Several results of increasing generality have been obtained. We discuss only the simplest case here, referring to the literature for the more general results. Assume that the extrinsic curvature is a pure trace term,
The momentum constraint (32) implies that
is constant while the hyperboloidal character of
implies that
. With these simplifications
and a rescaling of
with a constant factor, the Hamiltonian
constraint (31
) takes the form
Note also that is a solution of
Equation (34
) which, however, is
not useful for our purposes because it does not give rise to a
meaningful conformal factor
. Therefore, we require that
be strictly positive and bounded on the boundary.
Then the relation above determines the boundary values of
in terms of the function
. Taking derivatives of
Equation (34
), one finds that also
the normal derivative of
is fixed on the boundary in
terms of the second derivative of
.
A given metric does not fix a
unique pair
. Therefore, Equation (34
) has the property
that, for fixed
, rescaling the metric
with an arbitrary smooth non-vanishing function
on
according to
results in a rescaling of the solution
of Equation (34
) according to
and, hence, a change in the conformal factor
.
Now we define the trace-free part of the projection
of the trace-free part of the unphysical Ricci
tensor onto
, and consider the equations
Theorem 7: Suppose is a three-dimensional,
orientable, compact, smooth Riemannian manifold with boundary
. Then there exists a unique solution
of Equation (34
), and the following conditions are equivalent:
Condition (3) in
Theorem 7
is a weak restriction on the conformal class of the metric
on
, since it is only effective on the boundary. It is
equivalent to the fact that in the space-time that evolves from the
hyperboloidal data, null-infinity
is shear-free.
Interestingly, the theorem only requires
to be orientable and
does not restrict the topology of
any further.
This theorem gives the answer in a highly
simplified case because the freedom in the extrinsic curvature has
been suppressed. But there are also several other, less
restrictive, treatments in the literature. In [3, 4] the
assumption (30
) is dropped, allowing
for an extrinsic curvature that is almost general apart from the
fact that the mean curvature is required to be constant.
In [101] also this
requirement is dropped (but, in contrast to the other works, there
is no discussion of smoothness of the implied conformal initial
data), and in [103] the existence of
hyperboloidal initial data is discussed for situations with a
non-vanishing cosmological constant.
The theorem states that one can construct the
essential initial data for the evolution once Equation (34) has been solved. The
data are given by expressions that are formally singular at the
boundary because of the division by the conformal factor
. This is of no consequences for the analytical
considerations if Condition (3) in
Theorem 7
is satisfied. However, even then it is a problem for the numerical
treatments because one has to perform a limit process to get to the
values of the fields on the boundary. This is numerically
difficult. Therefore, it would be desirable to solve the conformal
constraints directly. It is clear from Equations (82
, 83
, 84
, 85
, 86
, 87
, 88
, 89
, 90
, 91
, 92
) that the conformal
constraints are regular as well. Some of the equations are rather
simple but the overall dependencies and interrelations among the
equations are very complicated. At the moment there exists no clear
analytical method (or even strategy) for solving this system. An
interesting feature appears in connection with Condition (3) in Theorem 7 and analogous conditions in the
more general cases. The necessity of having to impose this
condition seems to indicate that the development of hyperboloidal
data is not smooth but in general at most
. If the condition were not imposed then logarithms
appear in an expansion of the solution of the Yamabe equation near
the boundary, and it is rather likely that these logarithmic terms
will be carried along with the time evolution, so that the
developing null-infinity loses differentiability. Thus, the
conformal boundary is not smooth enough and, consequently, the Weyl
tensor need not vanish on
which, in addition, is not
necessarily shear-free. The Sachs peeling property is not
completely realized in these situations. One can show [3] that
generically hyperboloidal data fall into the class of
“poly-homogeneous” functions, which are (roughly) characterized by
the fact that they allow for asymptotic expansions including
logarithmic terms. This behaviour is in accordance with other
work [166] on the smoothness of
, in particular with the Bondi-Sachs type expansions,
which were restricted by the condition of analyticity (i.e. no
appearance of logarithmic terms). It is also consistent with the
work of Christodoulou and Klainerman.
Solutions of the hyperboloidal initial value
problem provide pieces of space-times that are semi-global in the
sense that their future or past developments are determined
depending on whether the hyperboloidal hypersurface intersects or
. However, the domain of dependence of a
hyperboloidal initial surface does not include space-like infinity
and one may wonder whether this fact is the reason for the apparent
generic non-smoothness of null-infinity. Is it not conceivable that
the possibility of making a connection between
and
across
to build up a
global space-time automatically excludes the non-smooth data? If we
let the hyperboloidal initial surface approach space-like infinity,
it might well be that Condition (3) in Theorem 7 imposes additional conditions
on asymptotically flat Cauchy data at spatial infinity. These
conditions would make sure that the development of such Cauchy data
is an asymptotically flat space-time, in particular that it has a
smooth conformal extension at null-infinity.
These questions give some indications about the
importance of gaining a detailed understanding of the structure of
gravitational fields near space-like infinity. One of the
difficulties in obtaining more information about the structure at
space-like infinity is the lack of examples that are general
enough. There exist exact radiative solutions with boost-rotation
symmetry [22]. They possess a
part of a smooth null-infinity, which, however, is incomplete. This
is a general problem because the existence of a complete
null-infinity with non-vanishing radiation restricts the possible
isometry group of a space-time to be at most one-dimensional with
space-like orbits [14]. Some of the
boost-rotation symmetric space-times even have a regular
; thus they have a vanishing ADM mass. Other examples
exist of space-times that are solutions of the
Einstein-Maxwell [39] or
Einstein-Yang-Mills [15] equations. They
have smooth and complete null-infinities. However, they were
constructed in a way that enforces the field to coincide with the
Schwarzschild or the Reissner-Nordström solutions near
. So they are not general enough to draw any
conclusions about the generic behaviour of asymptotically flat
space-times near
.
Recently, the still outstanding answer to the
question as to whether there exist at all global asymptotically
simple vacuum space-times with smooth null-infinity could be
answered in the affirmative. Corvino [35] has shown that
there exist asymptotically flat metrics on
with vanishing scalar curvature that are spherically
symmetric outside a compact set. His method of construction
consists of a gluing procedure by which a scalar flat and
asymptotically flat metric can be deformed in an annular region in
such a way that it can be glued with a predetermined degree of
smoothness to Schwarzschild data while remaining scalar flat. Very
recently a generalisation of this construction to
non-time-symmetric initial data has been presented by Corvino and
Schoen [36
].
Such a metric satisfies the constraint equations
implied by the vacuum Einstein equations on a space-like
hypersurface of time symmetry. It evolves into a space-time that is
identical to the Schwarzschild solution near space-like infinity.
Within this space-time, which exists for some finite time interval,
one can now find hyperboloidal hypersurfaces on which hyperboloidal
initial data are induced. This hypersurface can be chosen close
enough to the initial Cauchy surface so that it intersects the
domain of dependence of the asymptotic region of the initial
hypersurface where the data are Schwarzschild. This means that the
data implied on the hyperboloidal hypersurface are also
Schwarzschild close to so that the conditions of
Theorem 7
are clearly satisfied.
If one could now apply Friedrich’s stability
result (Theorem 6), then the existence of
global asymptotically flat space-times with smooth extension to
null-infinity would have been established. However, one assumption
in this theorem is that the data be “sufficiently small”.
Therefore, one needs to check that one can in fact perform
Corvino’s construction in the limit of vanishing ADM mass while
maintaining a sufficiently “large” Schwarzschild region around so that one can find enough hyperboloidal
hypersurfaces intersecting the Schwarzschild region in the time
development of the data. ChruĊciel and Delay [32]
(see also the erratum in [33])
have adapted Corvino’s construction to that situation, and show
that it is possible to construct time-symmetric initial data that
are Schwarzschild outside a fixed compact region and with a fixed
degree of differentiability in the limit
. From
these data one can construct hyperboloidal hypersurfaces of the
type required in Theorem 6 on which hyperboloidal
data are induced which satisfy the smallness criterion of that
theorem. Hence, there exists a complete space-time in the future of
the initial Cauchy surface that admits a conformal extension to
null-infinity as smooth as one wishes. Since the argument can also
be applied towards the past, one has shown the existence of global
space-times with smooth
.