Elements of Numerical Relativity

Abstract

We became involved with numerical relativity under very different circumstances. For one of us (C.B.) it dates back to about 1987, when the current Laser-Interferometer Gravitational Wave Observatories were just promising proposals. It was during a visit to Paris, at the Institut Henri Poincar´e, where some colleagues were pushing the VIRGO proposal with such a contagious enthusiasm that I actually decided to reorient my career. The goal was to be ready, armed with a reliable numerical code, when the first detection data would arrive. Allowing for my experience with the 3+1 formalism at that time, I started working on singularity-avoidant gauge conditions. Soon, I became interested in hyperbolic evolution formalisms. When trying to get some practical applications, I turned to numerical algorithms (a really big step for a theoretically oriented guy) and black hole initial data. More recently, I became interested in boundary conditions and, closing the circle, again in gauge conditions. The problem is that a reliable code needs all these ingredients to be working fine at the same time. It is like an orchestra, where strings, woodwinds, brass and percussion must play together in a harmonic way: a violin virtuoso, no matter how good, cannot play Vivaldi’s Four Seasons by himself. During that time, I have had many Ph.D. students. The most recent one is the other of us (C.P.). All of them started with some specific topic, but they needed a basic knowledge of all the remaining ones: you cannot work on the saxophone part unless you know what the bass is supposed to play at the same time. This is where this book can be of a great help. Imagine a beginning graduate student armed only with a home PC. Imagine that the objective is to build a working numerical code for simple black-hole applications. This book should first provide him or her with a basic insight into the most relevant aspects of numerical relativity. But this is not enough; the book should also provide reliable and compatible choices for every component: evolution system, gauge, initial and boundary conditions, even for numerical algorithms. This pragmatic orientation may cause this book to be seen as biased. But the idea was not to produce a compendium of the excellent work that has been made in numerical relativity during these years. The idea is rather to present a well-founded and convenient way for a beginner to get into the field. He or she will Quickly discover everything else. The structure of the book reflects the peculiarities of numerical relativity research: • It is strongly rooted in theory. Einstein’s relativity is a general-covariant theory. This means that we are building at the same time the solution and the coordinate system, a unique fact among physical theories. This point is stressed in the first chapter, which could be omitted by more experienced readers. • It turns the theory upside down. General covariance implies that no specific coordinate is more special than the others, at least not a priori. But this is at odds with the way humans and computers usually model things: as functions (of space) that evolve in time. The second chapter is devoted to the evolution (or 3+1) formalism, which reconciles general relativity with our everyday perception of reality, in which time plays such a distinct role. • It is a fertile domain, even from the theoretical point of view. The structure of Einstein’s equations allows many ways of building well-posed evolution formalisms. Chapter 3 is devoted to those which are of first order in time but second order in space. Chapter 4 is devoted instead to those which are of first order both in time and in space. In both cases, suitable numerical algorithms are provided, although the most advanced ones apply mainly to the fully first order case. • It is challenging. The last sections of Chaps. 5 and 6 contain frontedge developments on constraint-preserving boundary conditions and gauge pathologies, respectively. These are very active research topics, where new developments will soon improve on the ones presented here. The prudent reader is encouraged to look for updates of these front-edge areas in the current scientific literature.