Harmonic coordinate

A harmonic coordinate satisfies ${\square }_g{x}^{\lambda }=0$, equivalently $g^{\mu \nu }{\Gamma }_{\mu \nu }^{\lambda }=0$ or ${\partial }_{\mu }(\sqrt{-g}{g}^{\lambda \mu })=0$.

Harmonic slicing imposes the harmonic-coordinate condition only on time. In 3+1 form it obeys $({\partial }_t-{\beta }^i{\partial }_i)\alpha =-{\alpha }^{2}K$.

The Bona--Masso family is $({\partial }_t-{\beta }^i{\partial }_i)\alpha =-{\alpha }^{2}f(\alpha )K$. Harmonic slicing is the choice $f=1$.

In harmonic gauge, the vacuum Einstein equations have principal part $-g^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }{g}_{\alpha \beta }/2$, forming a quasilinear wave system supplemented by constraints.