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
that 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 that 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, that 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 that 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 [31] 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 that 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 that 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 [57] 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
that is in the future of
. The data that have to be prescribed are essentially
the so-called null data on
and
, i.e. those parts of the rescaled Weyl tensor
that are entirely intrinsic to the respective null hypersurfaces.
In the case of
, the null datum is exactly the
radiation field.
Theorem 3 (Kánnár [104]):
For given smooth null data on an ingoing null
hypersurface and a
smooth radiation field on the part
of
that is to the future of the
intersection
of
with
and certain data on
, there exists a smooth
solution of Einstein’s vacuum equations in the future of
that 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 [145].
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 [60, 62]. 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 that 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
, 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 that 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 [58]):
Smooth hyperboloidal initial data on a
hyperboloidal hypersurface determine a unique solution of Einstein’s vacuum
field equations that admits a smooth conformal boundary
at null-infinity in the future of
.
There exists also a stability result that states that there are solutions that behave exactly like Minkowski space near future time-like infinity:
Theorem 6 (Friedrich [61]):
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.