To obtain direct informations, see the site Web de Laurent Nottale :

http://www.daec.obspm.fr/users/nottale

see below one of the main pages :

version française

Scale Relativity

Warning:
these HTML pages are devoted to the concepts of Scale Relativity and Fractal Space-Time. For more detail, refer either to the book (Fractal Space-Time and Microphysics, World Scientific, 1993), or to a more recent (1996) review paper, or the popular books (in french) "La relativité dans tous ses états" (Hachette, 1998, Paris, 319 pp.) and "Les arbres de l'évolution" (L. Nottale, J. Chaline et P. Grou, Hachette Littératures, 2000). More generally, consult the bibliography published on the subject, or the author at the following e-mail address: Laurent.Nottale@obspm.fr. They have been developped from a first version achieved by E. Lefèvre.

The author:

This theory is proposed by Laurent Nottale, researcher at observatoire de Paris-Meudon. He worked for a long time in parallel on cosmology and gravitational lenses, (see his book in French "l'Univers et la Lumière", Flammarion, Nouvelle Bibliothèque Scientifique 1994, Champs 1998, Prix du livre d'Astronomie, Festival de Haute-Maurienne-Vanoise 1995). But he is now devoting most of his activity to the development of the theory of Scale Relativity (in french: Relativité d'Echelle) within DAEC department.

The origin

In this chapter, we first present the origin of the Scale Relativity (or ScR) theory: we will briefly evoke the reasons that led to its development.

The fundamental priciple of ScR:

It is an extension of Einstein's principle of relativity. It can be stated as follows:
The laws of nature must be valid in every coordinate systems, whatever their state of motion and of scale.
The results obtained show once again the extraordinary efficiency of this principle at constraining the laws of physics.

The method:

The formalism developed by L. Nottale for ScR is already sufficiently settled to be used "as is" to deal with a particular problem in many situations. The procedure is outlined in this chapter. The most general version of the theory is still under construction.

---

The new principles and their consequences:

There are plenty of them! Among the most important, one finds:

• the resolution becomes an explicit variable (which plays for scale transformations the role played by the velocity for motion laws), intrinsic to the fractal space-time.
  
• the principle of relativity is extended to scale transformations (on resolutions). Equations of physics must be covariant: i.e. they should keep their (simplest) form in scale transformations. Scale covariance is achieved by the introduction of a scale-covariant derivative, itself being an extension of the covariant derivative of general relativity.
• this covariant derivative transforms the classical mechanics into a quantum mechanics: Newton's equation of dynamics, once made scale-covariant by replacing the usual time derivative by the new covariant one, is integrated in terms of a Schrödinger equation.
• non-differentiable nature of Space-Time continuum, which implies its fractal character.
• existence of 2 asymptotic, unexceedable and invariant under dilation scales : lP: Planck length (minimal scale) and L: cosmological length (maximal scale), in the frame of new "lorentzian" scale laws.
• natural apparition of structures in some systems because of the theory itself: no need to invoke the growth of quantum fluctuations any more!
• Organization in chaotic systems (at very large time scales).
• formally quantum-like description of chaotic systems beyond their predictability horizon (t >>20 t_chaos)

The proofs/experimental and observational tests (Summary) :

The classification is only indicative since some problems belong to several categories. It is however not an arbitrary one because the 3 fields of microphysics, cosmology and chaotic systems correspond to privileged fields of application of ScR. Respectively: dx et dt tending towards zero, dx tending towards infinity and dt tending towards infinity. These are the three frontiers of today's physics: the infinitely small, the infinitely large and the infinity of complexity.

Particle physics:

• convergence of coupling constants, including the gravitational one, at Planck energy scale.
• prediction of grand unification and electroweak unification scales.
• prediction of the coupling constants values.
• relations between particle masses and charges, allowing the prediction of the mass of the electron from its charge.

Cosmology:

• solution to the problem of horizon/causality without inflation.
• large scale structures of the Universe: fractal dimension of the distribution of galaxies.
• prediction of the value and meaning of the cosmological constant.
• solution to the problem of vacuum energy density.
• explanation for Dirac's large numbers coincidence and achievement of Mach's principle.

Chaotic systems / gravitational structuration:

• morphogenesis of gravitational systems.
• prediction of the distributions of the distances, eccentricities and masses of the planets in the Solar System
• verification in extra-solar planetary systems (see catalog of exoplanets).
• quantification of velocity differences in binary galaxies.
• other quantification effects: stellar radii, obliquities and inclinations in the solar system, giant planet satellites, asteroids distribution, binary stars, star formation regions, galactic structures, local group of galaxies, compact groups, clusters and superclusters of galaxies, large scale structures... As predicted by the theory, the structures observed at all scales come under a unique fundamental constant having the dimension of a velocity
• . ...

The URL adress of this page is : http://www.daec.obspm.fr/users/nottale/ukmenure.htm