Algorithms for computer algebra calculations in spacetime I. The calculation of curvature

TL;DR

This paper compares different algorithms for calculating spacetime curvature, emphasizing that the efficiency depends on the simplification strategy rather than the algorithm itself, within the GRTensorII system.

Contribution

It demonstrates that the choice of simplification strategy significantly impacts computational performance in curvature calculations, challenging the notion of a universally best algorithm.

Findings

No single algorithm is fastest across all cases.

Simplification strategy is crucial for performance.

Proper simplification can make complex problems solvable instantly.

Abstract

We examine the relative performance of algorithms for the calculation of curvature in spacetime. The classical coordinate component method is compared to two distinct versions of the Newman-Penrose tetrad approach for a variety of spacetimes, and distinct coordinates and tetrads for a given spacetime. Within the system GRTensorII, we find that there is no single preferred approach on the basis of speed. Rather, we find that the fastest algorithm is the one that minimizes the amount of time spent on simplification. This means that arguments concerning the theoretical superiority of an algorithm need not translate into superior performance when applied to a specific spacetime calculation. In all cases it is the global simplification strategy which is of paramount importance. An appropriate simplification strategy can change an untractable problem into one which can be solved essentially instantaneously.