Computer algebra in gravity: Programs for (non-)Riemannian spacetimes. I
  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. 
Jose Socorro, Alfredo Macias, Friedrich W. Hehl