Dynamics of a self-gravitating lightlike matter shell: A gauge-invariant Lagrangian and Hamiltonian description

A complete Lagrangian and Hamiltonian description of the theory of self-gravitating lightlike matter shells is given in terms of gauge-independent geometric quantities. For this purpose the notion of an extrinsic curvature for a null-like hypersurface is discussed and the corresponding Gauss-Codazzi equations are proved. These equations imply Bianchi identities for spacetimes with null-like, singular curvature. The energy-momentum tensor density of a lightlike matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. The Noether identity and Belinfante-Rosenfeld theorem for such a tensor density are proved. Finally, the Hamiltonian dynamics of the interacting $“\mathrm{gravity}+\mathrm{matter}”$ system is derived from the total Lagrangian, the latter being an invariant scalar density.