One intriguing other possibility would be to implement a finite-element method on the basis of the Regge calculus.)

Before proceeding, it will be useful to understand the spectrum of the operator $\Delta_H$—the linearization of $R^H_{\mu\nu}$—about a metric satisfying $R_{\mu\nu}=0$ and $\xi^\mu=0$.

Spectrum of $\Delta_H$ {#sec:spectrum}
----------------------

With $R_{\mu\nu}=0$ and $\xi^\mu=0$, the operator $\Delta_H$ simplifies as follows; we will also need $\Delta_V$ (see , , , and ): $$\begin{aligned}
\Delta_Hh_{\mu\nu} &=& \Delta_Rh_{\mu\nu} + \nabla_{(\mu}t_{\nu)}
\,, \qquad
t^\mu \equiv -\delta\xi^\mu = -\nabla_\nu h^{\nu\mu} + \frac12\partial^\mu h + (\Gamma^\mu_{\nu\lambda}-\tilde\Gamma^\mu_{\nu\lambda})h^{\nu\lambda}\,.\label{deltaH} \\
\Delta_Vw^\mu &=& -\frac12\nabla^2w^\mu + (\Gamma^\mu_{\lambda\nu} - \tilde\Gamma^\mu_{\lambda\nu})\nabla^\lambda w^\nu\,.\end{aligned}$$

$\Delta_R$ is a self-adjoint operator on the space $\mathcal{H}$ of metric fluctuations, with respect to the inner product $\int_M g^{1/2}(h_{\mu\nu}h^{\mu\nu} - \frac12h^2)$, and therefore has a real spectrum. Let $\mathcal{H}_l\subset\mathcal{H}$ be the space of pure-gauge fluctuations, i.e. those of the form $h_{\mu\nu} = \nabla_{(\mu}w_{\nu)}$ for some vector field $w^\mu$. $\mathcal{H}_l$ is annihilated by $\Delta_R$, since the Ricci tensor remains zero under a gauge transformation. The quotient space $\mathcal{H}/\mathcal{H}_l$ is the space of physical metric fluctuations. On this space $\Delta_R$ may have a finite number of negative or zero eigenvalues. For example, if there is a moduli space of solutions, then each modulus will give rise 