on
, where
is the metric induced on
by the physical metric. Furthermore, let
be the extrinsic curvature of
in the physical space-time. Together with
it satisfies the vacuum constraint equations,
where
is the Levi-Civita connection of
and
is its scalar curvature.
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 (31) implies that
c
is constant while the hyperboloidal character of
implies that
. With these simplifications and a rescaling of
with a constant factor, the Hamiltonian constraint (30
) takes the form
A further consequence of the condition (29) is the vanishing of the magnetic part
of the Weyl tensor. For any defining function
of the boundary, the conformal factor has the form
. Expressing Equation (32
) in terms of the unphysical quantities
and
yields the single second-order equation
This equation is a special case of the Lichnérowicz equation
and is sometimes also referred to as the Yamabe equation. For a
given metric
and boundary defining function
it is a second-order, non-linear equation for the function
. Note that the principal part of the equation degenerates on
the boundary. Therefore, on the boundary, the Yamabe equation
degenerates to the relation
Note also that
is a solution of (33
) which, however, is not useful for our purposes because it would
correspond to a conformal factor with vanishing first derivative
on
. Therefore, we require that
be non-vanishing on the boundary, i.e. bounded from below by a
strictly positive constant. Then the relation above determines
the boundary values of
in terms of the function
. Taking derivatives of Equation (33
), 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 (33
) 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 (33
) 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
which follow from the Equations (20) and (16
), respectively. Together with the fields
,
,
they provide initial data for all the quantities appearing in
the evolution equations under the given assumptions. As they
stand, these expressions are formally singular at the boundary
and one needs to worry about the possibility of a smooth
extension of the field to
. This question was answered in [4], where the following theorem was proved:
Theorem 7:
Suppose
is a three-dimensional, orientable, compact, smooth Riemannian
manifold with boundary
. Then there exists a unique solution
of (33
), and the following conditions are equivalent:
Condition (
3
) is a weak restriction of the conformal class of the metric
h
on
, since it is only effective on the boundary. It is equivalent
to the fact that in the space-time which 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 [2,
3] the assumption (29
) is dropped allowing for an extrinsic curvature which is almost
general apart from the fact that the mean curvature is required
to be constant. In [85] 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 [87] 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 (33) has been solved. The data are given by expressions which 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 the theorem 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 (79
,
80
,
81
,
82
,
83
,
84
,
85
,
86
,
87
,
88
,
89
) that the conformal constraints are regular as well. Some of the
equations are rather simple but the overall dependencies and
interrelations between 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
) of the theorem 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 looses 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 [2] 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 [148] on the smoothness on
, 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 which are semi-global in the sense that
their future (or past) development is determined. 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
) 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 which are general
enough. There exist exact radiative solutions with boost-rotation
symmetry [20]. 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 [13]. 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 which are solutions of the Einstein-Maxwell [32] or Einstein-Yang-Mills [14] equations. They have smooth and complete null-infinities.
However, they were constructed in a way which 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
.
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Conformal Infinity
Jörg Frauendiener http://www.livingreviews.org/lrr-2000-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |