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One of the most discussed elements coming out of the identification of a field with a coordinate system is the vector field. The vectors are always related to an affine geometry. As Hermann Weyl noticed, they can be connected to the metric geometry only under the condition of defining a length (Weyl, 1952). Weyl does not elaborate in great detail on the subject, but we shall consider here that a certain vector defines a length, only as a result of a measurement. Indeed, a length involves matter in a specific space form, while the abstract notion of vector is actually connected to the action of a field of forces, or to quantities which don’t have an obvious space form. In this respect, the previous connection between analyticity and Laplace’s equation in space seems to point out to a specific interpretation of the elastic forces, leading to a natural generalization of them.

Assume indeed that the host space of the matter has a metric which, in some general coordinates, is given by the quadratic form

The length defined by a vector can be obtained from this metric just by replacing the differentials of coordinates with the components of the vector. The first known example was the Fresnel’s description of light, followed soon by Mac Cullagh gauge theory – the precursor of the modern electromagnetic theory. In this case, the Laplace’s equation takes itself the general form

(1)

which can be still written as

(2)

Here we used the notation

(3)

Now, the equations (1) and (3) show that we can write

(4)

thus offering a mathematical meaning to Γ^j. It turns out that they do not represent always a vector. However, if the coordinates are harmonic, i.e. three independent solutions of the Laplace equation, these quantities are zero, and this is an essential fact.

The equations (3) and (4) show that the use the harmonic coordinates actually comes down to a restriction of the space variation of the metric tensor. This gives, in fact, the true meaning of such a condition. At least in the case of constant curvature, the metric tensor contains the deformation of the space filled with matter, and in such a case it should also contain information on the forces acting inside the matter. Consequently, if we assume that the matter is in direct relation with the space, in such a way that a constant density of matter means constant curvature of space, the solutions of Laplace equation corresponding to the metric in matter can, and we claim that they should, be taken as harmonic coordinates.

Vladimir Fock insisted at length on the importance of harmonic coordinates from the point of view of natural philosophy, but in the framework of general relativity (Fock, 1964). There, these coordinates are four independent solutions of the homogeneous D’Alembert equation. Not all the physicists agree with this idea, and we think not without reason. Indeed, Fock insists on the fact that only the harmonic coordinates can confer legitimacy to some sound ideas about the physical universe, such as Copernicus’ idea: in any other coordinates it does not make sense to distinguish between the Copernican and the Ptolemaic systems. The general opinion of the opponents of the thesis of privileging the harmonic coordinates, can be simply summarized by the fact that, from the point of view of general relativity, it is immaterial which coordinates we use in the description of the universe. Therefore Copernicus or Ptolemy – it hardly matters! And if the general relativity is regarded as a straight denial of the classical Newtonian theory of gravitation, then it hardly matters indeed. However Fock insists on the point that, on the contrary, the Newtonian theory of gravitation is actually intimately involved in the general relativity as Einstein constructed it, which is, indeed, indisputably the case.

We recalled here this discussion, for it has a certain significance, but in the theory of space (not of space-time!), and it takes us, again, back to the founder of modern science – Newton. We start, as Fock himself does actually, with the observation of the obvious fact that the natural position of a point in space is described by harmonic coordinates – the Cartesian coordinates. Indeed, these are solutions of the Laplace equations – trivial solutions, it is true, for they are simply first degree polynomials, and these are by default, so to speak, solutions of a partial differential equation of second order – but still, we can count them as such solutions:

(5)

Here ∆ is the Euclidean Laplace operator. This statement may be put in a vector form. Indeed, let us take the identity vector function

(6)

simply representing the position vector. In view of the equation (5) it is a solution of the Laplace equation:

(7)

It turns out, therefore, that Newton wrote his equations in such coordinates, and we ask ourselves if there is not a natural connection between the Newtonian gravitational force and the coordinates we associate with material points. Well, if we take the property of the coordinates of being harmonic functions as fundamental, it seems that there is a natural connection after all, and this can be exhibited in the following manner.

There is a theorem of William Thomson, Lord Kelvin (Kelvin, 1847; Liouville, 1847), called the inversion theorem, and establishing the fact that if a certain function is a solution of the Laplace equation, then the function

(8)

is also a solution of the Laplace equation, but in the coordinates given by inversion:

(9)

For a neat demonstration of this theorem one can consult, besides the original works, the work of Gaston Darboux (Darboux, 1910). It is now pretty obvious that, according to this theorem, the Newtonian gravitational force is a solution of the Laplace equation, being the direct transformation of the identity vector function. Indeed it is the vector function

(10)

and necessarily satisfies the inversion theorem of Thomson, for it is of the form given in equation (8) referring to the particular identity function.

One can thus state that, at least in the classical limits of the mechanics, the Laplace equation is the one which generalizes the force fields by assimilating them with coordinates, or with the fields in general. This means that, if we now enlarge the idea of space coordinates, in order to include any three independent solutions of the Laplace equation, not just the trivial Cartesian ones, we can safely say that the space is a particular limit where the fields and the coordinates coincide. This idea has far-reaching consequences, and at some point we will actually need to recall that the whole classical mechanics, and even thermodynamics, was built upon it. To take an example, consider the ideal gas laws: they can be theoretically endorsed by the kinetic theory of molecular chaos, resulting in a certain definition of temperature. But once we relate that definition to the energy without any other limitations, it leads to contradictions, which have always been placed on the tab of quantization.

 

References

Darboux, G. (1910): Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes, Gauthier-Villars, Paris, p. 277

Fock, V. A. (1964): The Theory of Space, Time and Gravitation, Macmillan Company, New York

Kelvin, William Thomson, Lord (1847): Extraits de Deux Lettres Adressées a M. Liouville, Journal des Mathématiques Pures et Appliquées (Liouville), Tome 12, pp. 256 – 264

Liouville, J. (1847): Note au Sujet de l’Article Précédent, Journal des Mathématiques Pures et Appliquées (Liouville), Tome 12, pp. 265 – 290 (Liouville’s own commentaries to previous Kelvin paper)

Weyl, H. (1952): Space, Time, Matter, Dover Publications, Inc.