On the plane gravitational condensor with the positive gravitational constant

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

Section: Appendix B: Limit Casesmentioning

confidence: 81%

“…Hence, in this case, (25) turns out to be a vacuum solution with cosmological constant Λ = 2N (N +1) α; this is the higher-dimensional generalization of that found in [12].…”

Section: The Solutionmentioning

confidence: 87%

See 1 more Smart Citation

Higher-dimensional perfect fluids and empty singular boundaries

Saraví

2012

“…When C p = 0, it is clear from (11) that p(z) = −ρ, and the solution (17) turns out to be a vacuum solution with a cosmological constant = 8πρ [7,8] …”

Section: The Interior Solutionmentioning

confidence: 98%

Infinite slabs and other weird plane symmetric space–times with constant positive density

Saraví

2008

We present the exact solution of Einstein's equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below z = 0. This solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space-times finish down below at an inner singularity at finite depth. We show that for κ ≥ 0.3513 . . . the dominant energy condition is satisfied all over the space-time. We match this solution to the vacuum one and compute the external gravitational field in terms of slab's parameters. Depending on the value of κ, these slabs can be attractive, repulsive or neutral. In the first case, the space-time also finishes up above at an empty repelling singular boundary. In the other cases, they turn out to be semi-infinite and asymptotically flat when z → ∞. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss how to assemble a "gravitational capacitor" by inserting a slice of vacuum between two such slabs.

“…Taub [2], in 1951, found the general vacuum solution. A generalization to a spacetime with cosmological constant was found by Novotny and Horsky [3] in 1974, and the solution for a plane-symmetric scalar field was found by Singh [4] in the same year. The generalization of Singh's solution presented here can be shown to contain all previous solutions as special cases.…”

Section: Introductionmentioning

confidence: 95%

Exact solutions for the massless plane symmetric scalar field in general relativity, with cosmological constant

Vuille

2007

The Klein-Gordon equations are solved for the case of a planesymmetric static massless scalar field in general relativity with cosmological constant, generalizing the solutions found by Taub, Novotny and Horsky, and Singh. A separate class of solutions is obtained in which the metrics reduce to flat space in the limit that → 0. The static solutions can be used to generate time-dependent cosmological solutions, one of which exhibits rapid inflation followed by continued exponential expansion at all later times.