TensoriaCalc tackles (semi-)Riemannian tensor calculus problems encountered in general relativity, cosmology, and field theory. Currently, it calculates geometric objects – Christoffel symbols, the Riemann curvature tensor, Ricci tensor and scalar, etc. – given a metric and the relevant coordinates; and performs basic operations such as covariant derivatives of tensors.
xAct is a suite of free and actively maintained packages for tensor computer algebra in Mathematica. xAct implements state-of-the-art algorithms for fast manipulations of indices and has been modelled on the current geometric approach to General Relativity. It is highly programmable and configurable. Since its first public release in March 2004, xAct has been intensively tested and has solved a number of hard problems in GR.