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

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.Comment: 20 pages, Latex file using amstex, some references correcte

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

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.Comment: 26 pages, Latex file using amstex, some references correcte

Gravitational Constraint Combinations Generate a Lie Algebra

We find a first--order partial differential equation whose solutions are all ultralocal scalar combinations of gravitational constraints with Abelian Poisson brackets between themselves. This is a generalisation of the Kucha\v{r} idea of finding alternative constraints for canonical gravity. The new scalars may be used in place of the hamiltonian constraint of general relativity and, together with the usual momentum constraints, replace the Dirac algebra for pure gravity with a true Lie algebra: the semidirect product of the Abelian algebra of the new constraint combinations with the algebra of spatial diffeomorphisms.Comment: 10 pages, latex, submitted to Classical and Quantum Gravity. Section 3 is expanded and an additional solution provided, minor errors correcte

Geometry of crossing null shells

New geometric objects on null thin layers are introduced and their importance for crossing null-like shells are discussed. The Barrab\`es--Israel equations are represented in a new geometric form and they split into decoupled system of equations for two different geometric objects: tensor density ${\bf G}^a{_b}$ and vector field $I$. Continuity properties of these objects through a crossing sphere are proved. In the case of spherical symmetry Dray--t'Hooft--Redmount formula results from continuity property of the corresponding object.Comment: 24 pages, 1 figur

Pair of null gravitating shells I. Space of solutions and its symmetries

The dynamical system constituted by two spherically symmetric thin shells and their own gravitational field is studied. The shells can be distinguished from each other, and they can intersect. At each intersection, they exchange energy on the Dray, 't Hooft and Redmount formula. There are bound states: if the shells intersect, one, or both, external shells can be bound in the field of internal shells. The space of all solutions to classical dynamical equations has six components; each has the trivial topology but a non trivial boundary. Points within each component are labeled by four parameters. Three of the parameters determine the geometry of the corresponding solution spacetime and shell trajectories and the fourth describes the position of the system with respect to an observer frame. An account of symmetries associated with spacetime diffeomorphisms is given. The group is generated by an infinitesimal time shift, an infinitesimal dilatation and a time reversal.Comment: 28 pages, 9 figure included in the text, Latex file using amstex, epic and graphi

Action functionals of single scalar fields and arbitrary--weight gravitational constraints that generate a genuine Lie algebra

We discuss the issue initiated by Kucha\v{r} {\it et al}, of replacing the usual Hamiltonian constraint by alternative combinations of the gravitational constraints (scalar densities of arbitrary weight), whose Poisson brackets strongly vanish and cast the standard constraint-system for vacuum gravity into a form that generates a true Lie algebra. It is shown that any such combination---that satisfies certain reality conditions---may be derived from an action principle involving a single scalar field and a single Lagrange multiplier with a non--derivative coupling to gravity.Comment: 26 pages, plain TE