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IV.  Relation Between Planetary Motion and Inertial Field

Physically, the invariant metrics (2.19) and (3.15) open a possibility to describe the interaction between local Kepler motion and inertial field. The SL (2, R) structure involved here allows us to say that, as long as we consider the planetary motion in inertial field, it is this field that determines its parameters, and we can model this formally via Stoka Theorem. Indeed, the above invariance of the inertial field equations corresponds to a differential realization of SL (2, R) structure whose infinitesimals are the known Beltrami operators (Stoka 1968), which can be written at will, either as

or, in real terms h = u + iv as

The same can be said in terms of the variable z, characterizing the planetary motion. Thus we are in the position where we have two realizations of the SL (2, R) structure, one representing the planetary motion, the other representing the field, and we want to describe a relationship between the planetary motion and the field. This is the point where we can fruitfully use the Stoka Theorem. Writing the Stoka System (1.05) for the operators (4.01) and their correspondents in variable z for planetary motion

we notice that this system has the rank 3, so there is only one independent integral. This is the cross ratio generated by the two complex numbers:

where ρ is real and we took its square to account for the fact that the cross ratio (4.04) is always positive. Any joint invariant function is here a regular function of this cross ratio. This function does not say much by itself; however it is a valuable heuristic tool. Indeed, suppose that we know the cross ratio, in the sense that we are able to extract its values from our experience. As these values are always positive, we can assume them smaller than one, making thus legitimate the notation ρ º tanhψ where ψ is a solution of Laplace equation in Space. Then the number z is related to h by the linear relation

where h º u+iv and h₀ is given by (3.19). The complex number z related to Kepler motion in inertial field is thus practically obtained by replacing the imaginary unit by the complex number h characterizing this inertial field. As shown before ρ has the meaning of eccentricity of the Kepler motion, while α is its orientation ω. On the other hand, equation (4.05) can be interpreted as a Israel-Wilson condition given in equation (3.10). This fact enables one to draw the conclusion that there should be a relation between the electromagnetic field and the Kepler motion, as indeed historically has been the case. However we do not elaborate now on this conclusion without first analyzing the apriori possibility of quantization of the Kepler motion, and then showing that this quantization is intimately related to electromagnetic field as a transition field. This is, indeed, a determination of universality of the electromagnetic field, thereby relating this field with the inertia. The outcome has first been captured by Einstein himself, in the form of his celebrated conclusion about the equivalence between inertia and energy.

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