2.4 Example: Minkowski space2 General Background2.2 The emergence of the

2.3 Asymptotically flat space-times 

We have seen in Section  2.1 that the question of how to define isolated systems in general relativity has led to the mathematical idealization of asymptotically flat space-times. They are defined by the requirement that they allow the attachment of a smooth conformal boundary. The precise definition is:

Definition 1:  A smooth (time- and space-orientable) space-time tex2html_wrap_inline3883 is called asymptotically simple, if there exists another smooth Lorentz manifold tex2html_wrap_inline3885 such that

(i) tex2html_wrap_inline3887 is an open submanifold of tex2html_wrap_inline3889 with smooth boundary tex2html_wrap_inline3891 ;
(ii) there exists a smooth scalar field tex2html_wrap_inline3721 on tex2html_wrap_inline3889, such that tex2html_wrap_inline3897 on tex2html_wrap_inline3887, and so that tex2html_wrap_inline3901, tex2html_wrap_inline3903 on tex2html_wrap_inline3905 ;
(iii) every null geodesic in tex2html_wrap_inline3887 acquires a future and a past endpoint on tex2html_wrap_inline3905 .  

An asymptotically simple space-time is called asymptotically flat, if in addition tex2html_wrap_inline3911 in a neighbourhood of tex2html_wrap_inline3905 .  

Thus, asymptotically flat space-times are a subclass of asymptotically simple space-times, namely those for which the Einstein vacuum equations hold near tex2html_wrap_inline3905 . Examples for asymptotically simple space-times which are not asymptotically flat include the de Sitter and anti-de Sitter space-times, both solutions of the Einstein equations with non-vanishing cosmological constant. We will concentrate here on asymptotically flat space-times.

According to the first condition, the space-time tex2html_wrap_inline3883, which we call the physical space-time can be considered as part of a larger space-time tex2html_wrap_inline3885, the unphysical space-time . As a submanifold of tex2html_wrap_inline3889 the physical space-time can be given a boundary which is required to be smooth. The unphysical metric tex2html_wrap_inline3923 is well-defined on tex2html_wrap_inline3889 and, in particular, on tex2html_wrap_inline3887, while the physical metric tex2html_wrap_inline3929 is only defined on tex2html_wrap_inline3887 and cannot be extended in a well-defined sense to the boundary of tex2html_wrap_inline3887 or even beyond. The metrics generate the same conformal structure, they are conformally equivalent in the sense that on tex2html_wrap_inline3887 they define the same null-cone structure.

Note that although the extended manifold tex2html_wrap_inline3889 and its metric are called unphysical, there is nothing unphysical about this construction. As we shall see below, the boundary of tex2html_wrap_inline3887 in tex2html_wrap_inline3889 is uniquely determined by the conformal structure of tex2html_wrap_inline3887 and, therefore, it is just as physical as tex2html_wrap_inline3887 . The extension beyond the boundary, given by tex2html_wrap_inline3889 is not unique, as we have already seen in Section  2.2, but this is of no consequence for the physics in tex2html_wrap_inline3887 because the extension is causally disconnected from tex2html_wrap_inline3887 .

The second condition fixes the behaviour of the scaling factor on tex2html_wrap_inline3905 as being ``of the order 1/ r '' as one approaches tex2html_wrap_inline3905 from within tex2html_wrap_inline3887 . The condition  (iii) is a completeness condition to ensure that the entire boundary is included. In some cases of interest, this condition is not satisfied. In the Schwarzschild space-time, for instance, there are null-geodesics which circle around the singularity, unable to escape to infinity. This problem has led to a weakening of Definition  1 to weakly asymptotically simple space-times (see e.g. [110Jump To The Next Citation Point In The Article]). Such space-times are essentially required to be isometric to an asymptotically simple space-time in a neighbourhood of the boundary  tex2html_wrap_inline3905 . A different completeness condition has been proposed by Geroch and Horowitz [68]. In the following discussion of the analytic and geometric issues, weakly asymptotically simple space-times will not play a role so that we can assume our space-times to be asymptotically simple. Of course, for applications weakly asymptotically simple space-times are important because they provide interesting examples of space-times with black holes.

We defined asymptotically flat space-times by the requirement that the Einstein vacuum equation holds near the boundary, i.e., that asymptotically the physical space-time is empty. There are ways to relax this condition by imposing strong enough fall-off conditions on the energy-momentum tensor without violating any of the consequences. For example, it is then possible to include electro-magnetic fields. Since we are concerned here mainly with the asymptotic region, we are not really interested in including any matter fields. Therefore, we will assume henceforth that the physical space-time is a vacuum space-time. This does not mean that the following discussion is only valid for vacuum space-times, it simply allows us to make simpler statements.

The conformal factor tex2html_wrap_inline3721 used to construct the boundary tex2html_wrap_inline3905 is, to a large extent, arbitrary. It is fixed only by its properties on the boundary. This raises the important question about the uniqueness of the conformal boundary as a point set and as a differential manifold. If this uniqueness were not present, then the notion of ``points at infinity'' would be useless. It could then happen that two curves which approach the same point in one conformal boundary for a space-time reach two different points in another conformal completion. Or, similarly, that two conformal extensions which arise from two different conformal factors were not smoothly related. However, these problems do not arise. In fact, it can be shown that between two smooth extensions there always exists a diffeomorphism which is the identity on the physical space-time, so that the two extensions are indistinguishable from the point of view of their topological and differential structure. This was first proved by Geroch [62]. It also follows from Schmidt's so called b-boundary construction [131, 132, 134].

From the condition that the vacuum Einstein equation holds, one can derive several important consequences for asymptotically flat space-times:

   (a)    tex2html_wrap_inline3905 is a smooth null hypersurface in tex2html_wrap_inline3889 .  

   (b)    tex2html_wrap_inline3905 is shear-free.

   (c)    tex2html_wrap_inline3905 has two connected components, each with topology tex2html_wrap_inline3977 .  

   (d)   The conformal Weyl tensor vanishes on tex2html_wrap_inline3905 .  

The first part of statement  (a) follows from the fact that tex2html_wrap_inline3905 is given by the equation tex2html_wrap_inline3901 . Since tex2html_wrap_inline3721 has a non-vanishing gradient on tex2html_wrap_inline3905, regularity follows. Furthermore, from the Einstein vacuum equations one has tex2html_wrap_inline3989 on tex2html_wrap_inline3887 . Hence, Equation (111Popup Equation) implies on tex2html_wrap_inline3887 :


This equation can be extended smoothly to the boundary of tex2html_wrap_inline3887, yielding there the condition tex2html_wrap_inline3999 for the co-normal tex2html_wrap_inline4001 of tex2html_wrap_inline3905 . Hence, the gradient of the conformal factor is null, and tex2html_wrap_inline3905 is a null hypersurface.

As such it is generated by null geodesics. The statement (b) asserts that the congruence formed by the generators of tex2html_wrap_inline3905 has vanishing shear. To show this we look at Equation (110Popup Equation) and find from tex2html_wrap_inline4009 that


whence, on tex2html_wrap_inline3905 we get (writing tex2html_wrap_inline4015 for the degenerate induced metric on tex2html_wrap_inline3905)


whence the Lie-derivative of tex2html_wrap_inline4015 along the generators is proportional to tex2html_wrap_inline4015, which is the shear-free condition for null geodesic congruences with tangent vector tex2html_wrap_inline4023 (see [75, 118Jump To The Next Citation Point In The Article]).

To prove statement  (c) we observe that since tex2html_wrap_inline3905 is null, either the future or the past light cone of each of its points has a non-vanishing intersection with tex2html_wrap_inline3887 . This shows that there are two components of tex2html_wrap_inline3905, namely tex2html_wrap_inline3847 on which null geodesics attain a future endpoint, and tex2html_wrap_inline3849 where they attain a past endpoint. These are the only connected components because there is a continuous map from the bundle of null-directions over tex2html_wrap_inline3887 to tex2html_wrap_inline3867, assigning to each null direction at each point P of tex2html_wrap_inline3887 the future (past) endpoint of the light ray emanating from P in the given direction. If tex2html_wrap_inline3867 were not connected then neither would be the bundle of null-directions of tex2html_wrap_inline3887, which is a contradiction (tex2html_wrap_inline3887 being connected). To show that the topology of tex2html_wrap_inline3867 is tex2html_wrap_inline3977 requires a more sophisticated argument which has been given by Penrose [109Jump To The Next Citation Point In The Article] (a different proof has been provided by Geroch [65]). It has been pointed out by Newman [105] that these arguments are only partially correct. He rigorously analyzed the global structure of asymptotically simple space-times and he found that, in fact, there are more general topologies allowed for tex2html_wrap_inline3905 . However, his analysis was based on methods of differential topology not taking the field equations into account. Indeed, we will find later in Theorem  6 that the space-time which evolves from data close enough to Minkowski data will have a tex2html_wrap_inline3847 with topology tex2html_wrap_inline3977 .

The proof of statement  (d) depends in an essential way on the topological structure of tex2html_wrap_inline3905 . We refer again to [109Jump To The Next Citation Point In The Article]. The vanishing of the Weyl curvature on tex2html_wrap_inline3905 is the final justification for the definition of asymptotically flat space-times: Vanishing Ricci curvature implies the vanishing of the Weyl tensor and hence of the entire Riemann tensor on tex2html_wrap_inline3905 . The physical space-time becomes flat at infinity.

But there is another important property which follows from the vanishing of the Weyl tensor on tex2html_wrap_inline3905 . Consider the Weyl tensor tex2html_wrap_inline4069 of the unphysical metric tex2html_wrap_inline3923 which agrees on tex2html_wrap_inline3887 with the Weyl tensor tex2html_wrap_inline4075 of the physical metric tex2html_wrap_inline3929 because of the conformal invariance (107Popup Equation). On tex2html_wrap_inline3887, tex2html_wrap_inline4075 satisfies the vacuum Bianchi identity


This equation looks superficially like the zero rest-mass equation (8Popup Equation) for spin-2 fields. However, the conformal transformation property of (10Popup Equation) is different from the zero rest-mass case. The equation is not conformally invariant since the conformal rescaling of a vacuum metric generates Ricci curvature in the unphysical space-time by Equation (108Popup Equation), which then feeds back into the Weyl curvature via the Bianchi identity (cf. Equation (112Popup Equation)). However, we can define the field


on tex2html_wrap_inline3887 . As it stands, tex2html_wrap_inline4087 is not defined on tex2html_wrap_inline3905 . But the vanishing of the Weyl tensor there and the smoothness assumption allow the extension of tex2html_wrap_inline4087 to the boundary (and even beyond) as a smooth field on tex2html_wrap_inline3889 . It follows from Equation (10Popup Equation) that this field satisfies the zero rest-mass equation


on the unphysical space-time tex2html_wrap_inline3889 with respect to the unphysical metric. Therefore, the rescaled Weyl tensor tex2html_wrap_inline4087 is a genuine spin-2 field with the natural conformal behaviour. In fact, this is the field which most directly describes the gravitational effects, in particular its values on the boundary are closely related to the gravitational radiation which escapes from the system under consideration. It propagates on the conformal space-time in a conformally covariant way according to Equation (11Popup Equation) which looks superficially like the equation (8Popup Equation) for a (linear) spin-2 zero rest-mass field. However, there are highly non-linear couplings between the connection given by tex2html_wrap_inline4099 and the curvature given by tex2html_wrap_inline3877 . In the physical space-time, where the conformal factor is unity, the field tex2html_wrap_inline3877 coincides with the Weyl tensor which is the source of tidal forces acting on test particles moving in space-time. For these reasons, we will call the rescaled Weyl tensor tex2html_wrap_inline3877 the gravitational field .

From Equation (11Popup Equation) and the regularity on tex2html_wrap_inline3905 follows a specific fall-off behaviour of the field tex2html_wrap_inline4087 and hence of the Weyl tensor which is exactly the peeling property obtained by Sachs. It arises here from a reasoning similar to the one presented towards the end of Section  2.2 . It is a direct consequence of the geometric assumption that the conformal completion be possible and of the conformal invariance of Equation (11Popup Equation). This equation for the rescaled Weyl tensor is an important sub-structure of the Einstein equation because it is conformally invariant in contrast to the Einstein equation itself. In a sense it is the most important part also in the system of conformal field equations which we consider in the next section.

The possibility of conformal compactification restricts the lowest order structure of the gravitational field on the boundary. This means that all asymptotically flat manifolds are the same in that order, so that the conformal boundary and its structure are universal features among asymptotically flat space-times. The invariance group of this universal structure is exactly the BMS group. Differences between asymptotically flat space-times can arise only in a higher order. This is nicely illustrated by the Weyl tensor which necessarily vanishes on the conformal boundary, but the values of the rescaled Weyl tensor tex2html_wrap_inline4111 are not fixed there.

In summary, our qualitative picture of asymptotically flat space-times is as follows: Such space-times are characterized by the property that they can be conformally compactified. This means that we can attach boundary points to all null-geodesics. More importantly, these points together form a three-dimensional manifold which is smoothly embedded into a larger extended space-time. The physical metric and the metric on the compactified space are conformally related. Smoothness of the resulting manifold with boundary translates into asymptotic fall-off conditions for the physical metric and the fields derived from it. The boundary emerges here as a geometric concept and not as an artificial construct put in by hand. This is reflected by the fact that it is not possible to impose a ``boundary condition'' for solutions of the Einstein equations there. In this sense it was (and is) not correct to talk about a ``boundary condition at infinity'' as we and the early works sometimes did.

2.4 Example: Minkowski space2 General Background2.2 The emergence of the

image Conformal Infinity
Jörg Frauendiener
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de