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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!

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  • $\begingroup$ This is typically called vierbein, and used in many ways (e.g. to simplify certain calculations, in Einstein-Cartan theory or for defining spin on curved space times). More info: en.wikipedia.org/wiki/Tetrad_formalism $\endgroup$ Nov 3, 2021 at 21:25

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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.)

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I’ll take the above comment by Sebastian Riese as my answer:

This is typically called vierbein, and used in many ways (e.g. to simplify certain calculations, in Einstein-Cartan theory or for defining spin on curved space times). More info: https://en.wikipedia.org/wiki/Tetrad_formalism

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