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The Christoffel trace identity converts the covariant derivative of a vector field into a coordinate divergence weighted by the metric volume density. This is valid for the Levi-Civita connection. Applied to the raised gradient of a scalar field, it gives $\square \varphi =|g{|}^{-1/2}{\partial }_{\mu }(\sqrt{|g|}{g}^{\mu \nu }{\partial }_{\nu }\varphi )$ in Lorentzian signature, the curved d'Alembert operator.

Treat each $x^{\alpha }$ as a scalar field. Its second covariant derivative is ${\nabla }_{\beta }{\nabla }_{\gamma }{x}^{\alpha }=-{\Gamma }^{\alpha }{}_{\gamma \beta }$, so $\square x^{\alpha }=-H^{\alpha }$, where $H^{\alpha }=g^{\beta \gamma }{\Gamma }^{\alpha }{}_{\beta \gamma }$.

On the other hand, metric compatibility givesUsing the Christoffel trace identity in the last term yieldsSince the volume density never vanishes in a valid chart, the harmonic coordinate conditions are equivalent: