Evolving Geometries in General Relativity

Abstract: The problem of collisions of shockwaves in gravity is well known and has been studied extensively in the literature. Recently, the interest in this area has been revived trough the anti-de-Sitter space/Conformal Field Theory correspondence (AdS/CFT) with the difference that in this case the background geometry is Anti de Sitter in five dimensions. In a recent project that we have completed in the context of AdS/CFT, we have gained insight in the problem of shockwaves and our goal in this work is to apply the technique we have developed there in the case of ordinary gravity. In the current project, each of the shockwaves correspond to a point-like Stress-Energy tensor that moves with the speed of light while the collision is asymmetric and involves an impact parameter (b). Our method is to expand the metric $(g_{\mu \nu})$ in the background of flat space-time in the presence of the two shockwaves and compute corrections that satisfy causal boundary conditions taking into account back-reactions of the Stress-Energy tensor of the two point-like particles. Our solution respects causality as expected but this casual dependence takes place in an intuitive way. In particular, $g_{\mu \nu}$ at any given point $\vec{r}$ on the transverse plane at fixed $\tau$ evolves according from whether the propagation from the center of each of the shockwaves or from both shockwaves has enough proper time ($\tau$) to reach the point under consideration or not. Simultaneously around the center of each shockwave, the future metric develops a $\delta$-function profile with radius $\tau$; therefore this profile expands outwards from the centers (of the shockwaves) with the speed of light. Finally, we discuss the case of the zero impact parameter collision which results to the violation of conservation and we argue that this might be a signal for the formation of a black hole.