A trapped surface in the higher-dimensional self-similar Vaidya spacetime

TL;DR

This paper demonstrates the existence of trapped surfaces in higher-dimensional self-similar Vaidya spacetimes, establishing conditions for their formation and implications for naked singularities, with results extending previous four-dimensional analyses.

Contribution

It generalizes the analysis of trapped surfaces to D-dimensional spacetimes, identifying the critical mass function growth rate and linking trapped surfaces to the absence of naked singularities.

Findings

Trapped surfaces can extend into flat regions if the mass function grows faster than 0.4628.

The maximum radius of trapped surfaces approaches the Schwarzschild-Tangherlini radius as D increases.

No naked singularity exists if the trapped surface is present in the spacetime.

Abstract

We investigate a trapped surface and naked singularity in a $D$-dimensional Vaidya spacetime with a self-similar mass function. A trapped surface is defined as a closed spacelike $(D-2)$-surface which has negative both null expansions. There is no trapped surface in the Minkowski spacetime. However, in a four-dimensional self-similar Vaidya spacetime, Bengtsson and Senovilla considered non-spherical trapped surfaces and showed that a trapped surface can penetrate into a flat region, if and only if the mass function rises fast enough [I. Bengtsson and J. M. M. Senovilla, Phys. Rev. D \textbf{79}, 024027 (2009).]. We apply this result to a $D$-dimensional spacetime motivated by the context of large extra dimensions or TeV-scale gravity. In this paper, similarly to Bengtsson and Senovilla's study, we match four types of $(D-2)$-surfaces and show that a trapped surface extended into the flat region can be constructed in the $D$-dimensional Vaidya spacetime, if the increasing rate of the mass function is greater than 0.4628. Moreover, we show that the maximum radius of the trapped surface constructed here approaches the Schwarzschild-Tangherlini radius in the large $D$ limit. Also, we show that there is no naked singularity, if the spacetime has the trapped surface constructed here.