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 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 , 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
.
Condition (2) in
Definition 1 fixes the behaviour of the
scaling factor on as being “of the order
” as one approaches
from within
. 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. [127
]). 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 [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 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 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 [148, 149, 151].
From the condition that the vacuum Einstein
equation holds, one can derive several important consequences for
asymptotically flat space-times:
(1) |
|
(2) |
|
(3) |
|
(4) |
The conformal Weyl tensor vanishes on |
The first part of Statement (1) 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 (114
) 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 (2) asserts that the congruence
formed by the generators of has vanishing shear. To show
this we look at Equation (113
) and find from
that
whence, on we get (writing
for the degenerate induced metric on
)
To prove Statement (3) 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
of
the future (past) endpoint
of the light ray emanating from
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 [126
] (a different proof
has been provided by Geroch [78]). It has been
pointed out by Newman [122] 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
that evolves from data close enough to Minkowski data will have a
with topology
.
The proof of Statement (4) depends in an
essential way on the topological structure of . We refer again to [126
]. 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 that
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 (110
). On
,
satisfies the vacuum Bianchi identity,
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
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 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 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.