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Warning:  these HTML pages have been developped from a first version
achieved by E. Lef�vre. 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: Relativit� 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 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 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 Schr�dinger 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 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

          general 
          pulsars 
          galaxy pairs 


     Glossary: 

     Some definitions of the terms used (links in italic). 



     Other sites:

          Dr. Garnet Ord 
          Dr. Felix Herrmann 
          Rapha�l Hermann 
          Dr. G�rard Schumacher 


     Portrait gallery: 

     Some of the precursors: Galileo, Newton, etc. 

     Developed from an initial version due to Eric Lef�vre. To contact
him
     click here ou there

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

Glossary (designed by E.T. Lef�vre)

Gravitational Lenses 

                                               

Effect in which a visible or invisible mass acts as a lens to
"focalize" the light
coming from a background source. (see bending of light) On the image
above,
arcs are images distorted and demultiplied, by the galaxy cluster, of
more
distant objects.





Bending of light rays 
     When in classical mecanics one considers light as a material
object
     and not only as a wave, one can show that (the trajectories
followed by
     a body in the gravity field of a much more massive body do not
     depend on its mass) a light ray passing at a distance r of a body
of mass
     M undergoes a deviation of angle 

             .

     In general relativity, light follows the curvature of Space-Time
     induced by the mass M and undergoes a deviation of angle

             .

     This effect has been verified by Eddington in 1919 during an
eclipse.
     The measurement of the deviation angle predicted by GR
constituted
     the first experimental verification of Einstein's theory.





Planck length 
     Length obtained by combining the fundamental constants 
     (where h is Planck's constant), c (vacuum speed of light) and G
     (gravitation or Newton constant) to obtain a quantity having the
     dimension of a length. 



     This microscopic length constitutes in ScR an unexceedable,
     unreachable (it is an asymptot, an horizon) and invariant under
     dilation, scale (or resolution). Even if it is only very small,
it has all the
     physical properties of an infinitely small!

     It is the smallest scale of Nature. The fact that it is not null
has
     enormous repercussions on physics, like the finiteness of the
speed of
     light.

     The Planck mass MP and Planck Energy associated to it, EP = MP
     c2, are also computed from the 3 fundamental constants G, h and
c:
     MP= (hbar c / G)^(1/2)





Cosmological length 
     length linked to the cosmological constant by: 



     Second length invariable under dilation that appears in ScR. It
is the
     counterpart of the Planck length at large scale.

     Is is the largest scale of Nature




Fractal Curve 
     Let's consider a continuous  but
     non-differentiable  curve
     (function). In general these two limits are even not defined,
i.e. one
     cannot define the slope of the curve in any point. 

     An example of topological dimension 1 is the curve:



     5th iteration from the generator



     BE CAREFUL!: a (too) common error is to define a fractal curve or
     object as an object possessing the property to keep the same
     appearance under successive zooms. It applies only to
self-similar
     fractals like the preceeding curve!

     The word fractal refers, in ScR, to the appearance of a priori
new
     structures along successive zooms toward small or large scales,
which
     leads to a divergence for resolutions tending towards zero
(towards
     Planck's scale in special ScR) and the infinity (cosmological
scale in
     SScR).




Space-Time 

     A physical definition of Space-Time could take an entire book!
     However, giving a mathematical definition is relatively more
simple. 

     In relativity and quantum mechanics, an event involving 1 body is
     characterized by 4 and only 4 coordinates (x,y,z,t). One puts
apart
     properties like charge, spin, etc. and it does not mean that
knowing
     only one (x,y,z,t) is sufficient to determine the evolution of a
system but
     that the necessary quantities only depend on x, y, z, and t.

     Space-Time constitutes the set of all possible (x,y,z,t)
quadruplets
     and their transformations

     Relativity's Space-Time is continuous and curved (the case were
it is
     flat being not excluded a priori) and differentiable. This kind
of Space
     cannot account for the quantum properties of matter.

     Quantum Mechanics' Space-Time is in principle flat, minkovskian
     and differentiable. The contradiction with relativity as well as
other
     considerations leads to attempts at including another kind of
     Space-Time in quantum mechanics, that have been up to now
     unfruitful..

     In Scale Relativity, a criticism of measurement theory in
physics, in
     particular of the role played by the resolutions, (c.f. the
importance of
     the analysis of how we make measures of length and instant in the
     premisses of Einstein's theory) led L. Nottale to give up the
(implicit)
     hypothesis of the differentiability of Space-Time, which implies
its
     fractal and curved nature. The fractal Space-Time, explicitely
     dependent on resolutions, can be reduced to the definition of a
     "Space-Time-Zoom" with 5 dimensions (x,y,z,t,D). It is the
fractal
     dimension, became a variable, which plays the role of a 5th
dimension
     for scale laws (just as relativistic motion laws are implemented
by the
     interpretation of time as a 4th dimension).

     The concept of fractal space-time has also been independently
     introduced by Garnet Ord as a geometric analogue of relativistic
     quantum mechanics.




Chaos time 
     A chaotic system can be characterized by the behaviour of the
     separation between 2 of its trajectories. They are in general
solutions of
     a system of deterministic differential equations describing the
evolution
     of the system. When the system is chaotic, 2 solutions
corresponding to
     2 infinitely close initial conditions exponentially diverge one
from
     another. One speaks of an extreme sensibility to initial
conditions. 

     The deviation  follows an exponential law:



         is called the chaos time of the system. The inverse of the
chaos
     time is the Lyapunov exponent. If, having measured as precisely
as
     we want (but not enough to distinguish between the 2 initial
conditions)
     the system's parameters (initial conditions) at a time t0, we try
to
     determine its state after a time interval much greater than , we
realize
     that the deviation between the 2 solutions is now so big that one
cannot
     say anything on the state in which the system really is: knowing
that
     there are, in fact, an infinity of possible initial conditions
between the
     two that we have considered, the system can lie in an infinity of
states
     very different one from another. One has a sort of predictibility
     horizon which, like the one we are familiar with (due to the
roundness
     of the Earth), is not absolute: short term prediction (before the
horizon)
     remains entirely possible!

     Remarks: 

               At the present time, in "chaos theory", one analyses
the
               behaviour of a system whose differential equations,
               describing its evolution, are "classical". In such a
frame,
               chaotic behaviour (diverging errors / deviations) only
               appears when the equations are non-linear. In Scale
               Relativity, the situation is very different in that
               sense that the fractal nature of space-time induces an
other
               type of chaos, that could be called intrinsic, inherent
to
               space-time. This has the effect that a behaviour of the
               chaotic type appears at large time scale even in an
               ultra-simple like one or 2-body problems (test particle
               orbiting around a much more massive one or 2 bodies
               orbiting around their barycentre). Such a behaviour is
               completely incomprehensible in the frame of current
               theories. 
               Of course, if we could measure all the parameters with
an
               infinite precision at the same time, the problem would
               disappear. Unfortunately, besides the imprecisions of
the
               measuring apparatus, we have learned from quantum
               mechanics (Heisenberg inequalities) that it is
physically
               impossible (I mean that the impossibility does not lie
in the
               instruments but in Physics itself) to measure
               simultaneously to the desired precision the position
and
               the speed of a particle, for instance. In that case,
the
               knowledge of the exact position of a particle implies
an
               absolute uncertainty on its velocity. However, it is
not
               forbidden to measure individually some quantity with an
               absolute precision. In Scale Relativity, the Planck
               length constitutes an impassable (because it is an
               asymptot) lower limit to the resolution with which one
can
               measure a position or a length. At this
length/resolution,
               the concepts of position and of length become
               degenerated: the whole Universe, measured at Planck's
               resolution, as well as all it can contain, is of
extension
               equal to the Planck length itself!! 

          To make it short , the chaotic behaviour is unavoidable!

          Strangely, order happens in chaos! (attractors, etc.) How
does
          this happen? Scale Relativity brings some elements of an
answer
          to this question 





Covariant derivative 

     to be continued ...!


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