Harmonic Coordinates

Satisfy the condition

$$\Gamma^\lambda \equiv g^{\mu\nu} {\Gamma_{\mu\nu}}^\lambda=0,$$ (1)

or equivalently,

$${\partial \over \partial x^\kappa} \left({\sqrt{g} \,g^{\lambda\kappa}}\right)=0.$$ (2)

It is always possible to choose such a system. Using the d'Alembertian Operator,

$$\vbox{\hrule height.6pt\hbox{\vrule width.6pt height6pt \ker... ...pa} - \Gamma^\lambda {\partial \phi\over \partial x^\lambda}.$$ (3)

But since $\Gamma^\lambda\equiv 0$ for harmonic coordinates,

$$\vbox{\hrule height.6pt\hbox{\vrule width.6pt height6pt \kern6.4pt \vrule width.6pt} \hrule height.6pt}^2 x^\mu=0.$$ (4)