Spherically symmetric dust shell and the time problem in canonical relativity

The dynamics of a self-gravitating spherical dust shell is analyzed within the Hamiltonian formulation. Given any time variable on the shell, a complete reduction with respect to the momentum constraints is obtained and the true degrees of freedom of the composed “matter + gravity” system are found. The transition between two such formulations, corresponding to two different time parametrizations, is described as a canonical transformation in the reduced phase space. Explicit formulas for such a transformation are given. Implications for quantum gravity are discussed.