Simplification strategies for GRTensorII in Maple.

We have found that the global simplification strategy is of central importance for the computer algebra calculation of curvature in spacetime. Summarized here are some general rules of simplification appropriate to GRTensorII running under Maple. Although the general philosophy described here is more widely applicable, our comments are specific to GRTensorII under Maple.

i) There should be some default simplification procedure applied to every step of a calculation. Failure to do this can make a simple calculation intractable. The Maple routine normal is a good starting point for the default. If the calculation involves exponentials ( e.g. the Bondi metric ) the routine expand may be more appropriate. Only if very general functions are involved is it appropriate to consider no default simplification. In general cases this may be the optimal choice.

ii) For both coordinate and tetrad calculations the removal of trigonometric and like functions via elementary coordinate transformations will improve performance.

iii) Simplification of the metric tensor or tetrad components before further calculation will improve performance.

iv) Precalculation and further simplification of the spin coefficients ( and their complex conjugates ) will improve performance only in more complicated cases. The same holds for the Christoffel symbols in the coordinate approach.

v) For further simplification after an object has been calculated, the Maple routine simplify is seldom a good first choice. The routine expand followed by factor is often more appropriate. If the situation is sufficiently general ( e.g. the Debever-McLenaghan-Tariq metric ) there will be no further simplification if normal has been used as default.

vi) If complicated functions are involved, it can be advantageous to substitute the explicit forms of the functions after a more general calculation is completed.

vii) When a calculation is proceeding slowly, it should be halted, the simplification strategy altered, and the worksheet reexecuted.

For most situations these general rules will give adequate performance, and reduce the calculation of curvature for even complex spacetimes to an essentially trivial exercise. Usually, it is the answer that is of interest and not the fact that the simplification strategy is optimal. When optimal simplification strategies are the prime concern theproblem is more involved because of the large number of simplification procedures available* and the size of the resultant parameter space to beexplored.

* GRTensorII provides a menu of 12 distinct commonly used predefined simplification routines with the ability to introduce customized constraints and simplification routines. Any single parameter routine can be applied with gralter, and any multiple parameter routine can be applied with grmap.