Boosted Horizon of a Boosted Space-Time Geometry

We apply the ultrarelativistic boosting procedure to map the metric of Schwarzschild-de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface, by exploiting the picture of the embedding of an hyperboloid in a five-dimensional Minkowski spacetime. After reverting to the usual four-dimensional formalism, we also solve the geodesic equation and evaluate the Riemann curvature tensor of the boosted Schwarzschild-de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Eventually, the analysis of the Kretschmann invariant (and of the geodesic equation) shows the global structure of space- time, as we demonstrate the presence of a "scalar curvature singularity" within a 3-sphere and find that it is also possible to define what we have called "boosted horizon", a sort of elastic wall where all particles are surprisingly pushed away. This seems to suggest that such "boosted geometries" are ruled by a sort of "antigravity effect" since all geodesics seem to refuse entering the "boosted horizon" and are "reflected" by it, even though their initial conditions are aimed at driving the particles towards the "boosted horizon" itself.