Dynamics of a self-gravitating shell of matter

The dynamics of a self-gravitating shell of matter is derived from the Hilbert variational principle and then described as an (infinite-dimensional, constrained) Hamiltonian system. The method used here enables us to define a singular Riemann tensor of a noncontinuous connection via standard formulas of differential geometry, with derivatives understood in the sense of distributions. Bianchi identities for the singular curvature are proved. They match the conservation laws for the singular energy-momentum tensor of matter. The Rosenfed-Belinfante and Noether theorems are proved to be valid still in the case of these singular objects. The assumption about the continuity of the four-dimensional space-time metric is widely discussed.

Figure 1

Integrated portion of space-time.