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X = X

^{i}∂

_{i}

Now, I know that normally, the basis vectors ∂

_{i}can be derived by taking the derivatives of the position vector for the coordinate system with respect to all the axes like this:

∂

_{i}= ∂R/∂x

^{i}

where R is a position vector such as R = [rsin(θ)cos(∅), rsin(θ)sin(∅), rcos(θ)] for spherical coordinates.

This is how you normally derive such basis vectors.

However, how do you do it in GR where you don't have such R vectors (or at least they don't seem obvious to me)?

For instance, how would I derive the basis vectors for a metric like the Godel metric, which has the following line element?:

ds

^{2}= (1/(2ω

^{2}))[-c

^{2}dt

^{2}- 2ce

^{x}dzdt + dx

^{2}+ dy

^{2}- (1/2)e

^{2x}dz

^{2}]

I don't think that it would just be a matter of differentiating the position vector with respect to the axes, because in the case of this metric I'm about 95% sure that the position vector here is just [ ct, x, y, z ]. However, the metric tensor has terms of ω in it as well as terms of e. Obiviously, none of the derivatives of this position vector contain any such terms. This is problematic because you are supposed to be able to dot product two basis vectors together in order to get an element of the metric tensor (which I just noted to contain terms of ω and e).

This is not even to mention issues of sign convention and attaching a negative to at least one basis vector or metric tensor element?

Long story short:

How do I derive the basis vectors for spacetime metrics in GR where I don't have a clear cut R vector like I do with spherical coordinates in 3D?