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.

Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
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