For the following discussion we refer to the rescaled metric
which is the metric of the Einstein cylinder. The standard conformal diagram for Minkowski space [126] is shown in Figure 2
Each point in the interior of the
triangle corresponds to a 2-sphere. The long side of the triangle
consists of all the points in the centre, (i.e.
). The other two sides of the
triangle correspond to null-infinity
. The points
are points in the centre with
, while
is a point with
. The lines meeting at
are lines of constant
, while the lines
emanating from
and converging into
are lines of constant
. In the
four-dimensional space-time the lines of constant
correspond to three-dimensional space-like hypersurfaces that are
asymptotically Euclidean.
In the case of Minkowski space-time, the metric
can be extended in a regular way to three points representing
future and past time-like infinity and space-like infinity, but
this is not generally so. Already in the case of the Schwarzschild
metric, which is also an asymptotically flat metric, there are,
strictly speaking, no such points because any attempt to extend the
metric yields a singularity. However, it is common language to
refer to this behaviour by saying that “the points and/or
are singular”. The reason
for this is related to the presence of mass. For any space-time
that has a non-vanishing ADM mass, the point
is necessarily singular while the singularity of the
time-like infinities is, in general, related to the fall-off
properties of the energy-momentum tensor in time-like directions.
In the case of the Schwarzschild solution (as in any stationary
black hole solution), it is the presence of the static (stationary)
black hole that is responsible for the singularity of
.
Let us now assume that there is a
particle that moves along the central world-line , emitting radiation. For the sake of simplicity we
assume that it emits electro-magnetic radiation, which travels
along the outgoing null-cones to null-infinity. The null-cones are
symbolized in Figure 2
by the straight lines
going off the particle’s world-line at
. We are now
interested in the behaviour of the signal along various space-like
hypersurfaces. In Figure 3
we show again the
conformal diagram of Minkowski space. The generic features
discussed below will be the same for any asymptotically flat
space-time as long as we stay away from the corners of the diagram.
The reason for choosing Minkowski space is simply one of
convenience.
The vertical dashed line is the world-line of the
particle that defines the time axis. We have displayed two
asymptotically Euclidean space-like hypersurfaces intercepting the
time axis at two different points and reaching out to space-like
infinity. Furthermore, there are two hypersurfaces that intersect
the time axis in the same two points as the asymptotically flat
ones. They reach null-infinity, intersecting in a two-dimensional
space-like surface. This geometric statement about the behaviour of
the hypersurfaces in the unphysical space-time translates back to
the physical space-time as a statement about asymptotic fall-off
conditions of the induced (physical) metric on the hypersurfaces,
namely that asymptotically the metric has constant negative
curvature. This is, in particular, a property of the space-like
hyperboloids in Minkowski space. Thus, such hypersurfaces are
called hyperboloidal hypersurfaces. An important point to keep in
mind is that the conformal space-time does not “stop” at but that it can be extended smoothly beyond. The
extension is not uniquely determined as we have already discussed
in connection with the embedding of Minkowski space into the
Einstein cylinder (cf. Figure 1
). Thus, the extension
plays no role in the concept of null-infinity but it can be very
helpful for technical reasons, in particular when numerical issues
are discussed.
We now imagine that the central particle radiates
electro-magnetic waves of uniform frequency, i.e. proportional
to , where
is the particle’s
proper time. This gives rise to a retarded electro-magnetic field
on the entire space-time, which has the form
, where
is a retarded time coordinate on
Minkowski space with
on the central world-line. We
ignore the fall-off of the field because it is irrelevant for our
present purposes. Let us now look at the waves on the various
hypersurfaces.
In the physical space-time, the hypersurfaces
extend to infinity, and we can follow the waves only up to an
arbitrary but finite distance along the hypersurfaces. The
end-points are indicated in Figure 3 as little crosses.
The resulting waveforms are shown in Figure 4
.
The first diagram shows the situation on the asymptotically Euclidean surfaces. These are surfaces of constant Minkowski time, which implies that the signal is again a pure sine wave. Note, however, that this is only true for these special hypersurfaces. Even in Minkowski space-time we could choose space-like hypersurfaces that are not surfaces of constant Minkowski time but which nonetheless are asymptotically Euclidean. On such surfaces the wave would look completely different.
On the hyperboloidal surfaces the waves seem to “flatten out”. The reason for the decrease in frequency is the fact that these surfaces tend to become more “characteristic” as they extend to infinity, thus approaching surfaces of constant phase of the retarded field.
The final diagram shows the signal obtained by an
idealized observer who moves along the piece of between the two intersection points with the
hyperboloidal surfaces. The signal is recorded with respect to the
retarded time
which, in the present case, is a
so-called Bondi parameter (see Section 4.3). Therefore, the observer
measures a signal at a single frequency for a certain interval of
this time parameter. A different Bondi time would result in a
signal during a different time interval but with a single,
appropriately scaled, frequency. Using an arbitrary time parameter
would destroy the feature that only one frequency is present in the
signal. This is, in fact, the only information that can be
transmitted from the emitter to the receiver under the given
circumstances.
The waveforms of the signal as they
appear in the conformal space-time, i.e. with respect to a
coordinate system that covers a neighbourhood of , are shown in Figure 5
. In the specific case
of Minkowski space-time, we use the coordinates
and
on the Einstein cylinder. The signal on
the asymptotically Euclidean surfaces shows the “piling up” of the
waves as they approach space-like infinity. The signal on the
hyperboloidal surfaces looks very similar to the physical case.
Since the field and the surfaces are both smooth across
, the signal can continue on across null-infinity
without even noticing its presence. The points where
is crossed are indicated in the diagram as two
little crosses. The values of the field at these points are the
same as the boundary values of the signal in the third diagram.
Here the signal in the same region of
as in Figure 4
is displayed, but
with respect to the coordinate
which is
not a Bondi parameter. Accordingly, we see that the wavelength of
the signal is not constant.
What these diagrams teach us is the following: It
has been convenient in relativity to decompose space-time into
space and time by slicing it with a family of space-like
hypersurfaces. In most of the work on existence theorems of the
Einstein equations it has been convenient to choose them to be
Cauchy surfaces and thus asymptotically Euclidean. Also, in most
numerical treatments of Einstein’s equations the same method is
used to evolve space-times from one space-like hypersurface to the
next (see Section 4). Here the hypersurfaces used
are finite because the numerical grids are necessarily finite. In
the approaches based on the standard Einstein equations it makes no
difference whether the grid is based on a finite portion of an
asymptotically Euclidean or a hyperboloidal hypersurface. The fact
that the space-time should be asymptotically flat has to be
conveyed entirely by a suitable boundary condition, which has to be
imposed at the boundary of the finite portion of the hypersurface
(i.e. at the little crosses in Figure 3). However, this
implies that the accuracy of the waveform templates obtained with
such approaches depends to a large extent on the quality of that
boundary condition. So far there exists no suitable boundary
condition that would be physically reasonable and lead to stable
codes.
In the conformal approach one has the option to
“include infinity” by using the conformal field equations (see
Section 3). Then the type of the space-like
hypersurfaces becomes an issue. The diagrams show that the
hyperboloidal surfaces are very well suited to deal with the
radiation problems. They provide a foliation of the conformal
space-time on which one can base the evolution with the conformal
field equations. The solution obtained will be smooth near and we “only” need to locate
on each hypersurface to read off the value of the
radiation data (as indicated in the second diagram of Figure
5
).