Well-posed constrained evolution of 3+1 formulations of General Relativity

We present an analysis of well-posedness of constrained evolution of 3+1 formulations of GR. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of ADM, BSSN and other 3+1 formulations derived from ADM by adding combinations of constraints to the right-hand-side of ADM and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical Arnowitt--Deser--Misner (ADM) and many other 3+1 formulations of general relativity.