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 .
Another part of the investigation was concerned with the radiation at infinity. In Figure 13 (also from ) 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 was 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 has been the implementation of 2-D codes for the solution of A3-like space-times [39, 38]. 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 provide 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 [135, 80] 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 14 .
In both diagrams the solid line is the exact solution while the dots indicate the computed values. Note that this is the first time that a fully non-linear wave-form has been computed which agrees with an exact solution.
As a final example of the conformal method in numerical relativity we consider the Schwarzschild space-time which has recently been evolved with Hübner's 3-D code . Figure 15 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 U and V (see e.g. ) 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.
In all three 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 cannot only be used 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 wave-forms can reliably be computed. Together with the analysis of there is now good hope that the numerical computation of global space-times can be achieved in the near future.
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