Some numerical results

4.4 Some numerical results

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 [89 ], 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 system has been investigated rigorously in detail in earlier work by Christodoulou (see e.g. [30

] and the references therein). Thus, the numerical results can be judged against the very detailed information found by analytical work. In all cases considered, the numerical results agree with the analytical ones. As a specific example, let us look at Figure 14

(taken from [92

]) where the “upper part” of a space-time with a singularity is shown. It is obtained from the numerical evolution of initial data that 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 $𝜃out$ 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 ].

Figure 14: Upper corner of a space-time with singularity (thick line). The dashed line is $ℐ$ , 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 15 (also from [92 ]) the scalar radiation field at null-infinity as a function of proper time of an observer on $ℐ$ is shown.

Figure 15: Decay of the radiation at null-infinity.

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 $ℐ$ is not the physically distinguished $2 S × ℝ$ . 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 [151 , 93 ], one can again compare the numerical results with their exact counterparts. The radiation field $ψ4$ and the Bondi mass for a particular case are shown in Figure 16 .

Figure 16: The radiation field $ψ4$ 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 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.

Figure 17: 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. [163 ]) are equal. The dashed lines are “right going” null-geodesics, moving away from the left-hand $ℐ$ . 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 $ℐ$ ’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 [98 ] 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 $i+$ , 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 .

Figure 18: The rescaled Weyl tensor in a flat space-time obtained from flat initial data and random boundary data in the unphysical region.

This plot shows the square of the rescaled Weyl tensor depending on coordinate time $t$ and the distance $x$ from the symmetry axis in the equatorial plane $z = 0$ . Null-infinity is indicated by the black diagonal line running from $x ≈ 1.3$ to $x = 0$ . The computation is carried out up to time-like infinity, where $ℐ$ meets the $t$ -axis. The characteristic property of $ℐ$ 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 $i0$ , the numerical computation of global space-times can be achieved in the near future.