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 (ℳ^, &tidle;gab) is called asymptotically simple, if there exists another smooth Lorentz manifold (ℳ, gab) such that

(1) 

ℳ^ is an open submanifold of ℳ with smooth boundary ∂ℳ^ = ℐ;

   
(2) 

there exists a smooth scalar field Ω on ℳ, such that gab = Ω2g&tidle;ab on ^ℳ, and so that Ω = 0, dΩ ⁄= 0 on ℐ;

   
(3) 

every null geodesic in ^ ℳ acquires a future and a past endpoint on ℐ.

An asymptotically simple space-time is called asymptotically flat, if in addition &tidle;Rab = 0 in a neighbourhood of ℐ.

Thus, asymptotically flat space-times are a subclass of asymptotically simple space-times, namely those for which the Einstein vacuum equations hold near ℐ. Examples of asymptotically simple space-times that 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 Condition (1) in Definition 1, the space-time ( ^ℳ, &tidle;g ) ab, which we call the physical space-time, can be considered as part of a larger space-time (ℳ, gab), the unphysical space-time. As a submanifold of ℳ, the physical space-time can be given a boundary which is required to be smooth. The unphysical metric gab is well-defined on ℳ and, in particular, on ℳ^, while the physical metric &tidle;gab is only defined on ℳ^ and cannot be extended in a well-defined sense to the boundary of ℳ^ or even beyond. The metrics generate the same conformal structure; they are conformally equivalent in the sense that on ^ ℳ they define the same null-cone structure.

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

Condition (2) in Definition 1 fixes the behaviour of the scaling factor on ℐ as being “of the order 1∕r” as one approaches ℐ from within ^ℳ. Condition (3) in Definition 1 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 that 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. [126Jump To The Next Citation Point]). Such space-times are essentially required to be isometric to an asymptotically simple space-time in a neighbourhood of the boundary ℐ. A different completeness condition has been proposed by Geroch and Horowitz [81]. 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 Ω used to construct the boundary ℐ 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 that 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 that 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 [75]. It also follows from Schmidt’s so-called b-boundary construction [147148150].

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

(1) 

ℐ is a smooth null hypersurface in ℳ.

   
(2) 

ℐ is shear-free.

   
(3) 

ℐ has two connected components, each with topology S2 × ℝ.

   
(4) 

The conformal Weyl tensor vanishes on ℐ.

The first part of Statement (1) follows from the fact that ℐ is given by the equation Ω = 0. Since Ω has a non-vanishing gradient on ℐ, regularity follows. Furthermore, from the Einstein vacuum equations one has Λ&tidle; = 0 on ℳ^. Hence, Equation (114View Equation) implies on ℳ^:

Ω2Λ − 1Ω □ Ω + 1∇a Ω∇ Ω = 0. 4 2 a

This equation can be extended smoothly to the boundary of ^ℳ, yielding there the condition NaN a = 0 for the co-normal N = − ∇ Ω a a of ℐ. Hence, the gradient of the conformal factor is null, and ℐ is a null hypersurface.

As such it is generated by null geodesics. The Statement (2) asserts that the congruence formed by the generators of ℐ has vanishing shear. To show this we look at Equation (113View Equation) and find from &tidle;Φab = 0 that

1- Ω Φab + ∇a ∇bΩ − 4gab □Ω = 0,

whence, on ℐ we get (writing mab for the degenerate induced metric on ℐ)

1- 2∇aNb = ℒN mab = − 2mab □ Ω, (9 )
whence the Lie-derivative of mab along the generators is proportional to mab, which is the shear-free condition for null geodesic congruences with tangent vector N a (see [88134Jump To The Next Citation Point]).

To prove Statement (3) we observe that since ℐ is null, either the future or the past light cone of each of its points has a non-vanishing intersection with ℳ^. This shows that there are two components of ℐ, namely ℐ + on which null geodesics attain a future endpoint, and ℐ − 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 ℳ^ to ℐ ±, assigning to each null direction at each point P of ^ ℳ the future (past) endpoint of the light ray emanating from P in the given direction. If ± ℐ were not connected then neither would be the bundle of null-directions of ^ℳ, which is a contradiction (ℳ^ being connected). To show that the topology of ℐ ± is S2 × ℝ requires a more sophisticated argument, which has been given by Penrose [125Jump To The Next Citation Point] (a different proof has been provided by Geroch [78]). It has been pointed out by Newman [121] 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 ℐ. 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 that evolves from data close enough to Minkowski data will have a + ℐ with topology 2 S × ℝ.

The proof of Statement (4) depends in an essential way on the topological structure of ℐ. We refer again to [125Jump To The Next Citation Point]. The vanishing of the Weyl curvature on ℐ 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 ℐ. The physical space-time becomes flat at infinity.

But there is another important property that follows from the vanishing of the Weyl tensor on ℐ. Consider the Weyl tensor Cabcd of the unphysical metric gab, which agrees on ^ℳ with the Weyl tensor &tidle;Ca bcd of the physical metric &tidle;g ab because of the conformal invariance (110View Equation). On ^ℳ, &tidle;Ca bcd satisfies the vacuum Bianchi identity,

&tidle;∇ &tidle;Ca = 0. (10 ) a bcd
This equation looks superficially like the zero rest-mass equation (8View Equation) for spin-2 fields. However, the conformal transformation property of Equation (10View 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 (111View Equation), which then feeds back into the Weyl curvature via the Bianchi identity (cf. Equation (115View Equation)). However, we can define the field
Kabcd = Ω −1Cabcd

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

∇aKabcd = 0 (11 )
on the unphysical space-time ℳ with respect to the unphysical metric. Therefore, the rescaled Weyl tensor Kabcd is a genuine spin-2 field with the natural conformal behaviour. In fact, this is the field that most directly describes the gravitational effects; in particular, its values on the boundary are closely related to the gravitational radiation that escapes from the system under consideration. It propagates on the conformal space-time in a conformally covariant way according to Equation (11View Equation) which looks superficially like the equation (8View Equation) for a (linear) spin-2 zero rest-mass field. However, there are highly non-linear couplings between the connection given by ∇a and the curvature given by Kabcd. In the physical space-time, where the conformal factor is unity, the field Kabcd 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 Kabcd the gravitational field.

From Equation (11View Equation) and the regularity on ℐ follows a specific fall-off behaviour of the field Kabcd, 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 (11View 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 3.

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 −1 Kabcd = Ω Cabcd 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 that 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.


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