Asymptotically flat space-times

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. [126

]). 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 [147 , 148 , 150 ].

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 (114

) implies on $ℳ^$ :

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 (113

) and find from $&tidle;Φab = 0$ that

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

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 [125

] (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 [125

]. 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 (110

). On $^ℳ$ , $&tidle;Ca bcd$ satisfies the vacuum Bianchi identity,

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 (10

) that this field satisfies the zero rest-mass equation

From Equation (11

) 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 (11

). 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.

2.3 Asymptotically flat space-times