Formal structures, the concepts of covariance, invariance, equivalent reference frames, and the principle Relativity

Abstract

In this paper a given spacetime theoryT is characterized as the theory of a certainspecies of structure in the sense of Bourbaki [1]. It is then possible to clarify in a rigorous way the concepts ofpassive andactive covariance ofT under the action of the manifold mapping groupG M . For eachT, we define also aninvariance groupG I T and, in general,G I T ≠G M . This group is defined once we realize that, for eachτ ∈ModT, each explicit geometrical object defining the structure can be classified as absolute or dynamical [2]. All spacetime theories possess alsoimplicit geometrical objects that do not appear explicitly in the structure. These implicit objects are not absolute nor dynamical. Among them there are thereference frame fields, i.e., “timelike” vector fieldsX ∈TU,\(U \subseteq M\)M, whereM is a manifold which is part ofST, a substructure for eachτ ∈ModT, called spacetime. We give a physically motivated definition of equivalent reference frames and introduce the concept of theequivalence group of a class of reference frames of kind X according to T, G X T. We define thatT admits aweak principle of relativity (WPR) only ifG X T ≠ identity for someX. IfG X T =G I T for someX, we say thatT admits a strong principle of relativity (PR).