This is indicated by the appearance of trapped
surfaces and the subsequent formation of a singularity. The
boundary of the region where the trapped surfaces exist is
indicated by the thin line in the figure. It is the apparent horizon on which the divergence
of the outgoing light rays vanishes. Note that this
picture has been obtained by purely numerical methods. It should be
compared with Figure 1 in Christodoulou’s article [30].
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Another part of the investigation was
concerned with the radiation at infinity. In Figure 15 (also from [92]) the scalar radiation
field at null-infinity as a function of proper time of an observer
on
is shown.
In this example, the initial data are subcritical so that the scalar field, which initially collapses, subsequently disperses again. Note the long time-scale, ranging over approximately six orders of magnitude in proper time. This is a remarkable achievement because so far no other numerical method has been able to monitor the evolution of relativistic space-times for such a long period of time.
The next step in the application of the conformal
field equations to numerical problems was the implementation of 2D
codes for the solution of A3-like space-times [45, 46]. These provide the
first examples of vacuum space-times
with gravitational radiation. Of course, they cannot be taken
seriously as models of isolated systems because the topology of
their
is not the physically distinguished
. However, they provided important test cases for the
codes and in particular for methods to extract radiation. Since
exact solutions with this kind of global structure are
known [152, 93],
one can again compare the numerical results with their exact
counterparts. The radiation field
and the Bondi mass
for a particular case are shown in Figure 16
.
In both diagrams the solid line is the
exact solution while the dots indicate the computed values. Note
that this was the first time that a fully non-linear waveform that
agreed with an exact solution was computed. As a further example of
the conformal method in numerical relativity, we consider the
Schwarzschild space-time, which has recently been evolved with
Hübner’s 3D code [95]. Figure 17
is a numerical
version of the Kruskal diagram, i.e. a diagram for the
conformal structure of the Schwarzschild solution.
What is clearly visible here are the two
null-infinities (blue lines) and the horizons (red lines). The
green line is the “central” null-geodesic, i.e. the locus
where the Kruskal null-coordinates and
(see e.g. [163]) are equal. The dashed
lines are “right going” null-geodesics, moving away from the
left-hand
. The diagram shows the cross-over where
the two horizons (and the central line) intersect and, accordingly,
we see a large part of the region III, which is below the
cross-over, the regions I and IV with their corresponding
’s, and some part of region II where the future
singularity is located. The non-symmetric look of the diagram is,
of course, due to the fact that the coordinates used in the code
have nothing to do with the Kruskal coordinates with respect to
which one usually sees the Kruskal diagram of the extended
Schwarzschild solution.
Husa [100] has used the code
developed by Hübner to perform various parameter studies. Starting
from weak perturbations of flat data that evolve into complete
space-times with a regular
, he studies the evolution of
data obtained by increasing an “amplitude” and thereby increasing
the deviation from flat data. He reports that stronger data rather
quickly develop singularities which, however, are unphysical. This
is suggested by the fact that the radiation decays quickly and that
the news function still scales quadratically with the amplitude,
which indicates that the data are in fact still weak. The origin of
the singularities is due to an inappropriate choice of the gauge
source functions, which -- while adequate for the weak data --
leads to a rapid growth of the lapse function outside of the physical space-time in the
case of the stronger data. The cause of this growth is not known.
It might be related to the fact that in the exterior region the
constraints are not satisfied. In any case, this behaviour clearly
indicates the importance of understanding the gauges that are used
in the numerical implementations.
In [52] the
question was considered as to what extent the boundaries in the
unphysical region can influence the physical space-time. To this
end, flat initial data are prescribed together with random boundary
conditions on the grid boundary in the unphysical part. Then the
square of the rescaled Weyl tensor is monitored. This should vanish
everywhere inside the physical domain because the solution should
be conformal to Minkowski space-time. The result of this
calculation is shown in Figure 18.
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This plot shows the square of the
rescaled Weyl tensor depending on coordinate time
and the distance
from the symmetry axis in the equatorial
plane
. Null-infinity is indicated by the
black diagonal line running from
to
. The computation is carried out up to time-like
infinity, where
meets the
-axis. The
characteristic property of
is clearly visible.
In all the cases mentioned here, there is a clear
indication that long-time studies of gravitational fields are
feasible. All three cases have been checked against exact results
(exact solutions or known theorems) so that there is no doubt that
the numerical results are correct. These contributions show beyond
any reasonable doubt that the conformal field equations can be used
not only for the analytical discussion of global properties of
space-times, but also for the numerical determination of
semi-global solutions. Clearly the problems with the artificial
boundary have evaporated, the asymptotic region can accurately be
determined, and the waveforms can be reliably computed. There is
now good hope that, together with the analysis of , the numerical computation of global space-times can
be achieved in the near future.