will not be.) Therefore we need to control which representative of the diffeomorphism class we aim for. What will go wrong in practice if we do not? During the relaxation process, since short-wavelength pure-gauge modes are not damped out, small errors will accumulate, leading to a loss of control.

There are of course many different ways to fix a gauge in general relativity. One can fix certain metric components; for example, in the presence of spherical symmetry one can set the coordinate $r$ to equal the proper radius of the sphere. Alternatively, one can take advantage of certain special properties of the metric. If one is solving for a Kähler metric, then one can choose to employ complex coordinates; the only allowed gauge transformations are then holomorphic diffeomorphisms, which have no local degrees of freedom. (This is actually a special case of harmonic coordinates, which we will discuss below.) Our aim in this paper, however, is to find an all-purpose method for solving , one that requires as little special knowledge of the metric one is trying to find as possible.

Let us return to our local analysis. We wish to forbid the pure-gauge fluctuations; in other words we wish 