Geometrization of linear perturbation theory for diffeomorphism-invariant covariant field equations. I. The notion of a gauge-invariant variable

Abstract

Applying linear perturbation theory to the general-relativistic field equations, in a series of recent papers we have analyzed the gauge problem for an almost-Robertson-Walker universe. Mathematically, our analysis made use of a rather arbitrary choice of the background space-time geometry, and it turns out to possess the undersirable feature that the basic definitions and concepts are valid only for Einstein's gravity theory. The main purpose of this paper is to remedy all of the above deficiencies. Consequently, a new geometrical discussion of the notion of a gauge-invariant variable is presented with a view to demonstrating its usefulness in the context of an arbitrary diffeomorphism-invariant covariant field theory. Another welcome feature of this discussion is that, for linear perturbation theory, the proposed construction of gauge-invariant variables does not depend on the specific symmetry properties of the background “space-time” geometry chosen; in other words, it can be proven to hold for any possible choice of the background. In a companion paper, such an approach to the gauge problem will enable us to indicate in universal terms what geometrical objects are in fact essential if one is to obtain a fully satisfactory description of the equivalence classes of perturbations. A new example of the general structures, as compared with those already investigated for Einstein's gravity theory in the description of an almost-Robertson-Walker universe, is also given there. This example arises from consideration of the infinitesimal perturbation of the metric tensor itself (pure gravity) defined on a fixed background de Sitter space-time.