Dynamics of a self gravitating light-like matter shell: a gauge-invariant Lagrangian and Hamiltonian description

TL;DR

This paper develops a gauge-invariant Lagrangian and Hamiltonian framework for self-gravitating light-like matter shells, introducing geometric tools and proving key identities to describe their dynamics.

Contribution

It provides a complete gauge-invariant Lagrangian and Hamiltonian formulation for null matter shells, including extrinsic curvature, Gauss-Codazzi equations, and energy-momentum tensor definitions.

Findings

Derived gauge-invariant Lagrangian and Hamiltonian for light-like shells

Proved Gauss-Codazzi equations and Bianchi identities for null hypersurfaces

Defined energy-momentum tensor-density unambiguously in invariant form

Abstract

A complete Lagrangian and Hamiltonian description of the theory of self-gravitating light-like 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. Energy-momentum tensor-density of a light-like matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. Noether identity and Belinfante-Rosenfeld theorem for such a tensor-density are proved. Finally, the Hamiltonian dynamics of the interacting system: ``gravity + matter'' is derived from the total Lagrangian, the latter being an invariant scalar density.