us to a generalized form of harmonic gauge that is somewhat more covariant than the textbook version. Then in subsection 2.2 we advocate an *a posteriori* method of implementing the gauge-fixing, in which the gauge-fixing condition is not imposed from the outset but rather solved for at the same time as the Einstein equation. Specifically, we propose solving not itself but another PDE (see ) that we call the Einstein-DeTurck equation. The Einstein-DeTurck equation simultaneously encodes the Einstein equation and the gauge-fixing condition, and is strictly elliptic. In the next section, we will discuss algorithms for solving the Einstein-DeTurck equation.

Gauge condition
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If we make a small perturbation $\delta g_{\mu\nu} = h_{\mu\nu}$ to the metric, the change in the Ricci tensor is, to first order, $$\label{deltaR}
\delta R_{\mu\nu} \equiv \Delta_Rh_{\mu\nu} = 
\Delta_Lh_{\mu\nu} + \nabla_{(\mu}v_{\nu)}\,,$$ where $$\label{Lichvdef}
\Delta_Lh_{\mu\nu} \equiv 
-\frac12\nabla^2h_{\mu\nu} - R_\mu{}^\kappa{}_\nu{}^\lambda h_{\kappa\lambda} + {R_{(\mu}}^\kappa h_{\nu)\kappa} \,,\qquad v_\mu \equiv \nabla_\nu {h^\nu}_\mu - \frac12\partial_\mu h\,.$$ For the moment we are concerned with the local properties of $\Delta_R$, so we consider perturbations with wavelength much smaller than any curvature scale of the metric, and therefore take $g_{\mu\nu}$ to be constant; this yields the principal symbol $\Delta_R^{\rm principal}$ (the part of $R_{\mu\nu}$ with 