A generalization of the Zerilli master variable for a dynamical spherical spacetime

TL;DR

This paper extends the Zerilli master variable to dynamical spherical spacetimes with a scalar field, deriving decoupled, gauge-invariant variables and their evolution equations in a Hamiltonian framework.

Contribution

It introduces a generalized Zerilli master variable for dynamical backgrounds with matter, providing a Hamiltonian-based method to decouple and analyze perturbations.

Findings

Derived two master variables: one for gravitational waves, one for matter perturbations.

Obtained simplified evolution equations for the master variables.

Achieved complete decoupling of perturbative degrees of freedom.

Abstract

The evolution of polar perturbations on a spherical background spacetime is analyzed. The matter content is assumed to be a massless scalar field.This provides a nontrivial dynamics to the background and the linearized equations of motion become much more involved than in the vacuum case. The analysis is performed in a Hamiltonian framework, which makes explicit the dynamical role of each of the variables. After performing a number of canonical transformations, it is possible to completely decouple the different perturbative degrees of freedom into constrained, pure-gauge and gauge-invariant variables. In particular, two master variables are obtained: one corresponding to the polar mode of the gravitational wave, whereas the other encodes the complete physical information about the perturbative matter degree of freedom. The evolution equations for these master variables are obtained and simplified.