Action principle for numerical-relativity evolution systems

C. Bona and C.Bona-Casas

A Lagrangian density is provided, that allows to recover the Z4 evolution system. The Z4 system is 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 in the initial data. This opens the door to analogous results for other numerical-relativity formalisms, like BSSN, that can be derived from Z4 by a symmetry-breaking procedure.
The harmonic formulation can be easily recovered in this way, suggesting a mechanism for deriving both the field evolution equations and the gauge conditions from a coordinate-dependent action principle, with a view on using simplectic integrators for the numerical evolution.