The Rest-Frame Instant Form of Relativistic Perfect Fluids and of Non-Dissipative Elastic Materials

TL;DR

This paper develops a Hamiltonian and covariant formalism for relativistic perfect fluids and elastic materials, providing explicit mass formulas and coupling to gravity, advancing the theoretical understanding of these continuous media in relativistic settings.

Contribution

It introduces a rest-frame instant form for perfect fluids and elastic materials, with explicit Hamiltonian mass expressions and a unified formalism for coupling to gravity.

Findings

Closed-form Hamiltonian mass for dust and photon gas

Wigner-covariant rest-frame instant form established

Formalism applicable to non-dissipative elastic materials

Abstract

For perfect fluids with equation of state $\rho = \rho (n,s)$, Brown gave an action principle depending only on their Lagrange coordinates $\alpha^i(x)$ without Clebsch potentials. After a reformulation on arbitrary spacelike hypersurfaces in Minkowski spacetime, the Wigner-covariant rest-frame instant form of these perfect fluids is given. Their Hamiltonian invariant mass can be given in closed form for the dust and the photon gas. The action for the coupling to tetrad gravity is given. Dixon's multipoles for the perfect fluids are studied on the rest-frame Wigner hyperplane. It is also shown that the same formalism can be applied to non-dissipative relativistic elastic materials described in terms of Lagrangian coordinates.