Generalized Formalism in Gauge-Invariant Gravitational Perturbations

TL;DR

This paper develops a comprehensive gauge-invariant perturbation framework for higher-dimensional warped spacetimes, introducing new variables and equations that unify and extend previous approaches, with applications to Weyl tensor perturbations.

Contribution

It provides the most general gauge-invariant perturbation equations in higher-dimensional warped spacetimes, linking Kodama-Ishibashi variables to Weyl tensor perturbations and defining Teukolsky-like variables.

Findings

Derived general perturbation equations independent of spectral expansions.

Established relations between Teukolsky-like and Kodama-Ishibashi variables.

Demonstrated gauge invariance and completeness of the perturbation theory.

Abstract

By use of the gauge-invariant variables proposed by Kodama and Ishibashi, we obtain the most general perturbation equations in the $(m+n)$-dimensional spacetime with a warped product metric. These equations do not depend on the spectral expansions of the Laplace-type operators on the $n$-dimensional Einstein manifold. These equations enable us to have a complete gauge-invariant perturbation theory and a well-defined spectral expansion for all modes and the gauge invariance is kept for each mode. By studying perturbations of some projections of Weyl tensor in the case of $m=2$, we define three Teukolsky-like gauge-invariant variables and obtain the perturbation equations of these variables by considering perturbations of the Penrose wave equations in the $(2+n)$-dimensional Einstein spectime. In particular, we find the relations between the Teukolsky-like gauge-invariant variables and the Kodama-Ishibashi gauge-invariant variables. These relations imply that the Kodama-Ishibashi gauge-invariant variables all come from the perturbations of Weyl tensor of the spacetime.