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I. Introduction

The principle of equivalence, in the form first formulated by Einstein (Einstein, 1967) stipulates the equivalence of inertia with the gravitation. The formulation was clearly intended to facilitate the circumvention of classical dynamical principles. It is worth considering Einstein’s reasoning a little closer, for it gives us a perspective on the subject of the present essay. In short, Newton set the Space as an active cause in the problem of inertia. As Einstein put it, however,”…it is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) that acts but which cannot be acted upon” (Einstein, 1967, p. 54). Thus, Mach’s ideas should be placed on the position of an attempt to eliminate the Space as an active cause from Mechanics. Specifically, Mach’s contention is that the unaccelerated motion is always done not relatively to Space but to the center of mass of the matter in the Universe. This conception involves however the action of Matter through Space, so that we have to consider inertia as determined by ”…field properties of space, analogous to electromagnetic field” (Einstein, 1967, p. 55). But, as long as we abide by the principles of Classical Mechanics, as Mach did, we have no way of making this program work: ”The concepts of classical mechanics afford no way of expressing this” (Einstein, ibid.). It is here the point where Einstein calls on stage his formulation of the equivalence principle, in order to correct the deficiency of the classical principles that offer no explanation of the equality between the inertial and gravitational masses. In simple terms, because the two masses are equal (which is a sound experimental fact verified by Eötvös) we can very well disregard the cause of acceleration, so that any acceleration is equivalent to an acceleration due to gravitational field (or vice versa). Consequently, we can treat the equations of motion the way we treat the gravitational field and vice versa, etc, etc. Thus, the inertia must be considered a field, for it is the same with the gravitation field from this point of view. Based on this, Einstein created his famous program, a crowning jewel of human wisdom. However, this program never succeeded completely, so that even today its different aspects are under debate. An early signal was drawn by De Sitter, who was the first to notice ”field properties of pure space”, hardly accepted at that time by Einstein himself. (Reading the above quoted sequences in (Einstein, 1967) one wonders how an Einstein - De Sitter debate could ever take place. There is no other explanation to it but the fact that Einstein did not comprehend his own idea in all its implications. Only in time grew him on accepting the inherent, for the book is obviously written post debate. Nevertheless, he undoubtedly did never cope entirely with the situation.)

      However, this observation amplified along the time, so that today we are in possession of that unique feature allowing us to revive Mach’s ideas, namely to confer ” field properties to space, analogous to electromagnetic field ”. It turns out that these field properties are related to a SL (2, R) group structure of the Einstein equations in vacuum and electromagnetic vacuum. In the present essay we fully exploit this property of field equation, which we take as a property of space-time on the grounds that field equations in absence of matter simply refer to Space. However, in doing so, we simply disregard the gravitational properties of Space, and this would seem to take us back to the very starting reasoning of Einstein’s. This is, however not so.

      Dennis Sciama (Sciama, 1969) upheld the idea that the two forces – inertial and gravitational – are not equivalent, more precisely that they are only locally equivalent. In a positive way, Sciama’s theory offers an algebraic expression for the inertial force in the spirit of Mach’s principle. Indeed, this expression explicitly embodies the idea that the inertial forces are due to distant matter. First, they seem to be related to what we would like to call ensemble properties of the material forces. Secondly, they can be obtained as solutions of Einstein vacuum field equations. What is missing here is a treatment of this problem in terms of fields, and this essay seeks to fill the gap. Here we propose a ‘measurement’ approach of the relation between the local gravitational field and the long range inertial field: the actual form of a local planetary system is a measure of the gravitational field at a larger scale, here assimilated with the inertial field. Of course, this approach assumes a dichotomy of what we know as gravitational field, either classical or relativistic, akin to the dichotomy between inertial mass and gravitational mass. We take the idea that the Newtonian gravitational force is not of the same nature with the force due to distant bodies even deeper than it has ever been taken. The first of these forces is internal to a Keplerian system, the last one is due to a universal field (this term will be made more precise in due time). Even though their descriptions are different, the geometrical structure involved at the core of these descriptions is the same: both are intimately related to certain realizations of SL (2, R) group structure. This fact makes the problem easily manageable even though in a somewhat unusual manner.

      The chief detail of this work is the applicability of continuous group theory to two apparently different problems. It turns out that a group isomorphic to SL (2, R) describes the apriori variation of the physical parameters of a Kepler motion. Likewise, a group isomorphic to SL (2, R) describes the gravitational field, taken as solution of Einstein field equations for vacuum and electromagnetic vacuum. The manners of these descriptions are different; however the isomorphism is an attractive feature from the outset: could not one take it into account in order to show how the planetary motion comes formally across in an inertial field and thus add something to the problem of inertial field itself? We show here that this is possible. In order to do it we borrow the Kleinian point of view according to which Geometry is the science of some group invariants. Occasionally, this kind of approach has already been suggested in physical problems not a long time ago (Jaynes, 1973). The problem here is, however, a little more complex: on one hand, we have to build such invariants for two different theories and, on the other hand, we have to find how they can be related to one another. The solution seems to present itself in the form of joint invariants of isomorphic groups. We will base our analysis on a theorem of M. Stoka (Stoka, 1968) extracted here in a little modernized form as presented by M. Leuci and A. M. Pastore (Leuci, Pastore, 1994).

Let G_r (x¹, x², … , xⁿ) and H_r ( α¹, α², …, α^q ) be two isomorphic r – parameter Lie groups, with the infinitesimal generators

(1.01)

Then the following statements are equivalent:

1^o. There is a family of p – dimensional manifolds with equations

(1.02)

admitting G_r() as invariance group.

2^o.The matrix obtained by joining the components of X’s and A’s from equation (1):

(1.03)

has rank s < n + q.

If one of these statements holds true, then the family of invariant manifolds is given by the system of equations

(1.04)

where are the independent integrals of the system of partial differential equations

(1.05)

An arrow over a letter confers the meaning of vector to that letter, and we used the summation convention over repeated indices of different variance. One word of caution is now in order: from geometrical point of view H_r is the group of parameters associated to the group of variables G_r, indicating a possible subordination. From physical point of view it does not seem worth maintaining such a hierarchy: in different physical situations the variables may play the role of parameters as well. It is in order to avoid such a misunderstanding that we use here the above term joint invariant functions of two isomorphic groups and not invariant manifolds – the preferred geometrical term.

This theorem, referred to as the Stoka Theorem from now on, is the instrument we are using in the given theoretical environment as sketched above. Likewise, the partial differential system (1.05) will be referred to as Stoka System. In order to profitably use it we need to find physical realizations of SL (2, R) structure we are interested in here. The next two sections define these physical realizations.

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