symbol of $R^H_{\mu\nu}$ is simply $-\frac12\nabla^2h_{\mu\nu}$, and is strictly elliptic. In effect, rather than projecting out the pure-gauge modes (which would be the result of working in the space of gauge-fixed metrics), we have given them a kinetic term, putting them on the same footing with the physical transverse modes.

We now turn to the question, does a solution to necessarily solve ? We begin with a local analysis. Equation implies $$\label{xieqn}
\nabla^\mu R^H_{\mu\nu} - \frac12\partial_\nu R^H = 
-\frac12\left(\nabla^2\xi_\nu + {R_\nu}^\mu\xi_\mu\right) = 
0\,,$$ which locally has only the trivial solution $\xi^\mu = 0$. Thus $R^H_{\mu\nu} = 0$, which is $D(D+1)/2$ equations for the same number of metric components, is locally equivalent to the $D(D-1)/2$ equations of $R_{\mu\nu} = 0$ and the $D$ equations of $\xi^\mu = 0$.

As usual, the global question is more difficult. However, Perelman has shown that, on a compact manifold without boundary, implies [@Perelman1]. (In Perelman’s theorem, $\xi_\mu$ is a general one-form; it does not refer to the particular form .) It would be interesting to generalize his theorem to manifolds with boundary. (One would have to impose some boundary condition on $\xi^\mu$, such as the vanishing of its normal component. Otherwise one can easily construct 