Covariance, Geometricity, Setting, and Dynamical Structures on Cosmological Manifold

TL;DR

This paper advocates a coordinate-free, geometric approach to the principle of general covariance in cosmology, clarifying concepts like symmetry and invariance through structures on the manifold.

Contribution

It introduces the notion of setting elements on cosmological manifolds, emphasizing geometric and dynamical structures over group-theoretical methods.

Findings

Clarified the distinction between symmetry, covariance, and invariance.

Proposed a geometric framework based on manifold structures.

Introduced the concept of setting elements for dynamical analysis.

Abstract

The treatment of the principle of general covariance based on coordinate systems, i.e., on classical tensor analysis suffers from an ambiguity. A more preferable formulation of the principle is based on modern differential geometry: the formulation is coordinate-free. Then the principle may be called ``principle of geometricity.'' In relation to coordinate transformations, there had been confusions around such concepts as symmetry, covariance, invariance, and gauge transformations. Clarity has been achieved on the basis of a group-theoretical approach and the distinction between absolute and dynamical objects. In this paper, we start from arguments based on structures on cosmological manifold rather than from group-theoretical ones, and introduce the notion of setting elements. The latter create a scene on which dynamics is performed. The characteristics of the scene and dynamical structures on it are considered.