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 general relativity. 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 Arnowitt-Deser-Misner (ADM), Baumgarte-Shapiro-Shibata-Nakamura and other $3+1$ formulations derived from the ADM formulation by adding combinations of constraints to the right-hand side of the ADM formulation 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 ADM and many other $3+1$ formulations of general relativity.