Obviously, there exists a vast amount of material related to the subject of conformal infinity which cannot be covered adequately within this review article. A choice has to be made. We will discuss here those issues of ``conformal infinity'' which are relevant for numerical applications. On the one hand, this is a restriction to a subtopic which 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 which 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 which 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 which 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 which have been 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 which 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 which 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 survey the most important results relevant for the numerical application. The last section 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.
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