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 |