Euler-Poincaré Formulation of Barotropic Fluids Coupled with ADM Gravity

TL;DR

This work develops a geometric mechanics framework to derive Euler-Poincaré equations for a self-gravitating barotropic fluid by reducing covariant relativistic hydrodynamic variational principles to a 3D Eulerian description via a $3+1$ ADM decomposition. By gauge-fixing the fluid map and applying Lagrangian reduction, the authors obtain fluid momentum equations in fixed and moving frames, along with explicit expressions for the stress-energy tensor and the coupled constraint equations within ADM gravity. A Kelvin circulation theorem is established in both inertial and moving frames, highlighting the preserved variational structure across frames. The formalism paves the way for cross-pollination with Newtonian CFD techniques and offers a foundation for future extensions to GRMHD and multifluid systems in numerical relativity.

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-Poincaré 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-Poincaré 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.

Figures (2)

• Figure 1: Schematic representation of the Pull-back formalism of relativistic hydrodynamics. The world-history $W$ contains material space $(M)$ and time $(\mathbb{R})$ labels of the barotropic fluid parcels, and the time label is projected out using the trivial bundle $\pi:W\to M$. The fluid labels are mapped onto physical spacetime $\mathbf{M}$ by the embedding $\Psi\in Emb(W,\mathbf{M})$.
• Figure 2: Schematic representation of equivalent variational principles of a general relativistic barotopic fluid. a) and b) are covariant, field-theoretic descriptions with actions in the form of \ref{['taubpullback']} using the fluid map and metric, whereas c), d), and e) are $3+1$ formalisms with actions in the form of \ref{['S3+1']}, using ADM variables for the metric, and the Eulerian structure highlighted in \ref{['redlag']} in Section \ref{['Section 4']} to describe hydrodynamics. "$=$" signs connect a) to c) \ref{['S']}, b) to d), and d) to e) \ref{['uhat']}\ref{['J0hat']}, meaning they are related by a change of variables. "$\leftrightarrow$" signs connect a) to c) and b) to d), as they are related by spacetime gauge transformations, as shown in \ref{['rotgauge']}.