Has anybody used tensors in Mathematica? How to properly work with them? I find Mathematica not very friendly in this field, as I am defining my own functions for lowering & raising indices, multiplication and stuff like that. I was wondering if there is some good package or a secret way to use tensors more properly in Mathematica. For example, I need tensor analysis for general relativity kind of calculations.
Mathematica 9 contains some functionality for working with symbolic tensors.
Here's a list of packages in no particular order, that may have some functionality for working with symbolic tensors.
TensoriaCalc - intended for basic calculations in general relativity, but not finished (calculates only Christoffel symbols, Riemann and Ricci tensor). Parallel working with many metrics is possible. Symbolic calculations are not supported.
grt - intended for basic calculations in general relativity, but full of bugs (only Christoffel symbols fully function). Symbolic calculations are not supported.
NCAlgebra, for manipulating non-commuting algebraic expressions and computing non-commutative Gröbner bases. It allows working with symbolic matrices and symbolic block matrices (e.g. symbolic block matrix inversion).
xAct - a package designed by researchers for large scale projects in general relativity; subpackages capable of extensive tensor manipulation (xTensor, xCoba) as well as perturbation theory in general relativity to any order (xPert). Other subpackages can also work with tensor spherical harmonics, spinor computations as well as exterior calculus (diferential forms).
Ricci (last updated Sep 2011)
diffgeo (free) - a very simple package for differential geometry. Works only with given basis and metric
See also the Wikipedia page on Tensor Software.