Embedding variables in the canonical theory of gravitating shells

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.