Euler-Poincar\'e Formulation of Barotropic Fluids Coupled with ADM Gravity

Abstract

This paper develops a geometric mechanics framework for the reduction of general relativistic hydrodynamic variational principles, from the variation of worldlines approach in 4D spacetime to 3-dimensional Eulerian descriptions. We consider a self-gravitating, barotropic fluid and obtain the Euler-Poincar\'e equations of the system by Lagrangian reduction. Using the decomposition of general relativity into 3 + 1 dimensions, with a direction of time defined, the gauge invariance of the action over spacetime diffeomorphisms permits a 3-dimensional description of the fluid diffeomorphism by gauge fixing. The configuration space thus mirrors the Newtonian case, and by employing the Euler-Poincar\'e theorem, we derive the Eulerian equations of motion, in the same form as the PDEs from Newtonian fluid dynamics. The equations of motion are then derived, where the fluid variables are measured in a separate frame of reference from the variables involving gravity. Furthermore, the general relativistic barotropic fluid exhibits a Kelvin-Noether circulation conservation, which we derive in both the inertial and moving frames of reference. Finally, potential applications to Numerical Relativity are discussed.