Initial Data in General Relativity Described by Expansion, Conformal Deformation and Drift

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant mean curvature (CMC) solutions of the constraint equations, it is unknown how well it applies in the non-CMC setting, and there have been indications that it encounters difficulties there. We are therefore motivated to investigate alternative generalizations of the CMC conformal method. Introducing a densitized lapse into the ADM Lagrangian, we find that solutions of the momentum constraint can be described in terms of three parameters. The first is conformal momentum as it appears in the standard conformal method. The second is volumetric momentum, which appears as an explicit parameter in the CMC conformal method, but not in the non-CMC formulation. We have called the third parameter drift momentum, and it is the conjugate momentum to infinitesimal motions in superspace that preserve conformal class and volume form up to independent diffeomorphisms. This decomposition of solutions of the momentum constraint leads to extensions of the CMC conformal method where conformal and volumetric momenta both appear as parameters. There is more than one way to treat drift momentum, in part because of an interesting duality that emerges, and we identify three candidates for incorporating drift into a variation of the conformal method.