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 (ℳ, gab,Ω) such that

(1) 

(ℳ, gab) is a (time- and space-orientable) Lorentz manifold;

   
(2) 

Ω is a smooth scalar field on ℳ such that the set ^ℳ = {p ∈ ℳ : Ω (p) > 0} is non-empty and connected;

   
(3) 

the gravitational field Kabcd = Ω −1Cabcd extends smoothly to all of ℳ.

Two conformal space-times (ℳ, gab,Ω ) and (ℳˆ, ˆgab,Ωˆ) are equivalent if ℳ and ℳˆ are diffeomorphic and if, after identification of ℳ and ℳˆ with a suitable diffeomorphism, there exists a strictly positive scalar field 𝜃 on ℳ such that ˆΩ = 𝜃Ω and ˆgab = 𝜃2gab.

From this definition follows that ℳ^ is an open sub-manifold of ℳ on which a metric −2 &tidle;gab = Ω gab is defined, which is invariant in the sense that two equivalent conformal space-times define the same metric &tidle;gab.

The space-time (ℳ^, &tidle;gab) allows the attachment of a conformal boundary, which is given by ℐ = {p ∈ ℳ : Ω(p) = 0,d Ω ⁄= 0}. 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 ℐ because the gravitational field (i.e. the rescaled Weyl tensor) is smooth on ℳ. Note that we make no assumptions about the topology of ℐ. If each null geodesic which starts from the inside of ^ℳ has a future and a past endpoint on ℐ, then ^ ℳ is asymptotically simple in the sense of Definition 1. If, in addition, the metric &tidle;gab is a vacuum metric then ℐ has the implied topology S2 × ℝ. Note also that it is quite possible to have situations where &tidle;g ab is a vacuum metric and where the topology of ℐ is not 2 S × ℝ, but e.g. 2 T × ℝ. Then, necessarily, there must exist null geodesics that do not reach ℐ.

In the special case when ℐ is empty, the conformal factor Ω is strictly positive, i.e. ℳ^ = ℳ, and the conformal space-time is isometric to the physical space-time (choosing 𝜃 = Ω− 1).

Our goal is to express the vacuum equations in ^ℳ in terms of geometric quantities on the unphysical space-time. Consider first the Einstein vacuum equation for the metric g&tidle;ab = Ω2gab. When expressed in terms of unphysical quantities it reads (see the formulae of Appendix 7)

&tidle; 2- -3- c 0 = Gab = Gab − Ω (∇a ∇b Ω − gab□ Ω ) − Ω2 gab∇cΩ ∇ Ω. (13 )
This equation can be interpreted as the Einstein equation for the metric gab with a source term which is determined by the conformal factor. If we assume Ω to be known, then it is a second order equation for gab, which is formally singular on ℐ, where Ω vanishes. Therefore, it is very hard to make any progress towards the existence problem using this equation. To remedy this situation, Friedrich [5354Jump To The Next Citation Point55Jump To The Next Citation Point] suggested to consider a different system of equations on ℳ that can be derived from the geometric structure on ℳ, the conformal transformation properties of the curvature, and the vacuum Einstein equation on ^ℳ. It consists of equations for a connection ∇a, its curvature, and certain other fields obtained from the curvature and the conformal factor.

Let us assume that ∇a is a connection on ℳ that is compatible with the metric gab so that

∇ g = 0 (14 ) c ab
holds. This condition does not fix the connection. Let T abc and Rabcd denote the torsion and curvature tensors of ∇a. We will write down equations for the following unknowns:

We introduce the zero-quantity

Z = (Tbac,𝒬abcd,𝒫abc,ℬbcd,𝒮a,𝒮ab,𝒟a ), (15 )
where a Tbc is the torsion tensor of ∇a, and the other components of Z are defined in terms of the unknowns by
𝒬 ≡ R − ΩK + 2g Φ − 2 g Φ − 4 g g Λ, (16 ) abcd abcd abcd c[a b]d d[a b]c c[a b]d 𝒫abc ≡ 2∇ [cΦa ]b + 2gb[c∇a]Λ − KcabdΣd, (17 ) ℬ ≡ ∇ Ka , (18 ) bcd a bcd 𝒮a ≡ ∇a Ω − Σa, (19 ) 𝒮ab ≡ ∇a Σb − gabS + Ω Φab, (20 ) b 𝒟a ≡ ∇aS + ΦabΣ − Ω ∇a Λ − 2Λ Σa. (21 )
In addition, we consider the scalar field
𝒯 ≡ 2ΩS − 2 Ω2Λ − Σa Σa (22 )
on ℳ. The equations Z = 0 are the regular conformal vacuum field equations. They are first order equations. In contrast to Equation (13View Equation), this system is regular1 on ℳ, even on ℐ because there are no terms containing Ω− 1.

Consider the equation ℬbcd = ∇aKabcd = 0. 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 [5554]. Its importance lies in the fact that it splits naturally into a symmetric hyperbolic system of evolution equations2 and constraint equations. Energy estimates for the symmetric hyperbolic system naturally involve integrals over a certain component of the Bel–Robinson tensor [62Jump To The Next Citation Point], 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 ∇a is compatible with gab and that Z = 0 on ℳ. If 𝒯 = 0 at one point of ℳ, then 𝒯 = 0 everywhere and, furthermore, the metric −2 Ω gab is a vacuum metric on ^ ℳ.

Proof. From the vanishing of the torsion tensor it follows that ∇a is the Levi–Civita connection for the metric gab. Then, 𝒬abcd = 0 is the decomposition of the Riemann tensor into its irreducible parts which implies that the Weyl tensor a a C bcd = ΩK bcd, that Φab is the trace-free part of the Ricci tensor, and that Λ = 124R. The equation 𝒮a = 0 defines Σa in terms of Ω, and the trace of the equation 𝒮ab = 0 defines S = 14□ Ω. The trace-free part of that equation is the statement that Φ&tidle; = 0 ab, which follows from the conformal transformation property (113View Equation) of the trace-free Ricci tensor. With these identifications the equations ℬabc = 0 respectively 𝒫abc = 0 do not yield any further information because they are identically satisfied as a consequence of the Bianchi identity on (ℳ, g), respectively ( ^ℳ, g&tidle;).

Finally, we consider the field 𝒯. Taking its derivative and using 𝒮ab = 0 and 𝒟a = 0, we obtain ∇a𝒯 = 0. Hence, 𝒯 vanishes everywhere if it vanishes at one point. It follows from the transformation (114View Equation) of the scalar curvature under conformal rescalings that 𝒯 = 0 implies &tidle;Λ = 0. Thus, &tidle;gab 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 𝒯 is essentially this: If we impose the equation Φ&tidle;ab = 0 for the trace-free part of the Ricci tensor of a manifold, then by use of the contracted Bianchi identity we obtain &tidle;∇a &tidle;Λ = 0. Expressing this in terms of unphysical quantities leads to the reasoning in Theorem 1. The special case Ω = 1 reduces to the standard vacuum Einstein equations, because then we have Kabcd = Cabcd and Σa = 0. Then 𝒮ab = 0 implies Φ = 0 ab and S = 0, while 𝒯 = 0 forces Λ = 0. The other equations are identically satisfied.

Given a smooth solution of the conformal field equations on a conformal manifold, Theorem 1 implies that on ℳ^ we obtain a solution of the vacuum Einstein equation. In particular, since the Weyl tensor of gab vanishes on ℐ 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 [62Jump To The Next Citation Point], scalar wave [90Jump To The Next Citation Point91Jump To The Next Citation Point], etc.).


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