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IV. A Point of View

We may well accept philosophically the ephemeral point of view, for nothing is eternal in this world. This seems to be the case in Astrophysics nowadays. Still, maintaining the same philosophy even this point of view is not ultimate. Faute de mieux we pull back and think a little: there must be an attribute of eternal in the description of the ephemeral structure and vice versa, some attribute of ephemeral in the description of the eternal structure. The first observation, regarding the situation illustrated in Figure 1, is that the motion starts and ends somewhere, if we discuss on finite time intervals. But the real common characteristic of the two descriptions is that they are geodesic motions for certain space-time metrics. This is their permanent aspect. The ephemeral aspect is then the fact that the geodesics are not complete.

Indeed, equations (11) and (21) can be rewritten as

and this makes more obvious that they represent the geodesics of the indefinite metric

in the cases where this last one is positively definite. The azimuth angle then plays the part of an affine parameter for the geodesics. Exactly the same way the longitude angle is the parameter of the geodesics of (36) in the case where that metric is negative. Indeed, from equation (24) we have, instead of (35)

Note that here the expression of u is conveniently taken as analogous to that from equation (35); in general u is a ratio of two quadratic polynomials in the time of the dynamics just described. Summarizing the results thus far, the components of the motion of a partially free particle are geodesic motions of the same indefinite metric as given by equation (36). The spherical angles are then the affine parameters along two families of geodesics of that metric.

Not that easy to interpret as geodesic motion is the case of the Kepler motion. However, we are able to say that the last of equations (31) represents a geodesic motion. Indeed, consider the metric of the Poincaré representation of the Lobachewsky plane:

A Cartan-like approach of this geometry is the following (Flanders, 1989): the components of the first fundamental form of the surface are (du/v, dv/v). The connection form is du/v. Recently much work has been dedicated to this system, in view of its relationship with soliton dynamics. One of the basic results (Reyes, 2003) is that the algebraic structure generated by these differential forms is preserved by the following Bäcklund transformation:

The metric (38) of this new surface is obviously (ω²)² + (ω³)² and the connection form is ω¹. However we want to treat these differential forms on equal footing. It turns out that the metric

is an indefinite metric belonging to the same algebraic structure. Its geodesics are given by equations

where a is an integration constant. Here we are forced to take the arc length along the geodesics as the dynamical time, for there is no indication of what that time may be. The relationship with the angle φ is

This whole theory is to be described somewhere else. For now we just have the conclusion: the Kepler motion is the spherical image (see equation (31)) of one component of a geodesic motion of the metric (40). Thus the Kepler motion itself is not a geodesic motion but its spherical image is. In a spiral nebula model, choosing an appropriate sphere, this would mean that the star motion is the spherical image of a geodesic motion that takes shape inside the galactic bulge. By the very same token the motion of a planet around the Sun, or that of an electron around atomic nucleus for that matter, is the spherical image of a geodesic motion inside Sun or nucleus. The hard part of the problem is to prove that there is something physical corresponding to that geodesic motion. As to the geometry of the problem we just notice that it requires two geodesic motions in conjunction (equations (41)). This fact involves the extension of the radial coordinate to a complex variable whose imaginary part is the usual radial coordinate: a 2D geometry.

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