5 Acknowledgements4 Numerical Issues4.4 What has been done?

4.5 What should be done? 

Now we want to discuss some of the problems which still have to be dealt with. Since the conformal approach has been tested numerically so far mostly in 2D cases with unphysical global structure, it is necessary to implement full 3D codes to run on general enough data. Work on this is well underway and first results which confirm the expectations based on the two-dimensional codes are available [82].

Apart from the requirement of having a general 3D-code, there are other problems which need more consideration. Let us start with the boundary condition at the boundary of the computational domain. It would be interesting to see how the results of [59Jump To The Next Citation Point In The Article] translate to the conformal field equations. This would provide mathematically reasonable boundary conditions at the edge of the computational domain. Their implementation could result in stable codes which do not need any additional transition zones beyond tex2html_wrap_inline3905, and which are compatible with the evolution equation unlike the procedure used in [38Jump To The Next Citation Point In The Article].

Another problem has to do with the constraint equations. The necessity of dividing by the conformal factor to construct the initial data is annoying. The way around this problem is to solve the conformal constraints directly. While it has been possible to do this in the spherically symmetric case [77Jump To The Next Citation Point In The Article], no result is available for the general case. It would be very desirable to have another way of constructing the initial data, because then one could construct more easily data which evolve into space-times with multiple black holes.

As a last problem in connection with the conformal approach one should mention that Friedrich's work on tex2html_wrap_inline3683 provides a way to evolve Cauchy data specified on an asymptotically Euclidean hypersurface to hyperboloidal initial data. A code which does that kind of evolution can provide the initial data for an evolution code for the hyperboloidal initial value problem. This area is entirely unexplored. Surely there will be difficult problems in the numerical treatment of the transport equations related to the total characteristic at spatial infinity. But the work on this problem is worthwhile because it would provide the final step to the ultimate goal of a global simulation of an isolated system.

A problem which affects all the work in numerical relativity today is the obscure nature of the gauge conditions. Currently there is not much understanding of the effects of a gauge condition on the resulting nature of the coordinates (frame, conformal factor). Most of the work done on these problems is related to the choice of a lapse function; in particular several proposals have been made for selecting a time coordinate. These are mostly dictated by formal considerations like the need of making a system hyperbolic or of easy implementation. To some extent this is justified because the physics cannot depend on the coordinates which are used. But it is also well known that there are ``good'' and ``bad'' coordinates. What ``good'' and ``bad'' means depends to a large extent on what the goal is.

Ideally, coordinates should be tied to the geometry so that they obtain a more invariant nature. In 1D cases one can set up a system of double null-coordinates (and a derived system of time and space coordinates). This provides a gauge which is good as long as the geometry is well behaved [77]. But, unfortunately, this gauge cannot be generalized in a straightforward way to higher dimensions (some attempts have been made in [38]). Probably one should assume a pragmatic viewpoint towards the problem of finding appropriate coordinates in the sense that one should regard the gauge sources as knobs which have to be adjusted by trial and error. Maybe it is possible, at least to some extent, to let the code do the ``twiddling'' automatically. This requires that one should be able to formulate criteria for a ``good solution'' which can be checked by the computer. Furthermore, it is also necessary that a change in the gauge sources does not change the characteristics of the system, because otherwise it is easy to get into situations where the system is not hyperbolic anymore.

Finally, there is no doubt that the issue of the correct boundary condition for codes based on the standard Einstein equations also needs more attention. Such codes should try to implement the boundary conditions given in [59]. Although it is not clear what these conditions mean physically, chances are good that they will produce stable codes (see e.g. the discussion of numerical boundary conditions in [73]). If this is the case then one could try to check which ones work best by comparison with exact radiating solutions, like the boost-rotation symmetric solutions discussed in [20]. Another test should be a numerical comparison with the data computed by a hyperboloidal evolution code. Such tests are important because so far the standard codes have only been tested against linearized solutions. This is the regime where one would expect them to work because the boundary condition is still benign. In this way, one could not only select physically reasonable boundary conditions for the standard codes, one could also check how well they perform in comparison with the conformal codes. In particular, one could see how accurate the radiation extraction can be done with those codes and whether the accuracy is good enough for LIGO wave forms. Then one can compute them with a safe conscience using the standard codes, provided they are more efficient than the conformal codes.



5 Acknowledgements4 Numerical Issues4.4 What has been done?

image Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
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