Definition 2: A conformal space-time is a triple such that
(i) | is a (time- and space-orientable) Lorentz manifold; |
(ii) | is a smooth scalar field on such that the set is non-empty and connected; |
(iii) | the gravitational field extends smoothly to all of . |
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.)
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 |