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II. The Complete Group of Space

Whenever we state a problem regarding the Universe as a whole, we always have in mind the Space as an apriori condition of existence of the Universe. Within this premise, such a problem can be set as follows: the motion of Earth (our compulsory reference frame) should be a continuous motion as referred to the Space, and its nature is such as to make all the events possible as we perceive them. Bound to this reference frame the Man surveys the Space: he “goes” from place to place and compares the local properties of the Space. The quotation marks are explained by the fact that going does not necessarily imply here displacement. As a matter of fact Man’s displacement is limited only to Earth’s journey in Space, and this is quite a limitation! “Going” refers mostly to inferences according to the information as received from places. Sometimes even the location of the places that issued information is full of twists and turns. In a word, the surveying of Space is, again, a matter of imagination at least to a good extent. Anyway, we can geometrically characterize this survey by a motion reflecting the continuity with respect to a parameter:

Here x, y, z are the coordinates in Space with respect to our unavoidable reference frame and t is any time parameter we may use to characterize the continuity. Here the Space and Time are apriori intuitions – they do not have the conceptual Riemannian character. Fact is that we do not know the nature of motion, and this is why, while characterizing the motion as uniform, its velocity components are functions of the point in Space. The problem now is to find the most general form of these functions.

    One can find a solution to this problem starting from a certain criterion, and the best geometric criterion to be used in Physics seems to be that related to the metric of Space. This is always accessible to local evaluations. As we travel through the Space, this metric is liable to change. It is here the point where the Space isotropy has to enter the stage, and one can use this approach in order to find the functions X, Y, Z (Bromwich, 1901). Bromwich sets up the issue to determine the group of conformal transformations as defined by the fact that “the magnification is a function only of x, y, z and not of the direction of the line-element”. Of course, he talks of the magnification of the line-element (whose length is our metric), and the excerpt just quoted is nothing less of a primordial geometrical definition of isotropy.

    In short Bromwich’s result is the following: the continuous group of the problem is a group with ten parameters in three variables, x, y, z. In the good fashion of Lie group theory, the ten group generators are three translations as represented by the vectors

three rotations as represented by the vectors

three conformal inversions as represented by the vectors

where r is the position vector length , and R is the tenth vector of this algebra representing an extension of the position vector in its own direction

Of these transformations, only those generated by (2) and (3) are, as a rule, considered when one comes to discuss Space isotropy, and here is the best place to see that isotropy, conceived as group invariance of the metric, actually means more. The vectors from (4) generate inversions with respect to a sphere, and it is these inversions that impose scale considerations when one comes to discuss isotropy. Indeed, the finite transformation generated by (4) depends on three parameters – the components of some constant vector . It can be written as:

This transformation can be interpreted by decomposing it into three successive steps. Let us say that we discovered some physical properties of Space inside a sphere of radius R₀, this sphere representing the spatial extension of our knowledge in a certain phenomenological framework. The Space outside this sphere is represented by points – and the point is understood here as the agency by which matter accesses the Space – which can be reached directionally from those inside the sphere by inversions with origin as their pole:

Notice once again that the position vector and its transform are collinear: this transformation preserves direction. Suppose now that we want to explore the Space outside our sphere. There may be some kind of uncontrolled features that inconvenience this exploration (Otherwise we would not have limitations in distance along a direction). For instance we need to account for the fact that the origin of the reference frame moves, or the Space outside the sphere may actually stretch thus carrying points with it, or both these features at once. This is generally reflected in the fact that instead of studying what goes on in the point given by (7), we may end up studying the phenomena in the point

where is a vector cumulating the causes as above. Bringing the vector (8) back inside our sphere, the image is no more the initial position vector, but a new one as given by equation (6) with

The transformation generated by vector (5) – the isotropic extension of sphere – comes into play when the sphere itself with respect to which we perform the inversion of (8) changes its radius to R₀ + a₀, so that instead of equation (6) one has

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