Harmonic coordinate

A harmonic coordinate satisfies ${\Delta }_g{x}^{\lambda }=0$ for a Riemannian metric, or ${\square }_g{x}^{\lambda }=0$ for a Lorentzian metric. Equivalently $g^{\mu \nu }{\Gamma }_{\mu \nu }^{\lambda }=0$ or ${\partial }_{\mu }(\sqrt{|\det g|}{g}^{\lambda \mu })=0$. This common coordinate condition applies in both Riemannian geometry and general relativity.

With the Riemann curvature tensor convention $R^{\mu }{}_{\nu \rho \sigma }={\partial }_{\rho }{\Gamma }^{\mu }{}_{\nu \sigma }-{\partial }_{\sigma }{\Gamma }^{\mu }{}_{\nu \rho }+\cdots$, harmonic coordinates remove the mixed second derivatives from the Ricci tensor. The remainder $Q_{ab}$ contains at most first derivatives of the metric tensor. The vacuum Einstein field equations become a system of quasilinear partial differential equations with a Lorentzian wave equation principal part. Harmonic coordinate constraints must be maintained when using the reduced system.

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

For a positive lapse function $\alpha$, shift vector ${\beta }^i$, and the convention $K_{ij}=-({\partial }_t{\gamma }_{ij}-{\mathcal{L}}_{\beta }{\gamma }_{ij})/(2\alpha )$ for extrinsic curvature of a spatial hypersurface, the time-coordinate harmonic slicing condition yields the displayed advective lapse equation. Combining ${\partial }_t(\sqrt{\gamma }/\alpha )={\partial }_i(\sqrt{\gamma }{\beta }^i/\alpha )$ with the spatial volume evolution identity proves it without setting the shift vector to zero.

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

For the stationary conformally flat Schwarzschild slicing with areal radius $R=r+M$, lapse $\alpha =r/(r+M)$, and radial shift ${\beta }^r=Mr/(r+M{)}^2$, the Bona--Masso condition selects

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.