An “easy” solution to the Einstein equation (in free space) is the spherically symmetric Schwarzschild solution. The pdf bellow shows how one can use the diffgeom module in order to get the equations describing this solution.

One starts with the most general spherically symmetrical metric and by using Einstein equation \( R_{\mu \nu}=0\) deduces the equations that must be fulfilled by the components of the metric (in the chosen basis).