There have been several approaches over the years towards a treatment of space-like infinity. Geroch [79] 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 [76, 77], later to be generalized to stationary space-times by Hansen [87]. It was shown by Beig and Simon [20, 153] 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 [154], Ashtekar, and
Hansen [11, 7], and by
Ashtekar and Romano [12]. The difficulties
in all approaches that 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 that 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
that exhibit the claimed asymptotic behaviour. In a sense, all
these characterisations are implicit definitions of certain classes of
space-times (namely those that 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, 19] for such
attempts using formal power series).
Recently, Friedrich [68] 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
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
. Thus, the three-dimensional conformal structure
defined by
is required to be smoothly extensible
to the point
.
From these assumptions follows that the conformal
factor near is determined by two smooth functions
and
, where
is characterized by
the geometry near
, while
collects global
information because
, while
. With this information the rescaled Weyl tensor, the
most important piece of initial data for the conformal field
equations, near
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
, can be extended in a regular way to
, while the massive part always diverges at
as
in a normal coordinate system
at
unless the ADM mass vanishes.
In order to analyze the singular behaviour of the
initial data in more detail, the point is blown up to a
spherical set
essentially by replacing it with the
sphere of unit vectors at
. Roughly speaking, this
process yields a covering space
of (a suitable neighbourhood
of
in)
projecting down to
which has the following properties: The pre-image
of
is an entire sphere while any other
point on
has exactly two pre-image points. There exists a
coordinate
on
which vanishes on
, and which is such that on each pair of pre-image
points it takes values
and
, respectively. The actual
blow-up procedure involves a rather complicated bundle construction
that also takes into account the tensorial (respectively spinorial)
nature of the quantities in question. The reader is referred
to [68] 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 that are
conformally equivalent to the physical metric, but that one should
also allow for more general connections. Instead of using a
connection that is compatible with a metric in the conformal class,
one may use a connection compatible with the
conformal structure, i.e. that 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,
that 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 [73]. These curves 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
vector fields that are orthonormal for some metric in the conformal
class. This metric in turn defines a conformal factor
that 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. One may wonder whether the choice of
conformal “Gauß” coordinates is a practical one, since it is well
known that the usual Gauß coordinates are prone to develop
singularities very quickly. However, conformal geodesics are in
general not geodesics in the physical space-time, and parallel
transport with a Weyl connection is not the same as parallel
transport in the physical space-time. If one thinks of the usual
Gauß coordinates as being constructed from the world-lines of
freely falling particles, then one should think of the conformal
geodesics as the world-lines of some fiducial “particles” that
experience a force determined by the conformal structure of the
space-time. This “conformal force” could counteract the attractive
nature of the gravitational field under certain conditions, thereby
delaying the intersection of the world-lines and the formation of
coordinate singularities. In fact, Friedrich [70] has shown that
there exist global conformal Gauß coordinates on the Kruskal
extension of the Schwarzschild space-time that extend smoothly
beyond null-infinity. Thus, one would hope that the conformal
Gauß systems can be used to globally cover more general
space-times as well. This would make them an ideal tool for use in
numerical simulations because they are intrinsically defined so
that the numerical results would also have an intrinsic meaning.
Furthermore, as outlined below, the use of a conformal
Gauß system simplifies the field equations considerably.
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 on the initial surface has
been replaced by a sphere
, which is carried along the
conformal geodesics to form a finite cylinder
. 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
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 that determine the structure
of space-like infinity: The spheres are the places
where “
touches
” while the finite cylinder
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
is positive between the two null surfaces
corresponds to the physical space-time, while the
part with
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 that has properties similar
to 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 that 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 matricesThe fact that is a total characteristic
implies that one can determine all fields on
from data given on
.
is not a boundary on which
one could specify in- or outgoing fields. This is no surprise,
because the system (37
) yields an entirely
structural transport process that 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 that are necessary
for regularity through
. These are necessary
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
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.
In order to get more insight into the character
of the transport equations along the cylinder
and, in particular, to see whether these conditions are also
sufficient, one can take advantage of the group theoretical origin
of the blow-up procedure, and write all the relevant quantities
according to their tensorial character as linear combinations of a
complete set of functions on the sphere. (According to the
Peter-Weyl theorem, the matrix entries of irreducible
representations of
form a complete system of
functions on
, these functions being closely related
to the spin-spherical harmonics [134
].) Insertion into
the transport equations and restriction to the cylinder
yields a set of ODE’s for the coefficients along the conformal
geodesics that generate
. Taking successive
-derivatives and restricting to the cylinder yields a
hierarchy of such ODE’s. The system of ODE’s obtained at order
can be solved provided a solution up to order
is known. Thus, given initial conditions, one can
determine recursively the expansion coefficients and thereby obtain
a Taylor approximation of the fields in a neighbourhood of
. These calculations are rather cumbersome and can be
performed only with the aid of computer algebra.
In view of the unknown status of the above condition on the Cotton tensor, it is clear that a particularly interesting scenario is obtained for conformally flat initial data. For such data, the Cotton tensor vanishes identically, so that the conditions above are satisfied. If any logarithms appear in the solutions of the hierarchy of ODE’s, then this implies that the vanishing of the Cotton tensor and its symmetrized, trace-removed derivatives is only necessary and not sufficient.
In [160] Valiente Kroon has
carried out these calculations up to order in the general case and up to order
for data that were in addition assumed to be
axi-symmetric. The rather surprising result is this: Up to order
the solutions do not contain any logarithms. This
changes with
, where there are logarithmic terms that
contribute additively to the solution with coefficients
proportional to the Newman-Penrose constants. This
means that the solution is smooth up to
only if the Newman-Penrose constants vanish.
Assuming this, then for
the solutions will again
have an additive logarithmic term with some coefficients
(higher-order Newman-Penrose “constants”) whose
vanishing is necessary for smoothness of the solution. This picture
returns at the orders
and
, where again the vanishing of certain quantities
is necessary for the solution to be free of
logarithms.
From the explicit algebraic expressions for the
up to order
, one can guess a formula that
expresses them for arbitrary orders in terms of certain expansion
coefficients of the conformal factor on the initial hypersurface.
Then, the vanishing of the
implies that the expansion
coefficients of the conformal factor are correlated in a very
specific way. In the special case of the Schwarzschild space-time
(which is spherically symmetric, hence has conformally flat initial
data), one can compute the expansion coefficients of the conformal
factor explicitly, and one finds that they are correlated in exactly the way implied by the vanishing of
the
. This observation seems to indicate that the time
evolution of an asymptotically Euclidean, conformally flat, and
smooth initial data set admits a smooth extension to null-infinity
near space-like infinity if and only if it agrees with the
Schwarzschild solution (the only static solution in the conformal
class of the data) to all orders at the point
. This conjecture seems to imply that the
developments of the Brill-Lindquist and Misner initial data sets
have non-smooth null-infinities.
While the conditions on the Cotton tensor are
entirely local to , this is not so for the additional
conditions discussed above. They involve the part
of the conformal factor which is related to the ADM
mass of the system and which is obtained by solving an elliptic
equation. This part is therefore determined by properties of the
initial data that are not local to
. To what extent it is
really the global structure of the
initial data which is involved here is not so clear since the
results by Corvino [35] and
Corvino-Schoen [36] show that
one can deform initial data in certain annular regions of the
initial hypersurface while keeping the asymptotics and the inner
regions untouched.
These observations also shine a new light on the
static, respectively stationary, solutions. It might well be that
they play a fundamental role in the construction of asymptotically
flat space-times. If it turns out that such space-times have to be
necessarily static (or stationary in the non-time-symmetric case)
near in order to have a smooth conformal extension, then
this implies that there are no gravitational waves allowed in the
neighbourhood of space-like infinity. In particular, this means
that incoming radiation has to die off in the infinite future,
while the system cannot have emitted gravitational waves for all
times in the infinite past. Whether this is a reasonable scenario
for an isolated system is a question of physics. It has to be
answered by investigating whether it is possible to discuss
realistic physical processes like radiation emission and scattering
in a meaningful way such as whether one can uniquely define
physical quantities like energy-momentum, angular momentum,
radiation field, etc.
This touches upon the question as to how physically relevant the assumption of a smooth asymptotic structure will in the end turn out to be. It is after all an idealisation, which we use as a tool to describe an isolated system. If we want to find out whether this idealisation captures physically relevant scenarios, we first have to know the options, i.e. we need to have detailed knowledge about the mathematical situations that can arise. Whether these situations are general enough to admit all physical situations that one would consider as reasonable remains an open question.
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 that are defined independently at null and
space-like infinity. As one application of this kind, Friedrich and
Kánnár [71] 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 (37
). In a similar way,
one can relate the Bondi and ADM masses of a space-time.