Geometrization of linear perturbation theory for diffeomorphism-invariant covariant field equations. II. Basic gauge-invariant variables with applications to de sitter space-time

In a companion paper, a systematic treatment of linearized perturbations and a new geometric definition of gauge-invariant variables, based on the theory of vector bundles and applicable to the case of an arbitrary system of covariant field equations, were carefully presented. One of the purposes of the present paper is to specify a necessary and sufficient condition that a given, finite set of gaugeinvariant variables, denoted collectively by ω and referred to as the complete set of basic variables, can be used to extract the equivalence classes of perturbations from ω in a unique way. The above set is complete because it has the following property: a knowledge of ω is all one needs in the sense that ifx represents an arbitrary point of the “space-time” manifoldX andG denotes any gauge-invariant tensor field onX, then the value ofG atx∈X is uniquely specified by giving the germs of basic gauge-invariant variables atx∈X. Arguments are proposed that ω also has a stronger property which is more immediately useful: anyG is obtainable directly from the basic variables through purely algebraic and differential operations. These results are of practical interest, and one concrete setting where one is led to the explicit definition of ω occurs when considering the infinitesimal perturbation of the metric tensor itself (pure gravity) defined on a fixed background de Sitter space-time and obeying the linearized empty-space Einstein equations with nonnegative cosmological constant Λ; the case Λ=0 corresponds to linear perturbation theory in Minkowski space-time.