Higher-dimensional perfect fluids and empty singular boundaries

TL;DR

This paper investigates higher-dimensional static perfect fluid solutions in gravity, revealing the existence of empty, repulsive singular boundaries and analyzing geodesic and scalar field behaviors near these boundaries.

Contribution

It introduces new higher-dimensional solutions with empty singular boundaries and studies their properties, including geodesic motion and scalar wave propagation.

Findings

Existence of empty, repulsive singular boundaries in higher dimensions.

Null geodesics either bounce or are bounded, depending on their orientation.

Scalar waves are fully reflected at the singularity without additional boundary conditions.

Abstract

In order to find out whether empty singular boundaries can arise in higher dimensional Gravity, we study the solution of Einstein's equations consisting in a ($N+2$)-dimensional static and hyperplane symmetric perfect fluid satisfying the equation of state $\rho=\eta\, p$, being $\rho$ an arbitrary constant and $N\geq2$. We show that this spacetime has some weird properties. In particular, in the case $\eta>-1$, it has an empty (without matter) repulsive singular boundary. We also study the behavior of geodesics and the Cauchy problem for the propagation of massless scalar field in this spacetime. For $\eta>1$, we find that only vertical null geodesics touch the boundary and bounce, and all of them start and finish at $z=\infty$; whereas non-vertical null as well as all time-like ones are bounded between two planes determined by initial conditions. We obtain that the Cauchy problem for the propagation of a massless scalar field is well-posed and waves are completely reflected at the singularity, if we only demand the waves to have finite energy, although no boundary condition is required.