Definition 2:
A
conformal space-time is a triple
such that
(i) |
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(ii) |
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(iii) |
the gravitational field
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Two conformal space-times
and
are
equivalent
if
and
are diffeomorphic and if, after identification of
and
with a suitable diffeomorphism, there exists a strictly positive
scalar field
on
such that
and
.
From this definition follows that
is an open sub-manifold of
on which a metric
is defined, which is invariant in the sense that two equivalent
conformal space-times define the same metric
.
The space-time
allows the attachment of a conformal boundary which is given by
. The above definition of conformal space-times admits much more
general situations than those arising from asymptotically flat
space-times; this generality is sometimes needed for numerical
purposes.
Under the conditions of Definition
2, it follows that the Weyl tensor vanishes on
because the gravitational field (i.e. the rescaled Weyl tensor)
is smooth on
. Note that we make no assumptions about the topology of
. If each null geodesic which starts from the inside of
has a future and a past endpoint on
, then
is asymptotically simple in the sense of Definition
1
. If, in addition, the metric
is a vacuum metric then
has the implied topology
. Note also that it is quite possible to have situations, where
is a vacuum metric and where the topology of
is not
, but e.g.
. Then, necessarily, there must exist null geodesics which do
not reach
.
In the special case when
is empty, the conformal factor
is strictly positive, i.e.
, and the conformal space-time is isometric to the physical
space-time (choosing
).
Our goal is to express the vacuum equations in
in terms of geometric quantities on the unphysical space-time.
Consider first the Einstein vacuum equation for the metric
. When expressed in terms of unphysical quantities it reads (see
the formulae of Appendix
7)
This equation can be interpreted as the Einstein equation for
the metric
with a source term which is determined by the conformal factor.
If we assume
to be known, then it is a second order equation for
, which is formally singular on
, where
vanishes. Therefore, it is very hard to make any progress
towards the existence problem using this equation. To remedy this
situation, Friedrich [43,
44
,
45
] suggested to consider a different system of equations on
which can be derived from the geometric structure on
, the conformal transformation properties of the curvature and
the vacuum Einstein equation on
. It consists of equations for a connection
, its curvature and certain other fields obtained from the
curvature and the conformal factor.
Let us assume that
is a connection on
which is compatible with the metric
so that
holds. This condition does not fix the connection. Let
and
denote the torsion and curvature tensors of
. We will write down equations for the following unknowns:
where
is the torsion tensor of
and the other components of
Z
are defined in terms of the unknowns by
In addition, we consider the scalar field
on
. The equations
Z
=0 are the
regular conformal vacuum field equations
. They are first order equations. In contrast to Equation (13
) this system is regular
on
, even on
because there are no terms containing
.
Consider the equation
. This subsystem lies at the heart of the full system of
conformal field equations because it feeds back into all the
other parts. It was pointed out in Section
2.2
that the importance of the Bianchi identity had been realized by
Sachs. However, it was first used in connection with uniqueness
and existence proofs only by Friedrich [45,
44]. Its importance lies in the fact that it splits naturally into
a symmetric hyperbolic system of evolution equations
and constraint equations. Energy estimates for the symmetric
hyperbolic system naturally involve integrals over a certain
component of the Bel-Robinson tensor [52
], a well known tensor in general relativity which has certain
positivity properties.
The usefulness of the conformal field equations is documented in
Theorem 1:
Suppose that
is compatible with
and that
Z
=0 on
. If
at one point of
, then
everywhere and, furthermore, the metric
is a vacuum metric on
.
Proof:
From the vanishing of the torsion tensor it follows that
is the Levi-Civita connection for the metric
. Then,
is the decomposition of the Riemann tensor into its irreducible
parts which implies that the Weyl tensor
, that
is the trace-free part of the Ricci tensor, and that
. The equation
defines
in terms of
, and the trace of the equation
defines
. The trace-free part of that equation is the statement that
, which follows from the conformal transformation
property (110
) of the trace-free Ricci tensor. With these identifications the
equations
resp.
do not yield any further information because they are
identically satisfied as a consequence of the Bianchi identity on
, resp.
.
Finally, we consider the field
. Taking its derivative and using
and
, we obtain
. Hence,
vanishes everywhere if it vanishes at one point. It follows from
the transformation (111
) of the scalar curvature under conformal rescalings that
implies
. Thus,
is a vacuum metric.
It is easy to see that the conformal field equations are invariant under the conformal rescalings of the metric specified in Definition 2 and the implied transformation of the unknowns. The conformal invariance of the system implies that the information it contains depends only on the equivalence class of the conformal space-time.
The reason for the vanishing of the gradient of
is essentially this: If we impose the equation
for the trace-free part of the Ricci tensor of a manifold, then
by use of the contracted Bianchi identity we obtain
. Expressing this in terms of unphysical quantities leads to the
reasoning in Theorem
1
. The special case
reduces to the standard vacuum Einstein equations, because then
we have
and
. Then
implies
and
S
=0, while
forces
. The other equations are identically satisfied.
Given a smooth solution of the conformal field equations on a
conformal manifold, Theorem
1
implies that on
we obtain a solution of the vacuum Einstein equation. In
particular, since the Weyl tensor of
vanishes on
due to the smoothness of the gravitational field, this implies
that the Weyl tensor has the peeling property in the physical
space-time. Therefore, if existence of suitable solutions of the
conformal field equations on a conformal manifold can be
established, one has automatically shown existence of
asymptotically flat solutions of the Einstein equations. The main
advantage of this approach is the fact that the conformal
compactification supports the translation of global problems into
local ones.
Note that the use of the conformal field equations is not
limited to vacuum space-times. It is possible to include matter
fields into the conformal field equations provided the equations
for the matter have well-defined and compatible conformal
transformation properties. This will be the case for most of the
interesting fundamental field equations (Maxwell,
Yang-Mills [52], scalar wave [77
,
78
] etc.)
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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 |