Action principle for Numerical Relativity evolution systems

TL;DR

This paper introduces a Lagrangian density for the Z4 evolution system in Numerical Relativity, enabling derivation from an action principle and facilitating constraint-preserving numerical evolution.

Contribution

It provides a novel action principle for Z4, allowing derivation of evolution equations and gauge conditions, and suggests extensions to other formalisms like BSSN and harmonic formulation.

Findings

System is strongly hyperbolic with common gauge conditions

Constraints can be imposed on initial data only

Framework supports symplectic integrators for evolution

Abstract

A Lagrangian density is provided, that allows to recover 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, 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.