Initial Conditions in General Relativity: Lapse and Shift Formulation

We examine the system of coupled differential equations to which the constraints on the Cauchy data reduce if expressed in terms of the ``shift'' vector Nk and ``lapse'' N0. If (3)gij and ∂ (3)gij/∂t are given and Dirichlet boundary conditions are imposed, the solution Nk is found to be unique if 2 × (energy density) − (three-curvature) > 0, but need not be unique when this inequality is not satisfied. No general existence theorem is known, but we list some conditions which make solutions impossible.