Pair of null gravitating shells II. Canonical theory and embedding variables

TL;DR

This paper extends the analysis of null gravitating shells by generalizing the action to multiple shells, deriving the symplectic structure using embedding variables, and developing a method to solve the resulting equations.

Contribution

It introduces a generalized action for multiple null shells and constructs the canonical structure using embedding variables and Dirac observables.

Findings

Generalized action functional for multiple null shells.

Derived symplectic structure using Kuchař decomposition.

Developed a method to solve coupled PDEs in this context.

Abstract

The study of the two shell system started in our first paper ``Pair of null gravitating shells I'' (gr-qc/0112060) is continued. An action functional for a single shell due to Louko, Whiting and Friedman is generalized to give appropriate equations of motion for two and, in fact, any number of spherically symmetric null shells, including the cases when the shells intersect. In order to find the symplectic structure for the space of solutions described in paper I, the pull back to the constraint surface of the Liouville form determined by the action is transformed into new variables. They consist of Dirac observables, embeddings and embedding momenta (the so-called Kucha\v{r} decomposition). The calculation includes the integration of a set of coupled partial differential equations. A general method of solving the equations is worked out.