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).

The results of this paper generalize and clarify several results obtained by Anderson [2], Scheibe [3], Hiskes [4], Recami and Rodrigues [5], Friedman [6], Fock [7], and Scanavini [8]. Among the novelties here, there is the realization that the definitions ofG _I T andG _X T can be given only when certain boundary conditions for the equations of motion ofT can be physically realizable in the domainU $U \subseteq M$ M, where a given reference frame is defined. The existence ofphysically realizable boundary conditions for eachτ ∈ModT (in ∂U), in contrast with the mathematically possible boundary condition, is then seen to be essential for the validity of a principle of relativity forT.

The methodology of the present paper has been applied to several topics of spacetime physics with very interesting results. Here we mention:

• (i) The Newtonian concepts of absolute space and absolute time can be presented in a very elegant way as “species of structure”. One of the surprising results is that we succeeded in finding a Lorentzian structure [9] in Newtonian spacetime without introducing any new explict geometrical object in the original structure. The Newtonian spacetime structure and its relation to the relativistic spacetime structure and to the structure of the spacetime of the so-called Lorentz aether theories [11,12] is fully discussed in [13].

• (ii) It is possible to present in a novel and unified way the question concerning experiments designed to detect a possible breakdown of Lorentz invariance, a subject we already dedicated attention to in Rodrigues and Tiomno [11,12] and Rodrigues [14,15]. A full account of this subject will be published elsewhere.

• (iii) In Rodrigues and Scanavini [16], we proved that there are models of General Relativity that contain a canonical privileged locally inertial reference frame that can be physically distinguished from any other frame by experiments doneinside the frame.

Although the formalism of this paper may at first sight look very abstract, actually it is easy to aplly it to specific theories. We present an example at the end of the paper which is sufficiently general to show “in action” almost all concepts introduced in this paper.