Dynamics of a self-gravitating shell of matter

TL;DR

This paper derives the dynamics of a self-gravitating matter shell using a Hamiltonian framework, addressing singular curvature and energy-momentum tensors, and discusses the implications of metric continuity assumptions.

Contribution

It introduces a Hamiltonian formulation for self-gravitating shells and defines singular Riemann tensors via distributional derivatives, extending differential geometry tools.

Findings

Proves Bianchi identities for singular curvature match conservation laws.

Establishes validity of Rosenfeld-Belinfante and Noether theorems for singular objects.

Discusses the impact of metric continuity assumptions on the theory.

Abstract

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. A method used here enables us to define singular Riemann tensor of a non-continuous connection {\em via} standard formulae 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. Rosenfed-Belinfante and Noether theorems are proved to be still valid in case of these singular objects. Assumption about continuity of the four-dimensional spacetime metric is widely discussed.