High-order perturbations of a spherical collapsing star

TL;DR

This paper extends a formalism for high-order perturbations to spherical collapsing stars, providing explicit equations and matching conditions crucial for understanding complex stellar collapse dynamics.

Contribution

It applies a high-order perturbation formalism to perfect fluid stars, deriving explicit source decompositions and evolution procedures for the perturbations.

Findings

Explicit second-order source decomposition in tensor spherical harmonics

General evolution procedure for perturbations of perfect fluid stars

High-order matching conditions across stellar surfaces

Abstract

In Ref. [1, 2] a formalism to deal with high-order perturbations of a general spherical background was developed. In this article, we apply it to the particular case of a perfect fluid background. We have expressed the perturbations of the energy-momentum tensor at any order in terms of the perturbed fluid's pressure, density and velocity. In general, these expressions are not linear and have sources depending on lower order perturbations. For the second-order case we make the explicit decomposition of these sources in tensor spherical harmonics. Then, a general procedure is given to evolve the perturbative equations of motions of the perfect fluid for any value of the harmonic label. Finally, with the problem of a spherical collapsing star in mind, we discuss the high-order perturbative matching conditions across a timelike surface, in particular the surface separating the perfect fluid interior from the exterior vacuum.