High-order gauge-invariant perturbations of a spherical spacetime

TL;DR

This paper develops a comprehensive, gauge-invariant framework for analyzing high-order nonspherical perturbations in spherical spacetimes, including explicit formulas and addressing mode coupling issues.

Contribution

It introduces a general algorithm for constructing gauge-invariant perturbations and extends the formulation to second-order metric perturbations.

Findings

Mode coupling causes invariance issues for higher-order perturbations.

Explicit formulas for second-order metric invariants are provided.

Gauge invariance depends on specific conditions in the perturbation analysis.

Abstract

We complete the formulation of a general framework for the analysis of high-order nonspherical perturbations of a four-dimensional spherical spacetime by including a gauge-invariant description of the perturbations. We present a general algorithm to construct these invariants and provide explicit formulas for the case of second-order metric perturbations. We show that the well-known problem of lack of invariance for the first-order perturbations with l=0,1 propagates to increasing values of l for perturbations of higher order, owing to mode coupling. We also discuss in which circumstances it is possible to construct the invariants.