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

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

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

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

The second condition fixes the behaviour of the scaling factor
on
as being ``of the order 1/
*r*
'' as one approaches
from within
. 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. [110]). Such space-times are essentially required to be isometric to
an asymptotically simple space-time in a neighbourhood of the
boundary
. 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 used to construct the boundary 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) is a smooth null hypersurface in .

(b) is shear-free.

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

(d) The conformal Weyl tensor vanishes on .

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

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

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

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

whence the Lie-derivative of along the generators is proportional to , which is the shear-free condition for null geodesic congruences with tangent vector (see [75, 118]).

To prove statement
(c)
we observe that since
is null, either the future or the past light cone of each of its
points has a non-vanishing intersection with
. This shows that there are two components of
, namely
on which null geodesics attain a future endpoint, and
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
to
, assigning to each null direction at each point
*P*
of
the future (past) endpoint of the light ray emanating from
*P*
in the given direction. If
were not connected then neither would be the bundle of
null-directions of
, which is a contradiction (
being connected). To show that the topology of
is
requires a more sophisticated argument which has been given by
Penrose [109] (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
. 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
with topology
.

The proof of statement (d) depends in an essential way on the topological structure of . We refer again to [109]. The vanishing of the Weyl curvature on 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 . The physical space-time becomes flat at infinity.

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

This equation looks superficially like the zero rest-mass equation (8) for spin-2 fields. However, the conformal transformation property of (10) 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 (108), which then feeds back into the Weyl curvature via the Bianchi identity (cf. Equation (112)). However, we can define the field

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

on the unphysical space-time
with respect to the unphysical metric. Therefore, the rescaled
Weyl tensor
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 (11) which looks superficially like the equation (8) for a (linear) spin-2 zero rest-mass field. However, there are
highly non-linear couplings between the connection given by
and the curvature given by
. In the physical space-time, where the conformal factor is
unity, the field
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
the
*gravitational field*
.

From Equation (11) and the regularity on follows a specific fall-off behaviour of the field 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.

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 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.

Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |