1 Introduction

The notion of “conformal infinity” introduced by Penrose almost forty years ago is one of the most fruitful concepts within Einstein’s theory of gravitation. Most of the modern developments in the theory are based on or at least influenced in one way or another by the conformal properties of Einstein’s equations in general or, in particular, by the structure of null infinity

the study of radiating solutions of the field equations and the question of fall-off conditions for them;
the global structure of space-times;
the structure of singularities;
conserved quantities;
multipole moments;
asymptotic quantization procedures;
$H$ -spaces and the twistor programme;
the null hypersurface formulation of general relativity;
the conformal field equations and their importance for the numerical evolution of space-times.

Obviously, there exists a vast amount of material related to the subject of conformal infinity that cannot be covered adequately within this review article. A choice has to be made. We will discuss here those issues of “conformal infinity” that are relevant for numerical applications. On the one hand, this is a restriction to a subtopic that is reasonably narrow and yet broad enough to encompass the central ideas and new developments. On the other hand, questions concerning the numerical treatment of gravitational radiation and, in particular, the problems that arise from the attempt to numerically model infinitely extended systems, suggest that the conformal methods can be useful not only for rigorous arguments but also for numerical purposes.

Indeed, we will show that the conformal picture has matured enough to provide an approach to applications in numerical relativity that relies on a very sound theoretical basis not only with respect to the physical appropriateness, but also with respect to the mathematical well-posedness of the problems considered. What is even more remarkable is the fact that the numerical implementations of the conformal picture via the conformal field equations are numerically well-defined in the sense that there are no spurious instabilities in the codes (which so often are the stumbling blocks for the traditional approaches via the ADM equations), so that the computed solutions demonstrably converge to the order of the discretization scheme. The conformal approach based on the hyperboloidal initial-value problem allows us to compute (semi-)global space-times including their asymptotic structure which in turn enables us to rigorously, i.e. without any further approximation beyond the discretization, determine the radiation coming out from the system under consideration. The work that has been devoted to this approach up to now clearly shows its power. The results obtained have been checked against exact results (exact solutions or known theorems), and there is no doubt that the results are correct. The geometric concepts devised by Penrose now turn out to be very useful in practical applications in the sense that they provide the solution to all conceptual problems posed by the notion of “gravitational radiation” in connection with numerical computations. Already, it is obvious that with this tool one can achieve results that have not been feasible by any other numerical method. Furthermore, the analysis at space-like infinity (see Section 3.5) has finally put within reach the feasibility of computing the global structure of asymptotically flat space-times from asymptotically flat Cauchy data.

This article will begin with a survey of the general background, i.e. the physical motivations behind the idea of asymptotically flat space-times, and a short account of the historical development that led to our current understanding. Next, the necessary mathematical ideas will be introduced and the regular conformal field equations will be discussed. The equations have been applied to several initial value problems. We will survey the most important results relevant for numerical applications. The last Section 4 is concerned with current issues in the numerical implementation of the conformal methods. We will see that the conformal field equations provide a very powerful method to study global problems in numerical relativity, such as gravitational wave propagation and detection as well as the emergence of singularities and their horizons.