contribution

It provides explicit integration of metric functions, characterizes the nakedness and curvature strength of singularities, and introduces the gravity dominance condition in this context.

Findings

The relation R ≈ 2y₀ m^β characterizes naked singularities.

Strong curvature condition (SCC) holds only when β=1.

Gravity dominance condition (GDC) determines the validity of SCC and LFC for null and timelike geodesics.

Abstract

It was recently shown that the metric functions which describe a spherically symmetric space-time with vanishing radial pressure can be explicitly integrated. We investigate the nakedness and curvature strength of the shell-focusing singularity in that space-time. If the singularity is naked, the relation between the circumferential radius and the Misner-Sharp mass is given by $R\approx 2y_{0} m^{\beta}$ with $ 1/3<\beta\le 1$ along the first radial null geodesic from the singularity. The $\beta$ is closely related to the curvature strength of the naked singularity. For example, for the outgoing or ingoing null geodesic, if the strong curvature condition (SCC) by Tipler holds, then $\beta$ must be equal to 1. We define the ``gravity dominance condition'' (GDC) for a geodesic. If GDC is satisfied for the null geodesic, both SCC and the limiting focusing condition (LFC) by Kr\'olak hold for $\beta=1$ and $y_{0}\ne 1$, not SCC but only LFC holds for $1/2\le \beta <1$, and neither holds for $1/3<\beta <1/2$, for the null geodesic. On the other hand, if GDC is satisfied for the timelike geodesic $r=0$, both SCC and LFC are satisfied for the timelike geodesic, irrespective of the value of $\beta$. Several examples are also discussed.