Scale Relativity
these HTML pages have been
developped from a first version achieved by E. Lefèvre.
For more detail, refer either to the book (1993),
or to a more recent (1996) review paper,
more generally to the bibliography published
on the subject, or to the author at the following e-mail address: Laurent.Nottale@obspm.fr
The author:
This theory is proposed by Laurent Nottale,
researcher at observatoire
de Paris-Meudon. He worked for a long time in parallel on gravitational
lenses, but is now devoting most of his activity to the development
of the theory of Scale Relativity (in french: Relativité
d'Echelle) within DAEC
department.
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 principle
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 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)
(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:
- 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 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 / large time scales:
- structuring 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.
- ...
For a more complete list of the results and predictions of the scale-relativity
theory, see Sec. 9 of the review paper.
A html improved version of this section can be found here.
Non exhaustive bibliography,
author's list of publication (all subjects),
list of publication of the author and others
on scale relativity
Some definitions of the terms used (links in italic).
Other sites:
Some of the precursors: Galileo,
Newton, etc.
Developed from an initial version due to Eric
Lefèvre. To contact him click here
ou there
The URL of this page is: http://www.daec.obspm.fr/users/nottale/ukmenure.htm