Definition 2: A conformal
space-time is a triple such that
(1) |
|
(2) |
|
(3) |
the gravitational
field |
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 that 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)
Let us assume that is a connection
on
that is compatible with the metric
so that
We introduce the zero-quantity
whereConsider 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 [56, 55]. 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 [63
], a well-known
tensor in general relativity that has certain positivity
properties.
The usefulness of the conformal field equations is documented in
Theorem 1: Suppose that
is compatible with
and that
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 (113
) of the trace-free
Ricci tensor. With these identifications the equations
respectively
do not yield
any further information because they are identically satisfied as a
consequence of the Bianchi identity on
, respectively
.
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 (114
) 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
, 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 [63], scalar
wave [90
, 91
], etc.).