Pair of null gravitating shells III. Algebra of Dirac's observables

TL;DR

This paper advances the understanding of the algebra of Dirac observables in a two-shell gravitating system by deriving explicit formulas for the symplectic structure and Poisson brackets in a specific gauge, applicable to multiple shells.

Contribution

It provides a new explicit formula for the Poisson brackets of Dirac observables in a double-null gauge for any number of shells, and constructs the symplectic form for the physical phase space.

Findings

Derived the pull back of the Liouville form in DNEF gauge.

Expressed conjugate variables as simple combinations of DNEF coordinates.

Calculated the symplectic form and transformation for shell-crossing scenarios.

Abstract

The study of the two-shell system started in ``Pair of null gravitating shells I and II'' (gr-qc/0112060--061) is continued. The pull back of the Liouville form to the constraint surface, which contains complete information about the Poisson brackets of Dirac observables, is computed in the singular double-null Eddington-Finkelstein (DNEF) gauge. The resulting formula shows that the variables conjugate to the Schwarzschild masses of the intershell spacetimes are simple combinations of the values of the DNEF coordinates on these spacetimes at the shells. The formula is valid for any number of in- and out-going shells. After applying it to the two-shell system, the symplectic form is calculated for each component of the physical phase space; regular coordinates are found, defining it as a symplectic manifold. The symplectic transformation between the initial and final values of observables for the shell-crossing case is written down.