Computer algebra in gravity: Reduce-Excalc programs for (non-) Riemannian space-times. I

Abstract

Computer algebra is applied to general relativity, to electrodynamics, and to gauge theories of gravity. As mathematical formalism we use the calculus of exterior differential forms and as computer algebra system Hearn's Reduce with Schrüfer's exterior form package Excalc. As a nontrivial example we discuss a metric of Plebański & Demiański (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.