"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 18dS 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." -- TDX.

Einstein's theory of gravity, general relativity, describes space and time together as one entity-spacetime. The governing equations couple together the evolution of spacetime and matter in a highly nonlinear fashion, making closed-form analytic solutions only available for simple systems with a high degree of symmetry. Computer simulations are used to study general relativity and the dynamics of matter in strongly gravitating systems. Numerical simulations of binary neutron star inspiral and coalescence are very challenging and current codes will be unable to produce accurate models for understanding observations as gravitational wave and electromagnetic experiments improve over the next few years. In the first two chapters of this thesis we develop and implement new numerical methods that we expect will reduce the cost of binary neutron star simulations by a factor of three to ten. This reduction in computational cost will no doubt be used to increase the accuracy of the simulations in order to meet the demands of new and ongoing experiments. In the second part of this thesis we study the formation of microscopic black holes. In the late 80's and early 90's Choptuik used numerical relativity to answer the question "What happens at the threshold of black hole formation?" Studying black hole formation requires resolving six orders of magnitude in space and time. This is why before the present work there were no detailed studies of critical behavior in 3d. This work contributes to understanding how the symmetry of the spacetime affects the threshold of black hole formation. In the last two chapters of this thesis we turn our attention to studying the stability of anti-de Sitter space against the formation of black holes. Anti-de Sitter spacetime has seen a lot of interest recently thanks to the anti-de Sitter/conformal field theory conjecture, which relates black hole formation in anti-de Sitter space to thermalization of the conformal field theory on the boundary of the spacetime. We study massive scalar fields in anti-de Sitter space by solving the Einstein equations numerically, finding new evidence for chaos and rich dynamics compared to the massless scalar field case. Finally, we use perturbation theory to probe arbitrarily small perturbations, which cannot be done by numerically solving the Einstein equations. We present evidence that anti-de Sitter space in more than four spacetime dimensions is unstable against black hole formation.

Einstein's theory of general relativity is a theory of gravity and, as in the earlier Newtonian theory, much can be learnt about the character of gravitation and its effects by investigating particular idealised examples. This book describes the basic solutions of Einstein's equations with a particular emphasis on what they mean, both geometrically and physically. Concepts such as big bang and big crunch-types of singularities, different kinds of horizons and gravitational waves, are described in the context of the particular space-times in which they naturally arise. These notions are initially introduced using the most simple and symmetric cases. Various important coordinate forms of each solution are presented, thus enabling the global structure of the corresponding space-time and its other properties to be analysed. The book is an invaluable resource both for graduate students and academic researchers working in gravitational physics.

This book introduces the modern field of 3+1 numerical relativity. The book has been written in a way as to be as self-contained as possible, and only assumes a basic knowledge of special relativity. Starting from a brief introduction to general relativity, it discusses the different concepts and tools necessary for the fully consistent numerical simulation of relativistic astrophysical systems, with strong and dynamical gravitational fields. Among the topics discussed in detail are the following: the initial data problem, hyperbolic reductions of the field equations, gauge conditions, the evolution of black hole space-times, relativistic hydrodynamics, gravitational wave extraction and numerical methods. There is also a final chapter with examples of some simple numerical space-times. The book is aimed at both graduate students and researchers in physics and astrophysics, and at those interested in relativistic astrophysics.

Spurred by the current development of numerous large-scale projects for detecting gravitational radiation, with the aim to open a completely new window to the observable Universe, numerical relativity has become a major field of research over the past years. Indeed, numerical relativity is the standard approach when studying potential sources of gravitational waves, where strong fields and relativistic velocities are part of any physical scenario. This book can be considered a primer for both graduate students and non-specialist researchers wishing to enter the field. Starting from the most basic insights and aspects of numerical relativity, Elements of Numerical Relativity develops coherent guidelines for the reliable and convenient selection of each of the following key aspects: evolution formalism, gauge, initial and boundary conditions as well as various numerical algorithms. The tests and applications proposed in this book can be performed on a standard PC.

This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this the perfect reference companion on the topic.

Many large-scale projects for detecting gravitational radiation are currently being developed, all with the aim of opening a new window onto the observable Universe. As a result, numerical relativity has recently become a major field of research, and Elements of Numerical Relativity and Relativistic Hydrodynamics is a valuable primer for both graduate students and non-specialist researchers wishing to enter the field. A revised and significantly enlarged edition of LNP 673 Elements of Numerical Relativity, this book starts with the most basic insights and aspects of numerical relativity before it develops coherent guidelines for the reliable and convenient selection of each of the following key aspects: evolution formalism; gauge, initial, and boundary conditions; and various numerical algorithms. And in addition to many revisions, it includes new, convenient damping terms for numerical implementations, a presentation of the recently-developed harmonic formalism, and an extensive, new chapter on matter space-times, containing a thorough introduction to relativistic hydrodynamics. While proper reference is given to advanced applications requiring large computational resources, most tests and applications in this book can be performed on a standard PC.

Numerical relativity has emerged as the key tool to model gravitational waves - recently detected for the first time - that are emitted when black holes or neutron stars collide. This book provides a pedagogical, accessible, and concise introduction to the subject. Relying heavily on analogies with Newtonian gravity, scalar fields and electromagnetic fields, it introduces key concepts of numerical relativity in a context familiar to readers without prior expertise in general relativity. Readers can explore these concepts by working through numerous exercises, and can see them 'in action' by experimenting with the accompanying Python sample codes, and so develop familiarity with many techniques commonly employed by publicly available numerical relativity codes. This is an attractive, student-friendly resource for short courses on numerical relativity, as well as providing supplementary reading for courses on general relativity and computational physics.

Aimed at students and researchers entering the field, this pedagogical introduction to numerical relativity will also interest scientists seeking a broad survey of its challenges and achievements. Assuming only a basic knowledge of classical general relativity, the book develops the mathematical formalism from first principles, and then highlights some of the pioneering simulations involving black holes and neutron stars, gravitational collapse and gravitational waves. The book contains 300 exercises to help readers master new material as it is presented. Numerous illustrations, many in color, assist in visualizing new geometric concepts and highlighting the results of computer simulations. Summary boxes encapsulate some of the most important results for quick reference. Applications covered include calculations of coalescing binary black holes and binary neutron stars, rotating stars, colliding star clusters, gravitational and magnetorotational collapse, critical phenomena, the generation of gravitational waves, and other topics of current physical and astrophysical significance.