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In GR, one often uses harmonic (or wave) coordinates to simplify things. Now, one definition involves the coordinates themselves:

$$ \Box_g x^{\alpha} = 0 $$

where $ \Box_g = g_{\mu \nu}\nabla^{\mu}\nabla^{\nu} $ and the $ \nabla $'s stand for covariant derivatives.

Now, in recent studies, I often encountered an apparently equivalent definition, i.e. the following:

$$\partial_{\beta}g^{\alpha \beta} = 0$$

Does anybody now how to show this equivalence or can someone give me a hint? Thanks a lot in advance.

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  • $\begingroup$ These are not entirely equivalent, do you have a reference? $\endgroup$ – Ryan Unger Aug 25 '16 at 23:53

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