In more mathematical terms, this requires the solution of an initial value problem: We provide appropriate initial data, describing the initial configuration of the matter and the geometry, on a hyperboloidal hypersurface , and appropriate boundary data, describing the incoming gravitational radiation, on the piece of which is in the future of . Then we have to show that there is a unique solution of the conformal field equations coupled to the matter equations which exists for some time. If the situation is ``close enough'' to a Newtonian situation, i.e., the gravitational waves are weak and the matter itself is rather ``tame'', then one would expect that there is a solution, i.e. a space-time, which is regular on arbitrary hyperboloidal hypersurfaces intersecting . In general, however, we cannot expect to have a regular point representing time-like infinity.

So far, results of this kind are out of reach. The reason is not so much the incorporation of matter into the conformal field equations but a more fundamental one. Space-like infinity is a singularity for the conformal structure of any space-time, which has a non-vanishing ADM-mass. Without the proper understanding of there will be no way to bridge the gap between past and future null-infinity because provides the link between the incoming and the outgoing radiation fields.

The results obtained so far are concerned only with the
*pure radiation problem*, i.e. the vacuum case. In [29] Christodoulou and Klainerman prove the global non-linear
stability of Minkowski space, i.e. the existence of global
solutions of the Einstein vacuum equations for ``small enough''
Cauchy data which satisfy certain fall-off conditions at
space-like infinity. Their result qualitatively confirms the
expectations based on the concept of asymptotic flatness.
However, they do not recover the peeling property for the Weyl
tensor but a weaker fall-off, which implies that in this class of
solutions the conformal compactification would not be as smooth
as it was expected to be. This raises the question whether their
results are sharp, i.e., whether there are solutions in this
class which indeed have their fall-off behaviour. In that case,
one would probably have to strengthen the fall-off conditions of
the initial data at space-like infinity in order to establish the
correct peeling of the Weyl tensor. Then an interesting question
arises as to what the physical meaning of these stronger fall-off
conditions is. An indication that maybe more restrictive
conditions are needed is provided by the analysis of the initial
data on hyperboloidal hypersurfaces (see below).

The first result [46] obtained with the conformal field equations is concerned with the asymptotic characteristic initial value problem (see Figure 7) in the analytic case. It was later generalized to the case.

In this kind of initial value problem, one specifies data on
an ingoing null hypersurface
and that part of
which is in the future of
. The data which have to be prescribed are essentially the
so-called
*null data*
on
and
, i.e. those parts of the rescaled Weyl tensor which are
entirely intrinsic to the respective null hypersurfaces. In the
case of
the null datum is exactly the radiation field.

**Theorem 3: **
*[Kánnár [88]] For given smooth null data on an ingoing null
hypersurface
and a smooth radiation field on the part
of
which is to the future of the intersection
S
of
with
and certain data on
S, there exists a smooth solution of Einstein's vacuum equations
in the future of
which implies the given data on
.
*

The result is in complete agreement with Sachs' earlier analysis of the asymptotic characteristic initial value problem based on formal expansion methods [128].

Another case is concerned with the existence of solutions representing pure radiation. These are vacuum solutions characterized by the fact that they are smoothly extensible through past time-like infinity, i.e. by the regularity of the point . This case has been treated in [49, 51]. A solution of this kind is uniquely characterized by its radiation field, i.e. the intrinsic components of the rescaled Weyl tensor on . In the analytic case, a formal expansion of the solution at can be derived, and growth conditions on the coefficients can be given to ensure convergence of the formal expansion near . Furthermore, there exists a surprising relation between this type of solutions and static solutions, summarized in

**Theorem 4: **
*[Friedrich] With each asymptotically flat static
solution of Einstein's vacuum field equations can be associated
another solution of these equations which has a smooth conformal
boundary
and for which the point
is regular.
*

This result establishes the existence of a large class of purely radiative solutions.

For applications, however, the most important type of initial
value problem so far, in the sense that the asymptotic behaviour
can be controlled, has been the
*hyperboloidal*
initial value problem where data are prescribed on a
hyperboloidal hypersurface. This is a space-like hypersurface
whose induced physical metric behaves asymptotically like a
surface of constant negative curvature (see Section
2.4). In the conformal picture, a hyperboloidal hypersurface is
characterized simply by the geometric fact that it intersects
*transversely*
in a two-dimensional space-like surface. Prototypes of such
hypersurfaces are the space-like hyperboloids in Minkowski
space-time. In the Minkowski picture they can be seen to become
asymptotic to null cones which suggests that they reach
null-infinity. However, the picture is deceiving: The conformal
structure is such that the hyperboloids always remain space-like,
the null-cones and the hyperboloids never become tangent. The
intersection is a two-dimensional surface
*S*, a ``cut'' of
. The data implied by the conformal fields on such a
hypersurface are called hyperboloidal initial data. The first
result obtained for the hyperboloidal initial value problem
states that if the space-time admits a hypersurface which extends
smoothly across
with certain smooth data given on it, then the smoothness of
will be guaranteed at least for some time into the future. This
is contained in

**Theorem 5: **
*[Friedrich [47]] Smooth hyperboloidal initial data on a hyperboloidal
hypersurface
determine a unique solution of Einstein's vacuum field equations
which admits a smooth conformal boundary at null-infinity in the
future of
.
*

There exists also a stability result which states that there are solutions which behave exactly like Minkowski space near future time-like infinity:

**Theorem 6: **
*[Friedrich [50]] If the hyperboloidal initial data are in a sense
sufficiently close to Minkowskian hyperboloidal data, then there
exists a conformal extension of the corresponding solution which
contains a point
such that
is the past null cone of that point.
*

It should be emphasized that this result implies that the
physical metric of the corresponding solution is regular
*for all future times*
. Thus, the theorem constitutes a (semi-)global existence result
for the Einstein vacuum equations.

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