wish to solve the vacuum Einstein equation, $$\label{Einstein}
R_{\mu\nu} = 0\,,$$ for a Euclidean-signature metric on a given $D$-dimensional manifold $M$, with appropriate boundary conditions if $M$ has a boundary. For simplicity and concreteness, we are restricting our attention to the vacuum Einstein equation with zero cosmological constant. However, most of our considerations will be local, and including a non-zero cosmological constant and matter fields should not present any significant additional difficulties.

As we will explain below, equation is *weakly elliptic*; roughly speaking, it is elliptic for the physical degrees of freedom in the metric. Most methods for solving elliptic PDEs are relaxation methods, meaning that one starts with an initial guess for the unknown function, which presumably does not solve the PDE, and then iteratively deforms it to obtain functions that solve it to a better and better approximation. This is to be contrasted with methods for solving initial-value problems, associated with hyperbolic or parabolic PDEs, which usually proceed by building up the solution time-slice by time-slice.

Before being able to apply any of the standard numerical methods for elliptic PDEs, one must deal with the issue of gauge invariance. The problem is that is underdetermined. Naively it appears 