to consist of $D(D+1)/2$ equations, per point, for $D(D+1)/2$ unknowns (the metric components). However, as a consequence of diffeomorphism invariance, the Ricci tensor for any metric satisfies the Bianchi identity, $\nabla^\mu R_{\mu\nu} - \frac12\partial_\nu R = 0$, which has $D$ components. Hence actually represents only $D(D-1)/2$ non-trivial equations. Stated more precisely, the principal symbol of the Ricci tensor (considered as a nonlinear operator on the metric) annihilates pure gauge modes of the metric (corresponding to small diffeomorphisms), while it is positive definite on physical modes. This is the technical meaning of “weakly elliptic".

The same issue, of course, arises in the Lorentzian context, where the Einstein equation is only weakly hyperbolic, and our discussion will to a large extent parallel the standard treatment of the latter problem using harmonic coordinates (see for example section 10.2 of [@Wald]). As we will explain in subsection 2.1, the weak ellipticity makes the Einstein equation unsuitable for numerical solution without some form of gauge fixing. Of course, there are many different possible gauge-fixing schemes, many of them adapted to particular problems. Here we are looking for one that can be applied as generally as possible. A local analysis of the problem of gauge-fixing leads 