Numerical Relativity studies in Anti-de Sitter spacetimes: Gravitational Collapse and the AdS/CFT correspondence

In this thesis we study several open problems using Numerical Relativity on asymptotically Anti-de Sitter (AdS) spacetimes. The understanding of the dynamics of AdS is interesting not only because of pure theoretical reasons but also because of its importance in the correspondence gauge/gravity. In the thesis we present three different topics. The first is our research on the gravitational collapse of massless scalar fields in AAdS spacetimes. We have developed a new method that combines two different formulations of the Einstein Field Equations to get closer and with more accuracy to the collapse. The simulation starts with a Cauchy evolution with pseudo-spectral methods and when the collapse is taking place, it performs a change of coordinates to a characteristic one to track the formation of the apparent horizon. The collapse of the scalar field happens after a number of bounces with the critical points being the separation between the different branches. We have numerical evidence that in the separation of the branches there is a power law for subcritical configurations in addition to the one for supercritical ones. This new power law confirms that there is a gap in the mass of the apparent horizon. In the second part, we introduce a shock waves model in AdS to study the far-from-equilibrium regime in the heavy ion collisions through the holographic correspondence in a non-conformal theory. Holographic collisions have attracted a lot of attention in the last few years because of the possibility of simulating strongly coupled systems but, as a drawback, we do not know yet the exact dual of the QCD that should explain the phenomena. In the models used until now, the shock waves correspond to conformal gauge theories while QCD is not conformal. In order to get closer to a description of the actual physical collisions we present the first shock wave collisions in a non-conformal theory. With this, we show how the non-conformality increases the hydrodynamisation time and also that this can happen before the equation of state is fulfilled. In the last part, we propose the use of spectral methods as a very strong option for high precision computations. Arbitrary precision arithmetic has two main problems. The first is the necessity of increasing a lot the discretisation units to reach the precision we want. The other one is the slowing down in the computational performance due to the fact that we need to emulate the fundamental operations with software because current processors are not adapted to carry out computations with precision different from the standard one. The exponential convergence of spectral methods can approximate functions to a very high accuracy with a few hundred terms in our spectral expansion while in other numerical methods it would be a few orders of magnitude larger. This makes these methods very attractive because they facilitate the accessibility to very small error simulations, removes the bottleneck of the memory demand and also help in the computational speed because fewer points are needed for the computation. We have tested this idea with the ANETO library for simulations in AdS spacetimes and the gravitational collapse in an asymptotically flat spacetime with very promising results. This library has been developed as a direct result of this thesis and that can be downloaded as Free Software.