Empty singularities in higher-dimensional Gravity

TL;DR

This paper presents an exact higher-dimensional Einstein solution describing a thick matter slice with positive energy density, revealing an empty singular boundary and conditions for energy compliance.

Contribution

It provides a new exact solution in higher-dimensional gravity with detailed analysis of pressure, gravitational fields, and energy conditions, highlighting an empty singular boundary feature.

Findings

Solution with positive energy density and bounded pressure

Existence of an empty singular boundary in one vacuum

Conditions under which the dominant energy condition holds

Abstract

We study the exact solution of Einstein's field equations consisting of a ($n+2$)-dimensional static and hyperplane symmetric thick slice of matter, with constant and positive energy density $\rho$ and thickness $d$, surrounded by two different vacua. We explicitly write down the pressure and the external gravitational fields in terms of $\rho$ and $d$, the pressure is positive and bounded, presenting a maximum at an asymmetrical position. And if $\sqrt{\rho}\,d$ is small enough, the dominant energy condition is satisfied all over the spacetime. We find that this solution presents many interesting features. In particular, it has an empty singular boundary in one of the vacua.