On Relativistic Material Reference Systems

TL;DR

This paper explores material reference systems in general relativity, showing perfect fluids as a special case of relativistic elastic media, and analyzes the Hamiltonian structure when coupled to gravity, especially for dust.

Contribution

It demonstrates that perfect fluids are a subset of relativistic elastic media and provides a Hamiltonian formulation with explicit true Hamiltonian for dust.

Findings

Perfect fluids are a special case of relativistic elastic media.

The Hamiltonian analysis reveals constraints and the form of the true Hamiltonian.

Explicit true Hamiltonian derived for dust medium.

Abstract

This work closes certain gaps in the literature on material reference systems in general relativity. It is shown that perfect fluids are a special case of DeWitt's relativistic elastic media and that the velocity--potential formalism for perfect fluids can be interpreted as describing a perfect fluid coupled to a fleet of clocks. A Hamiltonian analysis of the elastic media with clocks is carried out and the constraints that arise when the system is coupled to gravity are studied. When the Hamiltonian constraint is resolved with respect to the clock momentum, the resulting true Hamiltonian is found to be a functional only of the gravitational variables. The true Hamiltonian is explicitly displayed when the medium is dust, and is shown to depend on the detailed construction of the clocks.