4.5 What should be done?4 Numerical Issues4.3 Evolution equations

4.4 What has been done? 

Let us now discuss some of the achievements of the approach based on the conformal field equations in some more detail. They range from investigations of gravitational fields coupled to a scalar field in spherical symmetry to pure gravitational interactions studied in two space dimensions and, recently, in the most general case of three space dimensions. The numerical treatment of the conformal field equations was started by Hübner [76] who analyzed the asymptotic structure of spherically symmetric space-times in which a scalar field propagates under the influence of the gravitational attraction due to its own energy density. This is a system which has been investigated rigorously in detail earlier by Christodoulou (see e.g. [28Jump To The Next Citation Point In The Article] and the references therein). So the numerical results can be judged against the very detailed information found by analytical work. In all cases considered, the numerical results agreed with the analytical ones. As a specific example, let us look at Figure  12 (taken from [79Jump To The Next Citation Point In The Article]) where the ``upper part'' of a space-time with a singularity is shown. It is obtained from the numerical evolution of initial data which are supercritical in the sense that the initial energy density (specified by a parameter A) is so large that the scalar field collapses down to a black hole.

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 tex2html_wrap_inline5352 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 [28].


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Figure 12: Upper corner of a space-time with singularity (thick line). The dashed line is tex2html_wrap_inline3905, while the thin line is the locus of vanishing divergence of outgoing light rays, i.e. an apparent horizon.

Another part of the investigation was concerned with the radiation at infinity. In Figure  13 (also from [79]) the scalar radiation field at null-infinity as a function of proper time of an observer on tex2html_wrap_inline3905 is shown.


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Figure 13: Decay of the radiation at null-infinity.

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, 38Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline3905 is not the physically distinguished tex2html_wrap_inline3977 . 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 tex2html_wrap_inline3697 and the Bondi mass for a particular case are shown in Figure  14 .


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Figure 14: The radiation field tex2html_wrap_inline3697 and the Bondi mass for a radiating A3-like space-time.

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 [82Jump To The Next Citation Point In The Article]. Figure  15 is a numerical version of the Kruskal diagram, i.e. a diagram for the conformal structure of the Schwarzschild solution.


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Figure 15: The numerically generated ``Kruskal diagram'' for 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. [145]) are equal. The dashed lines are ``right going'' null-geodesics, moving away from the left-hand tex2html_wrap_inline3905 . 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 tex2html_wrap_inline3905 '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 tex2html_wrap_inline3683 there is now good hope that the numerical computation of global space-times can be achieved in the near future.

4.5 What should be done?4 Numerical Issues4.3 Evolution equations

image Conformal Infinity
Jörg Frauendiener
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de