Action principle for Numerical Relativity evolution systems

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.

Revised version, includes explicit expresions for gauge sources corresponding to 1+log and gamma-driver gauge conditions ('punctures' gauge)

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• An Introduction to Well-posedness and Free-evolution
• GRAthena++: puncture evolutions on vertex-centered oct-tree AMR
• Numerical stability of the Z4c formulation of general relativity
• Introduction to Numerical Relativity
• Large Eddy Simulations of Magnetized Mergers of Black Holes and Neutron Stars
• Generalized Harmonic Equations in 3+1 Form
• Action Principle for the Generalized Harmonic Formulation of General Relativity
• Testing merging of Dark Exotic Stars from Gravitational Waves in the Multi-messenger approach
• Hypergraph Discretization of the Cauchy Problem in General Relativity via Wolfram Model Evolution