3.2 The reduction process for 3 The Regular Conformal Field 3 The Regular Conformal Field

3.1 General properties of the conformal field equations 

Before deriving the equations we need to define the arena where the discussion is taking place.

Definition 2:  A conformal space-time is a triple tex2html_wrap_inline4216 such that

(i) tex2html_wrap_inline3885 is a (time- and space-orientable) Lorentz manifold;
(ii) tex2html_wrap_inline3721 is a smooth scalar field on tex2html_wrap_inline3889 such that the set tex2html_wrap_inline4224 is non-empty and connected;
(iii) the gravitational field tex2html_wrap_inline4226 extends smoothly to all of tex2html_wrap_inline3889 .

Two conformal space-times tex2html_wrap_inline4216 and tex2html_wrap_inline4232 are equivalent if tex2html_wrap_inline3889 and tex2html_wrap_inline4236 are diffeomorphic and if, after identification of tex2html_wrap_inline3889 and tex2html_wrap_inline3887 with a suitable diffeomorphism, there exists a strictly positive scalar field tex2html_wrap_inline3875 on tex2html_wrap_inline3889 such that tex2html_wrap_inline3873 and tex2html_wrap_inline4248 .  

From this definition follows that tex2html_wrap_inline3887 is an open sub-manifold of tex2html_wrap_inline3889 on which a metric tex2html_wrap_inline4254 is defined, which is invariant in the sense that two equivalent conformal space-times define the same metric tex2html_wrap_inline3929 .

The space-time tex2html_wrap_inline3883 allows the attachment of a conformal boundary which is given by tex2html_wrap_inline4260 . 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 tex2html_wrap_inline3905 because the gravitational field (i.e. the rescaled Weyl tensor) is smooth on tex2html_wrap_inline3889 . Note that we make no assumptions about the topology of tex2html_wrap_inline3905 . If each null geodesic which starts from the inside of tex2html_wrap_inline3887 has a future and a past endpoint on tex2html_wrap_inline3905, then tex2html_wrap_inline3887 is asymptotically simple in the sense of Definition  1 . If, in addition, the metric tex2html_wrap_inline3929 is a vacuum metric then tex2html_wrap_inline3905 has the implied topology tex2html_wrap_inline3977 . Note also that it is quite possible to have situations, where tex2html_wrap_inline3929 is a vacuum metric and where the topology of tex2html_wrap_inline3905 is not tex2html_wrap_inline3977, but e.g. tex2html_wrap_inline4286 . Then, necessarily, there must exist null geodesics which do not reach tex2html_wrap_inline3905 .

In the special case when tex2html_wrap_inline3905 is empty, the conformal factor tex2html_wrap_inline3721 is strictly positive, i.e. tex2html_wrap_inline4294, and the conformal space-time is isometric to the physical space-time (choosing tex2html_wrap_inline4296).

Our goal is to express the vacuum equations in tex2html_wrap_inline3887 in terms of geometric quantities on the unphysical space-time. Consider first the Einstein vacuum equation for the metric tex2html_wrap_inline4300 . When expressed in terms of unphysical quantities it reads (see the formulae of Appendix  7)

  equation392

This equation can be interpreted as the Einstein equation for the metric tex2html_wrap_inline3923 with a source term which is determined by the conformal factor. If we assume tex2html_wrap_inline3721 to be known, then it is a second order equation for tex2html_wrap_inline3923, which is formally singular on tex2html_wrap_inline3905, where tex2html_wrap_inline3721 vanishes. Therefore, it is very hard to make any progress towards the existence problem using this equation. To remedy this situation, Friedrich [43, 44Jump To The Next Citation Point In The Article, 45Jump To The Next Citation Point In The Article] suggested to consider a different system of equations on tex2html_wrap_inline3889 which can be derived from the geometric structure on tex2html_wrap_inline3889, the conformal transformation properties of the curvature and the vacuum Einstein equation on tex2html_wrap_inline3887 . It consists of equations for a connection tex2html_wrap_inline4099, its curvature and certain other fields obtained from the curvature and the conformal factor.

Let us assume that tex2html_wrap_inline4099 is a connection on tex2html_wrap_inline3889 which is compatible with the metric tex2html_wrap_inline3923 so that

  equation405

holds. This condition does not fix the connection. Let tex2html_wrap_inline4326 and tex2html_wrap_inline4328 denote the torsion and curvature tensors of tex2html_wrap_inline4099 . We will write down equations for the following unknowns:

We introduce the zero-quantity

  equation423

where tex2html_wrap_inline4326 is the torsion tensor of tex2html_wrap_inline4099 and the other components of Z are defined in terms of the unknowns by

  equation432

  equation445

  equation456

  equation463

  equation467

  equation474

In addition, we consider the scalar field

  equation478

on tex2html_wrap_inline3889 . The equations Z =0 are the regular conformal vacuum field equations . They are first order equations. In contrast to Equation (13Popup Equation) this system is regular Popup Footnote on tex2html_wrap_inline3889, even on tex2html_wrap_inline3905 because there are no terms containing tex2html_wrap_inline4358 .

Consider the equation tex2html_wrap_inline4360 . 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 Popup Footnote and constraint equations. Energy estimates for the symmetric hyperbolic system naturally involve integrals over a certain component of the Bel-Robinson tensor [52Jump To The Next Citation Point In The Article], 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 tex2html_wrap_inline4099 is compatible with tex2html_wrap_inline3923 and that Z =0 on tex2html_wrap_inline3889 . If tex2html_wrap_inline4388 at one point of tex2html_wrap_inline3889, then tex2html_wrap_inline4388 everywhere and, furthermore, the metric tex2html_wrap_inline4394 is a vacuum metric on tex2html_wrap_inline3887 .  

Proof: From the vanishing of the torsion tensor it follows that tex2html_wrap_inline4099 is the Levi-Civita connection for the metric tex2html_wrap_inline3923 . Then, tex2html_wrap_inline4402 is the decomposition of the Riemann tensor into its irreducible parts which implies that the Weyl tensor tex2html_wrap_inline4404, that tex2html_wrap_inline4340 is the trace-free part of the Ricci tensor, and that tex2html_wrap_inline4408 . The equation tex2html_wrap_inline4410 defines tex2html_wrap_inline4336 in terms of tex2html_wrap_inline3721, and the trace of the equation tex2html_wrap_inline4416 defines tex2html_wrap_inline4418 . The trace-free part of that equation is the statement that tex2html_wrap_inline4009, which follows from the conformal transformation property (110Popup Equation) of the trace-free Ricci tensor. With these identifications the equations tex2html_wrap_inline4422 resp. tex2html_wrap_inline4424 do not yield any further information because they are identically satisfied as a consequence of the Bianchi identity on tex2html_wrap_inline4426, resp. tex2html_wrap_inline4428 .

Finally, we consider the field tex2html_wrap_inline4430 . Taking its derivative and using tex2html_wrap_inline4416 and tex2html_wrap_inline4434, we obtain tex2html_wrap_inline4436 . Hence, tex2html_wrap_inline4430 vanishes everywhere if it vanishes at one point. It follows from the transformation (111Popup Equation) of the scalar curvature under conformal rescalings that tex2html_wrap_inline4388 implies tex2html_wrap_inline3989 . Thus, tex2html_wrap_inline3929 is a vacuum metric.

tex2html_wrap_inline4446

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 tex2html_wrap_inline4430 is essentially this: If we impose the equation tex2html_wrap_inline4009 for the trace-free part of the Ricci tensor of a manifold, then by use of the contracted Bianchi identity we obtain tex2html_wrap_inline4452 . Expressing this in terms of unphysical quantities leads to the reasoning in Theorem  1 . The special case tex2html_wrap_inline4454 reduces to the standard vacuum Einstein equations, because then we have tex2html_wrap_inline4456 and tex2html_wrap_inline4458 . Then tex2html_wrap_inline4416 implies tex2html_wrap_inline4462 and S =0, while tex2html_wrap_inline4388 forces tex2html_wrap_inline4468 . The other equations are identically satisfied.

Given a smooth solution of the conformal field equations on a conformal manifold, Theorem  1 implies that on tex2html_wrap_inline3887 we obtain a solution of the vacuum Einstein equation. In particular, since the Weyl tensor of tex2html_wrap_inline3923 vanishes on tex2html_wrap_inline3905 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 [52Jump To The Next Citation Point In The Article], scalar wave [77Jump To The Next Citation Point In The Article, 78Jump To The Next Citation Point In The Article] etc.)



3.2 The reduction process for 3 The Regular Conformal Field 3 The Regular Conformal Field

image Conformal Infinity
Jörg Frauendiener
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