to require $\partial_\nu {h^\nu}_\mu - \frac12\partial_\mu h = 0$. We need to de-linearize this equation, i.e. to find a one-form $\xi_\mu$ depending on the metric (and of course *not* diffeomorphism-invariant) whose variation $\delta\xi_\mu$ about a constant $g_{\mu\nu}$ equals $\partial_\nu {h^\nu}_\mu - \frac12\partial_\mu h$. Fixing $\xi_\mu$ will then project out the pure-gauge modes. The simplest such one-form is $$\label{xidef1}
\xi_\mu = g^{\lambda\nu}\left(
\partial_\lambda g_{\nu\mu} - \frac12\partial_\mu g_{\lambda\nu}
\right)\,.$$ Here are three expressions for the dual vector field $\xi^\mu$: $$\xi^\mu = g^{\lambda\nu}\Gamma^\mu_{\lambda\nu} = -\nabla^2x^\mu = -g^{-1/2}\partial_\nu\left(g^{1/2}g^{\mu\nu}\right) \,,$$ where $x^\mu$ are the coordinates considered as scalar functions and $\nabla^2$ is the scalar Laplacian. The gauge choice $\xi^\mu = 0$ is the *harmonic gauge*, commonly employed in Lorentzian numerical relativity.

Although it does the job, the harmonic gauge is a bit inelegant insofar as it refers to a particular coordinate system. In particular, $\xi_\mu$ transforms in a complicated way on chart overlaps, and a metric that is in harmonic gauge in one patch will generally not be in that gauge in a neighboring one. This problem is easily fixed by replacing the coordinate derivatives in with any fixed covariant derivative. We thus choose an arbitrary background metric $\tilde g_{\mu\nu}$ on $M$, and define $$\label{xidef2}
