# Equivalence of definitions of harmonic (or wave) coordinates

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.

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