GEOMETRIZATION OF LINEAR PERTURBATION-THEORY FOR DIFFEOMORPHISM-INVARIANT COVARIANT FIELD-EQUATIONS .1. THE NOTION OF A GAUGE-INVARIANT VARIABLE

Applying linear perturbation theory to the general-relativistic field equations, in a series of recent papers we have analyzed the gauge pro blem for an almost-Robertson-Walker universe. Mathematically, our anal ysis made use of a rather arbitrary choice of the background space-tim e geometry, and it turns out to possess the undesirable feature that t he basic definitions and concepts are valid only for Einstein's gravit y theory. The main purpose of this paper is to remedy all of the above deficiencies. Consequently, a new geometrical discussion of the notio n of a gauge-invariant variable is presented with a view to demonstrat ing its usefulness in the context of an arbitrary diffeomorphism-invar iant covariant field theory. Another welcome feature of this discussio n is that, for linear perturbation theory, the proposed construction o f gauge-invariant variables does not depend on the specific symmetry p roperties of the background ''space-time'' geometry chosen; in other w ords, it can be proven to hold for any possible choice of the backgrou nd. In a companion paper, such an approach to the gauge problem will e nable us to indicate in universal terms what geometrical objects are i n fact essential if one is to obtain a fully satisfactory description of the equivalence classes of perturbations. A new example of the gene ral structures, as compared with those already investigated for Einste in's gravity theory in the description of an almost-Robertson-Walker u niverse, is also given there. This example arises from consideration o f the infinitesimal perturbation of the metric tensor itself (pure gra vity) defined on a fixed background de Sitter space-time.