Causal Propagation of Constraints and the Canonical Form of General Relativity

Studies of new hyperbolic systems for the Einstein evolution equations show that the “slicing density” α(t, x) can be freely specified while the lapse N = αg cannot. Implementation of this small change in the Arnowitt-Deser-Misner action principle leads to canonical equations that agree with the Einstein equations whether or not the constraints are satisfied. The constraint functions, independently of their values, then propagate according to a first order symmetric hyperbolic system whose characteristic cone is the light cone. This result follows from the twice-contracted Bianchi identity and constitutes the central content of the constraint “algebra” in the canonical formalism. In this paper I will show one way to follow a path to some small, but striking improvements in the Arnowitt-Deser-Misner (ADM) canonical form of general relativity. [1, 2] Arlen Anderson, collaborator in the primary reference [3] on this subject, and I followed a different track in the presentation given elsewhere. [3] Several aspects of the ADM canonical form of general relativity [1, 2] have been improved recently. [3] Slight modifications give rise to possibly significant changes of perspective. The modifications were suggested by the structure of the evolution equations written in new, manifestly hyperbolic, forms, not directly relevant to the present work, in which only physical characteristic speeds appear. [4, 5, 6, 7, 8, 9, 10, 11] One may see some aspects of those results as flowing to, or from, the action principle. Here I focus on the consequences of a direct demonstration that the constraint functions, whether they vanish or not, always propagate as dictated by a first order symmetric hyperbolic system whose only characteristic cone is the Dedicated to Richard Arnowitt