ARTICLE DE SYNTHESE RECENT

Article de synthèse (07/2000) publié à l'occasion du symposium international organisé en l'honneur de O. Rössler
ZKM Karlsruhe, 18-21 May 2000, Ed. H. Diebner - publié ou à paraître dans "Sciences of the interface

Abstract

The theory of scale relativity extends Einstein's principle of relativity to scale transformations of resolutions. It's based on the giving-up of the axiom of differentiability of the space-time continuum. Three consequences arise from this withdrawal.

1 - The geometry of space-time becomes fractal; i.e. explicitely resolution-dependant : this allows one to describe a non-differentiable physics in term of diffrential equations acting in the scale space. The requirement that these equations satisfy the principle of scale relativity leads to introduce scale laws having a Galilean forùm (constant fractal dimension), then a log-Lorentzian form. In this framework, the Planck length-time scale becomes a manimal impassable scale, invariant under dilatations, and the cosmic length-scale (related to the cosmological constant) a maximal one. Recent measurements of the cosmological constant have comfirmed the theoretically predicted value.
Then we attempt to construct a generalized scale relativity which includes opn-linear scale transformations and scale-motion coupling. In this last framework, one can reinterpret gauge invariance as scale invariance on the internal resolutions. This approach has allowed us to make theoretical predictions concerning coupling constants and elementary particle masses (electron, Higgs boson, vacuum energy of the Higgs field), which we update in the present contribution. These predictions are succesfully checked using recently improved experimental values.

2 - The geodesics on a non-differentiable space-time are fractal and in infinite number : this leads one to use a fluid-like description and implies adding new terms in the differential equations of mean motion.

3 - Time reversibility is broken at the infinitesimal level : this can be described in terms of a two-valuedness of the velocity vector, for which we use a complex representation.

These three effects can be combined to construct a covariant time derivative operator, which transforms the fundamental equations of classical dynamics into a generalized Schrödinger equation. This provides us with a theory of morphogenesis and self-organization, since the solutions of this equation yield probability densities, which are interpreted as a tendency for the system to make structures. Several new theoretical predictions can be made by applying this approach to the equations of motion of test-particles in various gravitational potentials of astrophysical relevance. These predictions are supported by a comparison with observational date on a wide range of scales, from planetary systems to cosmological structures.

Résumé