[Paper Review] The variational principle for dust shells

V. D. Gladush|arXiv (Cornell University)|Jan 24, 2000

Cosmology and Gravitation Theories1 citations

TL;DR

This paper formulates a variational principle for thin dust shells in General Relativity by combining York-Gibbons-Hawking actions for vacuum regions with a boundary term for dust matter and matching conditions. It derives natural boundary conditions that act as constraints, enabling full reduction of the action for spherically symmetric systems and yielding two equivalent effective actions—describing the shell from interior and exterior observer perspectives—thereby eliminating gravitational degrees of freedom and establishing Hamiltonian constraints via isometry conditions.

ABSTRACT

Variational principles for thin dust shells are considered in Newtonian theory of gravity, and for charged shells in Special Relativity. We propose the variational principle for thin massive dust shells in General Relativity and study the results following from it. The configuration, being considered, can be thought of as the two vacuum regions with the dust shell as the boundary surface between them. The full action of the system we take as the sum of actions in the form of the York-Gibbons-Hawking type for both regions and of the boundary term containing the action of the dust matter and of the corresponding matching surface term. The variational principle is compatible with the boundary-value problem of the Euler-Lagrange equations for either regions of the configuration, and leads to ”natural boundary conditions ” on the shell. These conditions can be considered as the constraints, and, together with the gravitational field equations, are used for elimination of the gravitational degrees of freedom. The full reduction of the action are performed here for spherically-symmetric systems. By transforming the variational formula it is shown that the obtained action with constraints is equivalent to the two variants of the effective action without constraints for a particle with self-action moving in the external gravitational field. One of these variants describes the shell from the interior resting observer’s point of view, another from the exterior one. The conditions of isometry of the exterior and interior sides of the shell lead to the Hamiltonian constraints. I.

Motivation & Objective

• To develop a consistent variational principle for thin massive dust shells in General Relativity.
• To address the boundary-value problem of the Euler-Lagrange equations across the shell by introducing natural boundary conditions.
• To eliminate gravitational degrees of freedom through constraints derived from the variational framework.
• To reduce the full action to effective, unconstrained actions for a particle with self-interaction in an external gravitational field.
• To establish Hamiltonian constraints via isometry conditions between the interior and exterior sides of the shell.

Proposed method

• Formulate the total action as the sum of York-Gibbons-Hawking actions for two vacuum regions and a boundary term for dust matter and matching conditions.
• Apply the variational principle to derive natural boundary conditions on the shell, which act as constraints.
• Perform full action reduction in the spherically symmetric case using symmetry and geometric constraints.
• Transform the variational formula to show equivalence between the constrained action and two effective unconstrained actions.
• Derive two equivalent effective actions: one from the interior observer's frame and one from the exterior observer's frame.
• Use isometry conditions between the interior and exterior sides of the shell to derive Hamiltonian constraints.

Experimental results

Research questions

• RQ1 How can a consistent variational principle be formulated for thin dust shells in General Relativity with proper boundary conditions?
• RQ2 What constraints arise naturally from the variational principle on the shell's worldsheet?
• RQ3 How can gravitational degrees of freedom be eliminated through these constraints in spherically symmetric systems?
• RQ4 What is the physical equivalence of the reduced action, and how does it manifest as effective particle actions with self-interaction?
• RQ5 How do isometry conditions between the interior and exterior sides of the shell lead to Hamiltonian constraints?

Key findings

• The variational principle yields natural boundary conditions on the dust shell, which act as constraints for the system.
• These constraints, combined with the gravitational field equations, allow for the complete elimination of gravitational degrees of freedom in spherically symmetric configurations.
• The full action reduces to two equivalent effective actions without constraints, describing the shell as a self-interacting particle in an external gravitational field.
• One effective action describes the shell from the interior observer’s frame, the other from the exterior observer’s frame.
• The isometry of the interior and exterior sides of the shell leads to Hamiltonian constraints, ensuring consistency of the dynamics across the boundary.
• The equivalence of the two effective actions confirms the physical consistency of the variational formulation across different reference frames.