Action principle for numerical-relativity evolution systems

A Lagrangian density is provided that allows the recovery of the Z4 evolution system from an action principle. The resulting system is then strongly hyperbolic when supplemented by gauge conditions like “$1+\log $” or “freezing shift,” suitable for numerical evolution. The physical constraint $Z_{\mu }=0$ can be imposed just on the initial data. The corresponding canonical equations are also provided. This opens the door to analogous results for other numerical-relativity formalisms, like BSSN (Baumgarte-Shapiro-Shibata-Nakamura), that can be derived from Z4 by a symmetry-breaking procedure. The harmonic formulation can be easily recovered by a slight modification of the procedure. This provides a mechanism for deriving both the field evolution equations and the gauge conditions from the action principle, with a view on using symplectic integrators for a constraint-preserving numerical evolution. The gauge sources corresponding to the “puncture gauge” conditions are identified in this context.