Higher-dimensional perfect fluids and empty singular boundaries

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.Comment: 16 page

Conceptual design project: Accelerator complex for nuclear physics studies and boron neutron capture therapy application at the Yerevan Physics Institute (YerPhI) Yerevan, Armenia

“…The first one corresponds to the Rindler spacetime and the second one is a higher-dimensional generalization of the well known Taub solution. By an appropriate choice of the integration constants the latter is given by ( 16) (see also [27]). It has a singularity at x = 1/σ that presents a repulsive wall for test particles.…”

“…( 26). The surface energy-momentum tensor on the separating brane is given by (27). The general solutions of the field equations for negative and positive CC are given by ( 30) and (34), respectively.…”

“…For D = 3 it is reduced to the Taub solution in General Relativity. The higher dimensional generalization of the Taub solution has also been considered in [27]. For a test particle at rest with the coordinate x, the acceleration in the geometry ( 16) is expressed as…”