The continuity equations are ##\nabla_a T^{ab} = 0##. In a coordinate basis, we can write out the resulting differential equations as:(adsbygoogle = window.adsbygoogle || []).push({});

##\nabla_a T^{ab} = \partial_a T^{ab} + \Gamma^a{}_{ac}T^{cb} + \Gamma^b{}_{ac}T^{ac}##

(modulo possible typos, though I tried to be careful-ish).

What do we do in a non-coordinate basis, ##T^{\hat{a}\hat{b}}## where we might have the ricci rotation coefficients ##\omega^a{}_{bc}## (or perhaps ##\omega_{abc}## if that's simpler), rather than the Christoffel symbols?

Right now, I'd just convert back to a coordinate basis, take the covariant deriatiave in that coordinate basis, then reconvert back to the non-coordinate basis, but I'd like to see if there is a more direct way.

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# Writing the continuity equations in a non-coordinate basis

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