I wrote a program that takes as input the basis vectors if electing to use an orthonormal basis, or metric components if using the coordinate basis, and outputs non-zero Christoffel symbols and components of Riemann, Ricci, and Einstein tensors, as well as the Ricci scalar.
I could include functionality to support a non-coordinate and non-orthonormal basis, but I don’t want to waste my time if that’s something that no one ever uses in GR. I know that I don’t know enough about GR yet to make a call on this, so I’m asking you all!
Best Answer
Coordinate bases are rarely orthonormal in GR. They're often orthogonal (in which case the metric is diagonal), but in general the basis vectors associated with each coordinate do not have a norm of $\pm1$. If you could truly establish an orthonormal set of coordinate basis vectors, then I'm pretty sure that your space would be flat.
Non-orthogonal coordinate bases are less common but are far from rare. Examples include "tortoise" or "Gullstrand-Painlevé" coordinates for Schwarzchild spacetime, or standard coordinates for Kerr spacetime (i.e., a rotating black hole.)