= Harmonic reduction of the vacuum Einstein equations
{title2=$R_{ab}=-\tfrac12g^{\mu\nu}\partial_\mu\partial_\nu g_{ab}+Q_{ab}(g,\partial g)$}
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.
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