Lagrangian and Hamiltonian formalism for discontinuous fluid and gravitational field

TL;DR

This paper develops a comprehensive Lagrangian and Hamiltonian formalism for discontinuous fluids coupled with gravity, including surface and volume formulations, applicable to astrophysical scenarios like star surfaces and thin shells.

Contribution

It introduces a novel covariant surface form of the Hamiltonian for discontinuous fluids in gravity, extending variational principles to non-symmetric, discontinuous systems.

Findings

Formulated covariant Hamiltonian with surface and volume terms.

Derived variational principles for discontinuous fluid-gravity systems.

Analyzed the geometrical and constraint structure of the Hamiltonian.

Abstract

The barotropic ideal fluid with step and delta-function discontinuities coupled to Einstein's gravity is studied. The discontinuities represent star surfaces and thin shells; only non-intersecting discontinuity hypersurfaces are considered. No symmetry (like eg. the spherical symmetry) is assumed. The symplectic structure as well as the Lagrangian and the Hamiltonian variational principles for the system are written down. The dynamics is described completely by the fluid variables and the metric on the fixed background manifold. The Lagrangian and the Hamiltonian are given in two forms: the volume form, which is identical to that corresponding to the smooth system, but employs distributions, and the surface form, which is a sum of volume and surface integrals and employs only smooth variables. The surface form is completely four- or three-covariant (unlike the volume form). The spacelike surfaces of time foliations can have a cusp at the surface of discontinuity. Geometrical meaning of the surface terms in the Hamiltonian is given. Some of the constraint functions that result from the shell Hamiltonian cannot be smeared so as to become differentiable functions on the (unconstrained) phase space. Generalization of the formulas to more general fluid is straifgtforward.