Covariant divergence formula (${\nabla }_{\mu }{A}^{\mu }=|g{|}^{-1/2}{\partial }_{\mu }(\sqrt{|g|}{A}^{\mu })$)

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.