only second derivatives of the metric): $$\Delta_R^{\rm principal}h_{\mu\nu} =
- \frac12\nabla^2h_{\mu\nu} - \frac12\partial_\mu\partial_\nu h + \partial_\lambda\partial_{(\nu} {h_{\mu)}}^\lambda\,.$$ Any such perturbation can be decomposed as $h_{\mu\nu} = \hat h_{\mu\nu} + \partial_{(\mu}w_{\nu)}$, where $\partial_\nu\hat h^\nu{}_\mu -\frac12\partial_\mu\hat h = 0$; $\hat h_{\mu\nu}$ is the transverse (physical) part, and $\partial_{(\mu}w_{\nu)}$ is the longitudinal (pure gauge) part. Specifically, the perturbation $\partial_{(\mu}w_{\nu)}$ is the change in the metric under the small diffeomorphism generated by the vector field $w^\mu$, and is annihilated by $\Delta_R^{\rm principal}$. On the other hand, $\Delta_R^{\rm principal}$ acts as $-\frac12\nabla^2$ on $\hat h_{\mu\nu}$.

We will now briefly explain why the weak ellipticity of is problematic for numerical purposes. For concreteness, suppose that we represent the metric by the values of its components on a set of lattice points in each coordinate patch (the argument does not change if we use some other representation). In general, to be representable the metric components must be smooth on the (coordinate) scale of the lattice spacing. However, this smoothness condition is not preserved by diffeomeorphisms. Thus, even if our lattice is sufficiently fine to represent some particular metric that solves , other metrics in its diffeomorphism class will not be representable. (In some sense, “most" such metrics 