There have been several approaches over the years towards a treatment of space-like infinity. Geroch [66] gave a geometric characterization along the same lines as for null-infinity based on the conformal structure of Cauchy surfaces. He used his construction to define multipole moments for static space-times [63, 64], later to be generalized to stationary space-times by Hansen [74]. It was shown by Beig and Simon [19, 136] that the multipole moments uniquely determine a stationary space-time and vice versa.

Different geometric characterizations of spatial infinity in
terms of the four-dimensional geometry were given by
Sommers [137], Ashtekar and Hansen [10,
6], and by Ashtekar and Romano [11]. The difficulties in all approaches which try to characterize
the structure of gravitational fields at space-like infinity in
terms of the four-dimensional geometry arise from the lack of
general results about the evolution of data near spatial
infinity. Since there are no radiating solutions which are
general enough at spatial infinity to provide hints, one is
limited more or less to one's intuition. So all these
constructions essentially impose ``reasonable'' asymptotic
conditions on the gravitational field at
and from them derive certain nice properties of space-times
which satisfy these conditions. But there is no guarantee that
there are indeed solutions of the Einstein equations which
exhibit the claimed asymptotic behaviour. In a sense, all these
characterisations are implicit
*definitions*
of certain classes of space-times (namely those which satisfy
the imposed asymptotic conditions). What is needed is an analysis
at space-like infinity which is not only guided by the geometry
but which also takes the field equations into account (see
e.g. [18,
17] for such attempts using formal power series).

Recently, Friedrich [57] has given such an analysis of space-like infinity which is
based exclusively on the initial data, the field equations and
the conformal structure of the space-time. In this representation
several new aspects come together. First, in order to simplify
the analysis, an assumption on the initial data (metric and
extrinsic curvature) on an asymptotically Euclidean hypersurface
is made. Since the focus is on the behaviour of the fields near
space-like infinity, the topology of
is taken to be
. It is assumed that the data are time-symmetric () and that on
a (negative definite) metric
with vanishing scalar curvature is given. Let
be the conformal completion of
which is topologically
, obtained by attaching a point
*i*
to
, and assume furthermore that there exists a smooth positive
function
on
with
,
and
negative definite. Furthermore, the metric
extends to a smooth metric on
*S*
. Thus, the
*three-dimensional*
conformal structure defined by
is required to be smoothly extensible to the point
*i*
.

From these assumptions follows that the conformal factor near
*i*
is determined by two smooth functions
*U*
and
*W*, where
*U*
is characterized by the geometry near
*i*
while
*W*
collects global information because
, while
*U*
(*i*)=1. With this information the rescaled Weyl tensor, the most
important piece of initial data for the conformal field
equations, near
*i*
is found to consist of two parts, a ``massive'' and a
``mass-less'' part. Under suitable conditions, the mass-less
part, determined entirely by the local geometry near
*i*, can be extended in a regular way to
*i*, while the massive part always diverges at
*i*
as
in a normal coordinate system
at
*i*
unless the ADM-mass vanishes.

In order to analyze the singular behaviour of the initial data
in more detail, the point
*i*
is blown up to a spherical set
essentially by replacing it with the sphere of unit vectors at
*i*
. Roughly speaking, this process yields a covering space
of (a suitable neighbourhood of
*i*
in)
projecting down to
which has the following properties: The pre-image
of
*i*
is an entire sphere while any other point on
has exactly two pre-image points. There exists a coordinate
*r*
on
which vanishes on
and which is such that on each pair of pre-image points it takes
values
*r*
and -
*r*, respectively. The actual blowup procedure involves a rather
involved bundle construction which also takes into account the
tensorial (respectively spinorial) nature of the quantities in
question. The reader is referred to [57] for details.

Consider now a four-dimensional neighbourhood of space-like infinity. The next important step is the realization that, in order to take full advantage of the conformal structure of space-time, it is not enough to simply allow for metrics which are conformally equivalent to the physical metric but that one should also allow for more general connections. Instead of using a connection which is compatible with a metric in the conformal class, one may use a connection which is compatible with the conformal structure, i.e. which satisfies the condition

for some one-form . If is exact, then one can find a metric in the conformal class for which is the Levi-Civita connection. Generally, however, this will not be the case. This generalization is motivated by the use of conformal geodesics as indicated below, and its effect is to free up the conformal factor, which we call to distinguish it from the conformal factor given on the initial surface , from the connection (recall that two connections which are compatible with metrics in the same conformal class differ only by terms which are linear in the first derivative of the conformal factor relating the metrics). As a consequence, the conformal field equations, when expressed in terms of a generalized connection, do not any longer contain an equation for the conformal factor. It appears, instead, as a gauge source function for the choice of conformal metric. Additionally, a free one-form appears which characterizes the freedom in the choice of the conformal connection.

To fix this freedom, Friedrich uses conformal geodesics [60]. These are curves which generalize the concept of auto-parallel curves. They are given in terms of a system of ordinary differential equations (ODE's) for their tangent vector together with a one-form along them. In coordinates this corresponds to a third-order ODE for the parameterization of the curve. Their crucial property is that they are defined entirely by the four-dimensional conformal structure with no relation to any specific metric in that conformal structure.

A time-like congruence of such curves is used to set up a ``Gauß'' coordinate system in a neighbourhood of and to define a conformal frame, a set of four vectorfields which are orthonormal for some metric in the conformal class. This metric in turn defines a conformal factor which rescales it to the physical metric. The one-form determined by the conformal geodesics defines a conformal connection , thus fixing the freedom in the connection. In this way, the gauge is fixed entirely in terms of the conformal structure.

If the physical space-time is a vacuum solution of the
Einstein equations then one can say more about the behaviour of
the conformal factor
along the conformal geodesics: It is a quadratic function of the
natural parameter
along the curves, vanishing at exactly two points if the initial
conditions for the curves are chosen appropriately. The vanishing
of
indicates the intersection of the curves with
. The intersection points are separated by a
*finite*
distance in the parameter
.

Now one fixes an initial surface
with data as described above, and the conformal geodesics are
used to set up the coordinate system and the gauge as above. When
the blow-up procedure is performed for
, a new
*finite*
representation of space-like infinity is obtained which is
sketched in Figure
8
.

The point
*i*
on the initial surface has been replaced by a sphere
which is carried along the conformal geodesics to form a finite
cylinder
*I*
. The surfaces
are the surfaces on which the conformal factor
vanishes. They touch the cylinder in the two spheres
, respectively. The conformal factor
vanishes with non-vanishing gradient on
*I*
and on
while on the spheres
its gradient also vanishes.

In this representation there is for the first time a clean
separation of the issues which go on at space-like infinity: The
spheres
are the places where ``
touches
'' while the finite cylinder
*I*
serves two purposes. On the one hand, it represents the
endpoints of space-like geodesics approaching from different
directions, while, on the other hand, it serves as the link
between past and future null-infinity. The part ``outside'' the
cylinder where
*r*
is positive between the two null surfaces
corresponds to the physical space-time, while the part with
*r*
negative is not causally related to the physical space-time but
constitutes a smooth extension. For easy reference, we call this
entire space an extended neighbourhood of space-like
infinity.

The conformal field equations, when expressed in the conformal Gauß gauge of this generalized conformal framework, yield a system of equations which has similar properties as the earlier version: It is a system of equations for a frame, the connection coefficients with respect to the frame, and the curvature, split up into the Ricci and the Weyl parts; they allow the extraction of a reduced system which is symmetric hyperbolic and propagates the constraint equations. Its solutions yield solutions of the vacuum Einstein equations whenever . The Bianchi identities, which form the only sub-system consisting of partial differential equations, again play a key role in the system. Due to the use of the conformal Gauß gauge, all other equations are simply transport equations along the conformal geodesics.

The reduced system is written in symbolic form as

with symmetric matrices
,
,
,
, and
which depend on the unknown
and the coordinates
. This system as well as initial data for it, first defined only
on the original space-time, can be extended in a regular way to
an extended neighbourhood of space-like infinity which allows for
the setup of a
*regular initial value problem at space-like infinity*
. Its properties are most interesting: When restricted to
, the initial data coincide necessarily with Minkowski data,
which together with the vanishing of
implies that on the entire cylinder
*I*
the coefficient matrix
vanishes. Thus, the system (35) degenerates into an interior symmetric hyperbolic system on
*I*
. Therefore, the finite cylinder
*I*
is a total characteristic of the system. The two null-infinities
are also characteristics, and at the intersections
between them and
*I*
the system degenerates: The coefficient matrix
which is positive definite on
*I*
and
looses rank on
.

The fact that
*I*
is a total characteristic implies that one can determine all
fields on
*I*
from data given on
.
*I*
is not a boundary on which one could specify in- or outgoing
fields. This is no surprise, because the system (35) yields an entirely structural transport process which picks up
data delivered from
via
and moves them to
via
. It is also consistent with the standard Cauchy problem where
it is known that one cannot specify any data ``at infinity''.

The degeneracy of the equations at
means that one has to take special precautions to make sure that
the transitions from and to
are smooth. In fact, not all data ``fit through the pipe'':
Friedrich has derived restrictions on the initial data of
solutions of the finite initial value problem which are necessary
for regularity through
. They are conditions on the conformal class of the initial
data, stating that the Cotton tensor and all its symmetrized and
trace-removed derivatives should vanish at the point
*i*
in the initial surface. If this is not the case, then the
solution of the intrinsic system will develop logarithmic
singularities which will then probably spread across
null-infinity, destroying its smoothness. So here is another
concrete indication that initial data have to be restricted
albeit in a rather mild way in order for the smooth picture of
asymptotic flatness to remain intact. It is not known whether
this condition is also sufficient nor what its physical
implications are.

Note that the conditions on the Cotton tensor are entirely local-at-infinity. This is the first time that such local conditions have been derived. It is rather surprising that the equations should render this possible.

The setting described in the above paragraphs certainly provides the means to analyze the consequences of the conformal Einstein evolution near space-like infinity and to understand the properties of gravitational fields in that region. The finite picture allows the discussion of the relation between various concepts which are defined independently at null and space-like infinity. As one application of this kind Friedrich and Kánnár [58] have related the Newman-Penrose constants which are defined by a surface integral over a cut of to initial data on . The cut of is pushed down towards where it is picked up by the transport equations of system (35). In a similar way, one can relate the Bondi- and ADM-masses of a space-time.

Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
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