Embedding variables in the canonical theory of gravitating shells

TL;DR

This paper reformulates the canonical theory of spherical gravitating shells with light-like dust, introducing embedding variables and Dirac observables, and clarifies the background manifold's gauge invariance and uniqueness.

Contribution

It reduces the action functional to Kuchař form using embeddings and Dirac observables, providing a gauge-invariant and unique background manifold framework.

Findings

Reduced dynamics describes motion on a unique background manifold.

Reformulation ensures gauge invariance of the background structure.

Clarifies the role of embeddings and Dirac observables in the system.

Abstract

A thin shell of light-like dust with its own gravitational field is studied in the special case of spherical symmetry. The action functional for this system due to Louko, Whiting, and Friedman is reduced to Kucha\v{r} form: the new variables are embeddings, their conjugate momenta, and Dirac observables. The concepts of background manifold and covariant gauge fixing, that underlie these variables, are reformulated in a way that implies the uniqueness and gauge invariance of the background manifold. The reduced dynamics describes motion on this background manifold.