Hamiltonian theory for the axial perturbations of a dynamical spherical background

TL;DR

This paper develops a Hamiltonian framework for axial gravitational perturbations on dynamic spherical backgrounds, identifying gauge-invariant variables and deriving a scalar quantity analogous to the Gerlach and Sengupta scalar.

Contribution

It introduces a Hamiltonian approach to axial perturbations on time-dependent backgrounds and constructs a gauge-invariant scalar, extending previous static analyses.

Findings

Derived the Hamiltonian formulation for axial perturbations.

Identified gauge-invariant variables and constructed a scalar quantity.

Established a foundation for analyzing polar perturbations in future work.

Abstract

We develop the Hamiltonian theory of axial perturbations around a general time-dependent spherical background spacetime. Using the fact that the linearized constraints are gauge generators, we isolate the physical and unconstrained axial gravitational wave in a Hamiltonian pair of variables. Then, switching to a more geometrical description of the system, we construct the only scalar combination of them. We obtain the well-known Gerlach and Sengupta scalar for axial perturbations, with no known equivalent for polar perturbations. The strategy suggested and tested here will be applied to the polar case in a separate article.