Cantorian space-time in Cartan, de Broglie and field theories

Some implications of the Cantorian fractal space-time in the study of the inertial motion of one and two material points, de Broglie and field theories, using the external forms and the contraforms are analysed. For a material point the isotropy of space implies the dependence of mass, momentum and energy on the speed and on an arbitrary limit speed. For two material points the spatial isotropy implies the dependence of the momenta and energy on speed, an arbitrary limit speed and on an arbitrary function of the distance between the material points, functions introduced through the interaction mass. In particular, the gravitational interaction mass depends on two diffusion coefficients in Nottaleʹs way (intra- and extragalactic). Through the intragalactic diffusion coefficient the external metrics weakly Lorentz covariant are induced, which verify the “classical” tests of General Relativity. Through the extragalactic diffusion coefficient the internal metrics are induced, the Lorentz transformations are generalized and the Hubble effect is explained as an “appearance” of the momentum conservation law. The “removal” of the repulsive mass induces the intragalactic fractal, the dependence of the extragalactic diffusion coefficient on position implies the extragalactic fractal, and the vanishing of the gravitational charge of the Universe leads to the proportionality of two fractal scales. The proportionality of the undulatory and inertial 1-forms implies the de Broglie theory, and through the momentum and Hamilton–Jacobi equations, the nonlinear theory in the de Broglie–Takabayasi representation. The use of the potential 1-form and of a motion principle leads to the linear theory of the field, and the non-equivalence of the inertial 1-form and the field 1-form to the nonlinear theory, which by means of the energy tensor conservation, leads to the Born–Infeld equations. Contraforms in the non-relativistic formulation of electromagnetism conclude at the end, the entire picture. In all these models, the fractal character of space-time implies the simultaneous existence, either of two inertial 1-forms, as it is the case for Cartan theories, or of two undulatory 1-forms, as it is the case for undulatory theories, or of two potential 1-forms, as it is the case for the field theories. The Cantorization of space-time implies the two-dimensional Toda lattices. In such a context from the analysis of the fractal dimension of a vortex street and of the superconducting pair it results that these physical objects can be considered as 2D projections of a higher-dimensional fractal string.