one may simply impose the gauge-fixing condition from the outset. Assuming the gauge fixing is complete, the Einstein equation will be a strictly elliptic operator on the space of metrics obeying the gauge-fixing condition. In our case, however, the space of gauge-fixed metrics is not so easy to describe explicitly.

Instead of imposing the gauge-fixing condition a priori, we therefore propose to solve $\xi^\mu = 0$ simultaneously with $R_{\mu\nu} = 0$. This can be done quite elegantly by combining the two equations as follows: $$\label{EDT}
R^H_{\mu\nu} \equiv R_{\mu\nu} - \nabla_{(\mu}\xi_{\nu)} = 0\,,$$ which we will refer to as the *Einstein-DeTurck equation*. The tensor $R^H_{\mu\nu}$ appears in work of DeTurck [@MR697987]; he gave a simple proof of short-time existence for the Ricci-flow, $\partial g_{\mu\nu}/\partial\lambda = - 2R_{\mu\nu}$, by pointing out that it is diffeomorphically equivalent to the strictly parabolic PDE $\partial g_{\mu\nu}/\partial\lambda = -2R^H_{\mu\nu}$. Roughly speaking, the term $\nabla_{(\mu}\xi_{\nu)}$ supplies the longitudinal components that are missing in the Ricci tensor. The linearization of $R^H_{\mu\nu}$ is $$\label{deltaRH}
\delta R^H_{\mu\nu} \equiv \Delta_Hh_{\mu\nu} = 
\Delta_Lh_{\mu\nu} -\frac12\pounds_\xi h_{\mu\nu} + \nabla_{(\mu}u_{\nu)} \,, \qquad
u^\mu \equiv (\Gamma^\mu_{\nu\lambda}-\tilde\Gamma^\mu_{\nu\lambda})h^{\nu\lambda} \,.$$ The second and third terms on the right-hand side do not contain second derivatives of $h_{\mu\nu}$, so the principal 