Computer algebra in gravity: Programs for (non-)Riemannian spacetimes. I

TL;DR

This paper introduces computer algebra programs using exterior differential forms to analyze (non-)Riemannian spacetimes in gravity, electrodynamics, and gauge theories, demonstrated through an exact solution in a metric-affine gauge theory.

Contribution

It presents new computer algebra programs for decomposing curvature, torsion, and nonmetricity in complex gravitational models using Reduce and Excalc.

Findings

Verified that the exact solution satisfies the field equations.

Demonstrated the application of programs to a Petrov type D metric with electromagnetic and post-Riemannian forms.

Provided a sample session illustrating the computational approach.

Abstract

Computer algebra programs are presented for application in general relativity, in electrodynamics, and in gauge theories of gravity. The mathematical formalism used is the calculus of exterior differential forms, the computer algebra system applied Hearn's Reduce with Schruefer's exterior form package Excalc. As a non-trivial example we discuss a metric of Plebanski & Demianski (of Petrov type D) together with an electromagnetic potential and a triplet of post-Riemannian one-forms. This whole geometrical construct represents an exact solution of a metric-affine gauge theory of gravity. We describe a sample session and verify by computer that this exact solution fulfills the appropriate field equations.-- Computer programs are described for the irreducible decomposition of (non-Riemannian) curvature, torsion, and nonmetricity.