Geometrical properties of the trans-spherical solutions in higher dimensions

TL;DR

This paper explores the geometric features of higher-dimensional static vacuum p-brane solutions in Einstein gravity, revealing how key properties depend on combined tension parameters and comparing them to five-dimensional cases.

Contribution

It provides a detailed analysis of the geometrical properties of higher-dimensional p-brane solutions, highlighting their dependence on combined tension parameters and extending understanding beyond five dimensions.

Findings

Geometric properties depend on mass density and combined tension parameters.

Most solutions feature naked singularities, except for specific black p-branes and bubbles.

Comparison with five-dimensional solutions shows similar causal structures.

Abstract

We investigate the geometrical properties of static vacuum $p$-brane solutions of Einstein gravity in $D=n+p+3$ dimensions, which have spherical symmetry of $S^{n+1}$ orthogonal to the $p$-directions and are invariant under the translation along them. % The solutions are characterized by mass density and $p$ tension densities. % The causal structure of the higher dimensional solutions is essentially the same as that of the five dimensional ones. Namely, a naked singularity appears for most solutions except for the Schwarzschild black $p$-brane and the Kaluza-Klein bubble. % We show that some important geometric properties such as the area of $S^{n+1}$ and the total spatial volume are characterized only by the three parameters such as the mass density, the sum of tension densities and the sum of tension density squares rather than individual tension densities. These geometric properties are analyzed in detail in this parameter space and are compared with those of 5-dimensional case.