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 almostRobertson-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 undesirable 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 almostRobertson-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.

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

“…Here it should perhaps be noticed that the tensor bundle S A* was originally introduced to define S* and S~r) by equations (2.15b) and (2.16b) appearing in Banach and Piekarski (1997), and that the possibility to describef~r~ in terms offa,p is a direct consequence of these equations. In exactly the same way it will be immediate to find the explicit form of Drs '.…”

“…-< q, where q is an integer which has exactly the same meaning as in equations (2.21) of Banach and Piekarski (1997). 5 Assume further that to every gauge-invariant perturbation…”

“…In Banach and Piekarski (1997), we constructed the gauge-invariant variables in such a way that for each choice offo e F~r).r, the object G(',f~r), ") defines a mapping from the quotient space F/FL into a set of cross sections of T. Denoting this set by F(T), we thus have…”

“…Therefore, it should not be surprising that, to avoid the risk of appearing to be repetitive and even trite, without further comment we shall use here those symbols and notions which either appear for the first time in Banach and Piekarski (1997) or are reasonably standard, and the analysis proceeds in a way similar to that already made familiar. Recalling the viewpoint presented in Banach and Piekarski (1997, Sections 3.1 and 4.1), one can define the gauge-invariant perturbation [s'] ~ F/FL associated with s' ~ F as the equivalence class of tangents ~' F, not necessarily satisfying the linearized field equations, which are equivalent to s' E F. This definition does not tell us directly how to use [s'] in practical calculations, or whether such calculations are possible at all.…”

“…Based on a geometrical foundation, in Banach and Piekarski (1997), we formulated linear perturbation theory for an arbitrary system of diffeomorphism-invariant, covariant field equations. In particular, we discussed the gauge problem, which has been known for a long time and whose consideration has led to so much of the current work in cosmology (Bardeen, 1980;Ellis and Bruni, 1989;Mukhanov et al, 1992), and examined, from essentially new points of view, the notion of a gauge-invariant variable of order r. Among many other things, it was possible to present the construction of various gaugeinvariant variables without ever specifying the detailed form of covariant field equations and without ever analyzing the symmetry properties, if any, of the background "space-time" geometry chosen.…”