On Pseudospherically Symmetric Repulsive Gravitational Field

Solutions of Einstein vacuum equations, for a static pseudospherically symmetric system, are presented. They describe a naked singularity and a singular solution with many resemblances to the Schwartzschild solution but with two major differences: its static region, lying inside the null horizon, sees the singularity, and its effective gravitational field is repulsive. We shortly discuss on the phenomenological plausibility of this last solution as a self-consistent system living on a space-time domain, and discuss some features of particle geodesics in its gravitational field. As a fundamental physical theory, General Relativity is well known for not imposing any constraints on spacetime geometry just but a Lorentzian-manifold structure and Einstein field equations. In specific problems, it may be appropriate to impose additional model-dependent constraints, such as a specific symmetry or asymptotic conditions on the geometry. Thus, the study of Einstein equations and their solutions have attracted so much attention from the point of view of numerical analysis [1] as well as exact solutions coupled to matter [2, 3], solutions in diverse dimensions [4], solitonic solutions [5], etc. In this contribution, we shall assume the existence of a region in the Universe having spatial pseudospherical symmetry. This region could be thought of as a kind of space-time domain possibly originated in an anisotropy occured at the Universe expansion. In any case, we shall solve under this hypothesis the Einstein vacuum equations inside the domain, reaching fairly interesting results. We obtain two possible singular solutions depending upon the sign of an integration constant ̟: (i) ̟ < 0 leads to a naked singularity, while (ii) ̟ > 0 gives rise to a singular space-time with a null-horizon and a repulsive gravitational field. The static region lies inside the horizon and, thus, sees the singularity. We shall analyse these singular solutions, study particle geodesics in the background metric of solution (ii), and argue on the self-consistency of the resulting effective repulsive gravitational field. Let us start by considering the Einstein vacuum equations in a space-time domain with spatial pseudospherical symmetry. The solution we shall develope, choosing a pseudospherical gauge, is CONICET