Scale relativity is a geometrical and fractal
space-time physical theory.
Relativity theories (special relativity and
general relativity) are based on the notion
that position, orientation, movement and acceleration
cannot be defined in an absolute way, but
only relative to a system of reference. The
scale relativity theory proposes to extend
the concept of relativity to physical scales
(time, length, energy, or momentum scales),
by introducing an explicit "state of scale"
in coordinate systems.
This extension of the relativity principle
was originally introduced by Laurent Nottale,
based on the idea of a fractal space-time
theory first introduced by Garnet Ord, and
by Nottale and Jean Schneider.To describe
scale transformations requires the use of
fractal geometries, which are typically concerned
with scale changes. Scale relativity is thus
an extension of relativity theory to the concept
of scale, using fractal geometries to study
scale transformations.
The construction of the theory is similar
to previous relativity theories, with three
different levels: Galilean, special and general.
The development of a full general scale relativity
is not finished yet.
== History ==
=== 
Feynman's paths in quantum mechanics ===
Richard Feynman developed a path integral
formulation of quantum mechanics before 1966.
Searching for the most important paths relevant
for quantum particles, Feynman noticed that
such paths were very irregular on small scales,
i.e. infinite and non-differentiable. This
means that in between two points, a particle
can have not one path, but an infinity of
potential paths.
This can be illustrated with a concrete example.
Imagine that you are hiking in the mountains,
and that you are free to walk wherever you
like. To go from point A to point B, there
is not just one path, but an infinity of possible
paths, each going through different valleys
and hills.Scale relativity hypothesizes that
quantum behavior comes from the fractal nature
of spacetime. Indeed, fractal geometries allow
to study such non-differentiable paths. This
fractal interpretation of quantum mechanics
has been further specified by Abbot and Wise,
showing that the paths have a fractal dimension
2. Scale relativity goes one step further
by asserting that the fractality of these
paths is a consequence of the fractality of
space-time.
There are other pioneers who saw the fractal
nature of quantum mechanical paths. Also,
as much as the development of general relativity
required the mathematical tools of non-Euclidean
(Riemannian) geometries, the development of
a fractal space-time theory would not have
been possible without the concept of fractal
geometries developed and popularized by Benoit
Mandelbrot. Fractals are usually associated
with the self-similar case of a fractal curve,
but other more complicated fractals are possible,
e.g. considering not only curves, but also
fractal surfaces or fractal volumes, as well
as investigating fractal dimensions which
have other values than 2, and which also vary
with scale.
=== Independent discovery ===
Garnet Ord and Laurent Nottale both connected
fractal space-time with quantum mechanics.
Nottale coined the term "scale relativity"
in 1992. He developed the theory and its applications
with more than one hundred scientific papers,
two technical books in English, and three
popular books in French.
== Basic concepts ==
=== Principle of scale relativity ===
The principle of relativity says that physical
laws should be valid in all coordinate systems.
This principle has been applied to states
of position (the origin and orientation of
axes), as well as to the states of movement
of coordinate systems (speed, acceleration).
Such states are never defined in an absolute
manner, but relatively to one another. For
example, there is no absolute movement, in
the sense that it can only be defined in a
relative way between one body and another.
Scale relativity proposes in a similar manner
to define a scale relative to another one,
and not in an absolute way. Only scale ratios
have a physical meaning, never an absolute
scale, in the same way as there exists no
absolute position or velocity, but only position
or velocity differences.
The concept of resolution is re-interpreted
as the "state of scale" of the system, in
the same way as velocity characterizes the
state of movement. The principle of scale
relativity can thus be formulated as:
the laws of physics must be such that they
apply to coordinate systems whatever their
state of scale.
The main goal of scale relativity is to find
laws which mathematically respect this new
principle of relativity. Mathematically, this
can be expressed through the principle of
covariance applied to scales, that is, the
invariance of the form of physics equations
under transformations of resolutions (dilations
and contractions).
=== Including resolutions in coordinate systems
===
Galileo introduced explicitly velocity parameters
in the observational referential. Then, Einstein
introduced explicitly acceleration parameters.
In a similar way, Nottale introduces scale
parameters explicitly in the observational
referential. The core idea of scale-relativity
is thus to include resolutions explicitly
in coordinate systems, thereby integrating
measure theory in the formulation of physical
laws.
An important consequence is that coordinates
are not numbers anymore, but functions, which
depend on the resolution. For example, the
length of the Brittany coast is explicitly
dependent on the resolution at which one measures
it.If we measure a pen with a ruler graduated
at a millimetric scale, we should write that
it is 15 ± 0.1 cm. The error bar indicates
the resolution of our measure. If we had measured
the pen at another resolution, for example
with a ruler graduated at the centimeter scale,
we would have found another result, 15 ± 1
cm. In scale relativity, this resolution defines
the "state of scale". In the relativity of
movement, this is similar to the concept of
speed, which defines the "state of movement".
The relative state of scale is fundamental
to know about for any physical description.
For example, if we want to describe the movement
and properties of a sphere, we may as well
use classical mechanics or quantum mechanics
depending on the size of the sphere in question.In
particular, information on resolution is essential
to understand quantum mechanical systems,
and in scale relativity, resolutions are included
in coordinate systems, so it seems a logical
and promising approach to account for quantum
phenomena.
=== Dropping the hypothesis of differentiability
===
Scientific theories usually do not improve
by adding complexity, but rather by starting
from a more and more simple basis. This fact
can be observed throughout the history of
science. The reason is that starting from
a less constrained basis provides more freedom
and therefore allows richer phenomena to be
included in the scope of the theory. Therefore,
new theories usually do not contradict the
old ones, but widen their domain of validity
and include previous knowledge as special
cases. For example, releasing the constraint
of rigidity of space led Einstein to derive
his theory of general relativity and to understand
gravitation. As expected, this theory naturally
includes Newton's theory, which is recovered
as a linear approximation under weak fields.
The same type of approach has been followed
by Nottale to build the theory of scale relativity.
The basis of current theories is a continuous
and two-times differentiable space. Space
is by definition a continuum, but the assumption
of differentiability is not supported by any
fundamental reason. It is usually assumed
only because it is observed that the first
two derivatives of position with respect to
time are needed to describe motion. Scale
relativity theory is rooted in the idea that
the constraint of differentiability can be
relaxed and that this allows quantum laws
to be derived.
In terms of geometry, differentiability means
that a curve is sufficiently smooth and can
be approximated by a tangent. Mathematically,
two points are placed on this curve and one
observes the slope of the straight line joining
them as they become closer and closer. If
the curve is smooth enough, this process converges
(almost) everywhere and the curve is said
to be differentiable. It is often believed
that this property is common in nature. However,
most natural objects have instead a very rough
surface, or contour. For example, the bark
of trees and snowflakes have a detailed structure
that does not become smoother when the scale
is refined. For such curves, the slope of
the tangent fluctuates endlessly or diverges.
The derivative is then undefined (almost)
everywhere and the curve is said to be nondifferentiable.Therefore,
when the assumption of space differentiability
is abandoned, there is an additional degree
of freedom that allows the geometry of space
to be extremely rough. The difficulty in this
approach is that new mathematical tools are
needed to model this geometry because the
classical derivative cannot be used. Nottale
found a solution to this problem by using
the fact that nondifferentiability implies
scale dependence and therefore the use of
fractal geometry. Scale dependence means that
the distances on a nondifferentiable curve
depend on the scale of observation. It is
therefore possible to maintain differential
calculus provided that the scale at which
derivatives are calculated is given, and that
their definition includes no limit. It amounts
to saying that nondifferentiable curves have
a whole set of tangents in one point instead
of one, and that there is a specific tangent
at each scale.
To abandon the hypothesis of differentiability
does not mean abandoning differentiability.
Instead, this leads to a more general framework,
where both differentiable and non-differentiable
cases are included. Combined with motion relativity,
scale relativity by definition thus extends
and contains general relativity. As much as
general relativity is possible when we drop
the hypothesis of euclidian space-time, allowing
the possibility of curved space-time, scale
relativity is possible when we abandon the
hypothesis of differentiability, allowing
the possibility of a fractal space-time. The
objective is then to describe a continuous
space-time which is not everywhere differentiable,
as it was in general relativity.
Abandoning differentiability doesn't mean
abandoning differential equations. The concept
of fractal allows work with the nondifferentiable
case with differential equations. In differential
calculus, we can see the concept of limit
as a zoom, but in this generalization of differential
calculus, one doesn't look only at the limit
zooms (zero and infinity) but also everything
in between, that is, all possible zooms.
In sum, we can drop the hypothesis of the
differentiability of space-time, keeping differential
equations, provided that fractal geometries
are used. With them, we can still deal with
the nondifferentiable case with the tools
of differential equations. This leads to a
double differential equation treatment: in
space-time and in scale space.
=== Fractal space-time ===
If Einstein showed that space-time was curved,
Nottale shows that it is not only curved,
but also fractal. Nottale has proven a key
theorem which shows that a space which is
continuous and non-differentiable is necessarily
fractal. It means that such a space depends
on scale.
Importantly, the theory does not merely describe
fractal objects in a given space. Instead,
it is space itself which is fractal. To understand
what a fractal space means requires to study
not just fractal curves, but also fractal
surfaces, fractal volumes, etc.
Mathematically, a fractal space-time is defined
as a nondifferentiable generalization of Riemannian
geometry. Such a fractal space-time geometry
is the natural choice to develop this new
principle of relativity, in the same way that
curved geometries were needed to develop Einstein's
theory of general relativity.In the same way
that general relativistic effects are not
felt in a typical human life, the most radical
effects of the fractality of spacetime appear
only at the extreme limits of scales: micro
scales or at cosmological scales. This approach
therefore proposes to bridge not only the
quantum and the classical, but also the classical
and the cosmological, with fractal to non-fractal
transitions (see Fig. 1). More plots of this
transition can be seen in the literature.
=== Minimum and maximum invariant scales ===
A fundamental and elegant result of scale
relativity is to propose a minimum and maximum
scale in physics, invariant under dilations,
in a very similar way as the speed of light
is an upper limit for speed.
==== Minimum invariant scale ====
In special relativity, there is an unreachable
speed, the speed of light. We can add speeds
without end, but they will always be less
than the speed of light. The sums of all speeds
are limited by the speed of light. Additionally,
the composition of two velocities is inferior
to the sum of those two speeds.
In special scale relativity, similar unreachable
observational scales are proposed, the Planck
length scale (lP) and the Planck time scale
(tP). Dilations are bounded by lP and tP,
which means that we can divide spatial or
temporal intervals without end, but they will
always be superior to Planck's length and
time scales. This is a result of special scale
relativity (see section 2.7 below). Similarly,
the composition of two scale changes is inferior
to the product of these two scales.
==== Maximum invariant scale ====
The choice of the maximum scale (noted L)
is less easy to explain, but it mostly consists
to identify it with the cosmological constant:
L = 1/(Λ2). This is motivated in parts because
a dimensional analysis shows that the cosmological
constant is the inverse of the square of a
length, i.e. a curvature.
=== Galilean scale relativity ===
The theory of scale relativity follows a similar
construction as the one of the relativity
of movement, which took place in three steps:
galilean, special and general relativity.
This is not surprising, as in both cases the
goal is to find laws satisfying transformation
laws including one parameter that is relative:
the speed in the case of the relativity of
movement; the resolution in the case of the
relativity of scales.
Galilean scale relativity involves linear
transformations a constant fractal dimension,
self-similarity and scale invariance. This
situation is best illustrated with self-similar
fractals. Here, the length of geodesics varies
constantly with resolution. The fractal dimensions
of free particles doesn't change with zooms.
These are self-similar curves.
In Galilean relativity, recall that the laws
of motion are the same in all inertial frames.
Galileo famously concluded that "the movement
is like nothing". In the case of self-similar
fractals, paraphrasing Galileo, one could
say that "scaling is like nothing". Indeed,
the same patterns occur at different scales,
so scaling is not noticeable, it is like nothing.
In the relativity of movement, Galileo's theory
is an additive Galilean group:
X' = X - VT
T' = THowever, if we consider scale transformations
(dilations and contractions), the laws are
products, and not sums. This can be seen by
the necessity to use units of measurements.
Indeed, when we say that an object measures
10 meters, we actually mean the object measures
10 times the definite predetermined length
called "meter". The number 10 is actually
a scale ratio of two lengths 10/1m, where
10 is the measured quantity, and 1m is the
arbitrary defining unit. This is the reason
why the group is multiplicative.
Moreover, an arbitrary scale e doesn't have
any physical meaning in itself (like the number
10), only scale ratios r = e' / e have a meaning,
in our example, r = 10 / 1. Using the Gell-Mann-Lévy
method, we can use a more relevant scale variable,
V = ln(e' / e), and then find back an additive
group for scale transformations by taking
the logarithm – which converts products
into sums.
When in addition to the principle of scale
relativity, one adds the principle of relativity
of movement, there is a transition of the
structure of geodesics at large scales, where
trajectories do not depend on the resolution
anymore, where trajectories become classical.
This explains the shift of behavior from quantum
to classical. See also Fig. 1.
=== Special scale relativity ===
Special scale relativity can be seen as a
correction of galilean scale relativity, where
Galilean transformations are replaced by Lorentz
transformations. The "corrections remain small
at 'large' scale (i.e. around the Compton
scale of particles) and increase when going
to smaller length scales (i.e. large energies)
in the same way as motion-relativistic corrections
increase when going to large speeds".In Galilean
relativity, it was considered "obvious" that
we could add speeds without limit (w = u +
v). This composition laws for speed was not
challenged. However, Poincaré and Einstein
did challenge it with special relativity,
setting a maximum speed on movement, the speed
of light. Formally, if v is a velocity, v
+ c = c. The status of the speed of light
in special relativity is a horizon, unreachable,
impassable, invariant under changes of movement.
Regarding scale, we are still within a Galilean
kind of thinking. Indeed, we assume without
justification that the composition of two
dilations is ρ * ρ = ρ2. Written with logarithms,
this equality becomes lnρ + lnρ = 2lnρ.
However, nothing guarantees that this law
should hold at quantum or cosmic scales. As
a matter of fact, this dilation law is corrected
in special scale relativity, and becomes:
ln ρ + ln ρ = 2 ln ρ / (1 + ln ρ2).
More generally, in special relativity the
composition law for velocities differs from
the Galilean approximation and becomes (with
the speed of light c = 1):
u ⊕ v = (u + v) / (1 + u * v)Similarly,
in special scale relativity, the composition
law for dilations differs from our Galilean
intuitions and becomes (in a logarithm of
base K which includes a possible constant
C = ln K, which plays the same role as c):
logρ1 ⊕ logρ2 = (logρ1 + logρ2) / (1
+ logρ1 * logρ2)The status of the Planck
scale in special scale relativity plays a
similar role as the speed of light in special
relativity. It is a horizon for small scales,
unreachable, impassable, invariant under scale
changes, i.e. dilations and contractions.
The consequence for special scale relativity
is that applying two times the same contraction
ρ to an object, the result is a contraction
less strong than contraction ρ x ρ. Formally,
if ρ is a contraction, ρ * lP = lP.
As noted above, there is also an unreachable,
impassable maximum scale, invariant under
scale changes, which is the cosmic length
L. In particular, it is invariant under the
expansion of the universe.
=== General scale relativity ===
In Galilean scale relativity, spacetime was
fractal with constant fractal dimensions.
In special scale relativity, fractal dimensions
can vary. This varying fractal dimension remains
however constrained by a log-Lorentz law.
This means that the laws satisfy a logarithmic
version of the Lorentz transformation. The
varying fractal dimension is covariant, in
a similar way as proper time is covariant
in special relativity.
In general scale relativity, the fractal dimension
is not constrained anymore, and can take any
value. In other words, it is the situation
where there is curvature in scale space. Einstein's
curved space-time becomes a particular case
of the more general fractal spacetime.
General scale relativity is much more complicated,
technical, and less developed than its Galilean
and special versions. It involves non-linear
laws, scale dynamics and gauge fields. In
the case of non self-similarity, changing
scales generates a new scale-force or scale-field
which needs to be taken into account in a
scale dynamics approach. Quantum mechanics
then needs to be analyzed in scale space.Finally,
in general scale relativity, we need to take
into account both movement and scale transformations,
where scale variables depend on space-time
coordinates. More details about the implications
for abelian gauge fields and non-abelian gauge
fields can be found in the literature. Nottale's
2011 book provides the state of the art.
To sum up, one can see some structural similarities
between the relativity of movement and the
relativity of scales in Table 1:
Table 1. Comparison between relativity of
movement and relativity of scales. In both
cases, there are two kinds of variables linked
to the coordinate systems: variables which
define the coordinate system, and variables
that characterize the state of the coordinate
system. In this analogy, the resolution can
be assimilated to a speed; acceleration to
a scale acceleration; space to the length
of a fractal; and time, to the variable fractal
dimension. Table adapted from this paper.
== Consequences for quantum mechanics ==
=== Introduction ===
The fractality of space-time implies an infinity
of virtual geodesics. This remark already
means that a fluid mechanics is needed. Note
that this view is not new, as many authors
have noticed fractal properties at quantum
scales, thereby suggesting that typical quantum
mechanical paths are fractal. See this article
for a review. However, the idea to consider
a fluid of geodesics in a fractal spacetime
is an original proposal from Nottale.
In scale relativity, quantum mechanical effects
appear as effects of fractal structures on
the movement. The fundamental indeterminism
and nonlocality of quantum mechanics are deduced
from the fractal geometry itself.
There is an analogy between the interpretation
of gravitation in general relativity and quantum
effects in scale relativity. Indeed, if gravitation
is a manifestation of space-time curvature
in general relativity, quantum effects are
manifestations of a fractal space-time in
scale relativity.
To sum up, there are two aspects which allows
scale relativity to better understand quantum
mechanics. On the one side, fractal fluctuations
themselves are hypothesized to lead to quantum
effects. On the other side, non-differentiability
leads to a local irreversibility of the dynamics
and therefore to the use of complex numbers.
Quantum mechanics thus receives not only a
new interpretation, but a firm foundation
in relativity principles.
=== Quantum-classical transition ===
As Philip Turner summarized:
the structure of space has both a smooth (differentiable)
component at the macro-scale and a chaotic,
fractal (non-differentiable) component at
the micro-scale, the transition taking place
at the de Broglie length scale.
This transition is explained with Galilean
scale relativity (see also above).
=== Derivation of quantum mechanics' postulates
===
Starting from scale relativity, it is possible
to derive the fundamental "postulates" of
quantum mechanics. More specifically, building
on the result of the key theorem showing that
a space which is continuous and non-differentiable
is necessarily fractal (see section 2.4),
Schrödinger's equation, Born's and von Neumann's
postulate are derived.
To derive Schrödinger's equation, Nottale
started with Newton's second law of motion,
and used the result of the key theorem. Many
subsequent works then confirmed the derivation.Actually,
the Schrödinger equation derived becomes
generalized in scale relativity, and opens
the way to a macroscopic quantum mechanics
(see below for validated empirical predictions
in astrophysics). This may also help to better
understand macroscopic quantum phenomena in
the future.
Reasoning about fractal geodesics and non-differentiability,
it is also possible to derive von Neumann's
postulate and Born's postulate.With the hypothesis
of a fractal space-time, the Klein-Gordon,
and the Dirac equation can then be derived.The
significance of these fundamental results
is immense, as the foundations of quantum
mechanics which were up to now axiomatic,
are now logically derived from more primary
relativity theory principles and methods.
=== Gauge transformations ===
Gauge fields appear when scale and movements
are combined. Scale relativity proposes a
geometric theory of gauge fields. As Turner
explains:
The theory offers a new interpretation of
gauge transformations and gauge fields (both
Abelian and non-Abelian), which are manifestations
of the fractality of space-time, in the same
way that gravitation is derived from its curvature.
The relationships between fractal space-time,
gauge fields and quantum mechanics are technical
and advanced subject-matters elaborated in
details in Nottale's latest book.
== Consequences for elementary particles physics
==
=== 
Introduction ===
Scale relativity gives a geometric interpretation
to charges, which are now "defined as the
conservative quantities that are built from
the new scale symmetries". Relations between
mass scales and coupling constants can be
theoretically established, and some of them
empirically validated. This is possible because
in scale relativity, the problem of divergences
in quantum field theory is resolved. Indeed,
in the new framework, masses and charges become
finite, even at infinite energy. In special
scale relativity, the possible scale ratios
become limited, constraining in a geometric
way the quantization of charges. Let us compare
a few theoretical predictions with their experimental
measures.
=== Fine-structure constant ===
Nottale's latest theoretical prediction of
the fine-structure constant at the Z0 scale
is:
α−1(mZ) = 128.92By comparison, a recent
experimental measure gives:
α−1(mZ) = 128.91 ± 0.02At low energy,
the theoretical fine-structure constant prediction
is:
α−1 = 137.01 ± 0.035;which is within the
range of the experimental precision:
α−1 = 137.036
=== SU (2) coupling at Z scale ===
Here the SU(2) coupling corresponds to rotations
in a three-dimensional scale-space. The theoretical
estimate of the SU (2) coupling at Z scale
is:
α−12 Z = 29.8169 ± 0.0002While the experimental
value gives:
α−12 Z = 29.802 ± 0.027.
=== Strong nuclear force at Z scale ===
Special scale relativity predicts the value
of the strong nuclear force with great precision,
as later experimental measurements confirmed.
The first prediction of the strong nuclear
force at the Z energy level was made in 1992:
αS (mZ) = 0.1165 ± 0.0005A recent and refined
theoretical estimate gives:
αS (mZ) = 0.1173 ± 0.0004,which fits very
well with the experimental measure:
αS (mZ) = 0.1176 ± 0.0009
=== Mass of the electron ===
As an application from this new approach to
gauge fields, a theoretical estimate of the
mass of the electron (me) is possible, from
the experimental value of the fine-structure
constant. This leads a very good agreement:
me(theoretical) = 1.007 me (experimental)
== Astrophysical applications ==
=== 
Macroquantum mechanics ===
Some chaotic systems can be analyzed thanks
to a macroquantum mechanics. The main tool
here is the generalized Schrödinger equation,
which brings statistical predictability characteristic
of quantum mechanics into other scales in
nature. The equation predicts probability
density peaks. For example, the position of
exoplanets can be predicted in a statistical
manner. The theory predicts that planets have
more chances to be found at such or such distance
from their star. As Baryshev and Teerikorpi
write:
With his equation for the probability density
of planetary orbits around a star, Nottale
has seemingly come close to the old analogy
which saw a similarity between our solar system
and an atom in which electrons orbit the nucleus.
But now the analogy is deeper and mathematically
and physically supported: it comes from the
suggestion that chaotic planetary orbits on
very long time scales have preferred sizes,
the roots of which go to fractal space-time
and generalized Newtonian equation of motion
which assumes the form of the quantum Schrödinger
equation.
However, as Nottale acknowledges, this general
approach is not totally new:
The suggestion to use the formalism of quantum
mechanics for the treatment of macroscopic
problems, in particular for understanding
structures in the solar system, dates back
to the beginnings of the quantum theory
=== 
Gravitational systems ===
==== 
Space debris ====
At the scale of Earth's orbit, space debris
probability peaks at 718 km and 1475 km have
been predicted with scale relativity, which
is in agreement with observations at 850 km
and 1475 km. Da Rocha and Nottale suggest
that the dynamical braking of the Earth's
atmosphere may be responsible for the difference
between the theoretical prediction and the
observational data of the first peak.
==== Solar system ====
Scale relativity predicts a new law for interplanetary
distances, proposing an alternative to the
nowadays falsified Titius-bode "law". However,
the predictions here are statistical and not
deterministic as in Newtonian dynamics. In
addition to being statistical, the scale relativistic
law has a different theoretical form, and
is more reliable than the original Titius-Bode
version:
The Titius-Bode "law" of planetary distance
is of the form a + b × c n, with a = 0.4
AU, b = 0.3 AU and c = 2 in its original version.
It is partly inconsistent — Mercury corresponds
to n = −∞, Venus to n = 0, the Earth to
n = 1, etc. It therefore "predicts" an infinity
of orbits between Mercury and Venus and fails
for the main asteroid belt and beyond Saturn.
It has been shown by Herrmann (1997) that
its agreement with the observed distances
is not statistically significant. ... [I]n
the scale relativity framework, the predicted
law of distance is not a Titius-Bode-like
power law but a more constrained and statistically
significant quadratic law of the form an = a0n2.
==== Extrasolar systems ====
The method also applies to other extrasolar
systems. Let us illustrate this with the first
exoplanets found around the pulsar PSR B1257+12.
Three planets, A, B and C have been found.
Their orbital period ratios (noted PA/PC for
the period ratio of planet A to C) can be
estimated and compared to observations. Using
the macroscopic Schrödinger equation, the
recent theoretical estimates are:
(PA/PC)1/3 = 0.63593 (predicted)
(PB/PC)1/3 = 0.8787 (predicted),which fit
the observed values with great precision:
(PA/PC)1/3 = 0.63597 (observed)
(PB/PC)1/3 = 0.8783 (observed).The puzzling
fact that many exoplanets (e.g. hot Jupiters)
are so close to their parent stars receives
a natural explanation in this framework. Indeed,
it corresponds to the fundamental orbital
of the model, where (exo)planets are at 0.04
UA / solar mass of their parent star.More
validated predictions can be found regarding
orbital periods and the distances of planets
from their parent star.
==== Galaxy pairs ====
Daniel da Rocha studied the velocity of about
2000 galaxy pairs, which gave statistically
significant results when compared to the theoretical
structuration in phase space from scale relativity.
The method and tools here are similar to the
one used for explaining the structure in solar
systems.
Similar successful results apply at other
extragalactic scales: the local group of galaxies,
clusters of galaxies, the local supercluster
and other very large scale structures.
=== Dark matter ===
Scale relativity suggests that the fractality
of matter contributes to the phenomenon of
dark matter. Indeed, some of the dynamical
and gravitational effects which seem to require
unseen matter are suggested to be consequences
of the fractality of space on very large scales.In
the same way as quantum physics differs from
the classical at very small scales because
of fractal effects, symmetrically, at very
large scales, scale relativity also predicts
that corrections from the fractality of space-time
must be taken into account (see also Fig.
1).
Such an interpretation is somehow similar
in spirit to modified Newtonian dynamics (MOND),
although here the approach is founded on relativity
principles. Indeed, in MOND, Newtonian dynamics
is modified in an ad hoc manner to account
for the new effects, while in scale relativity,
it is the new fractal geometric field taken
into consideration which leads to the emergence
of a dark potential.
On the largest scale, scale relativity offers
a new perspective on the issue of redshift
quantization. With a reasoning similar to
the one which allows to predict probability
peaks for the velocity of planets, this can
be generalized to larger intergalactic scales.
Nottale writes:
In the same way as there are well-established
structures in the position space (stars, clusters
of stars, galaxies, groups of galaxies, clusters
of galaxies, large scale structures), the
velocity probability peaks are simply the
manifestation of structuration in the velocity
space. In other words, as it is already well-known
in classical mechanics, a full view of the
structuring can be obtained in phase space.
== Cosmological applications ==
=== 
Large numbers hypothesis ===
Nottale noticed that reasoning about scales
was a promising road to explain the large
numbers hypothesis. This was elaborated in
more details in a working paper. The scale-relativistic
way to explain the large numbers hypothesis
was later discussed by Nottale and by Sidharth.
=== Prediction of the cosmological constant
===
In scale relativity, the cosmological constant
is interpreted as a curvature. If one does
a dimensional analysis, it is indeed the inverse
of the square of a length. The predicted value
of the cosmological constant, back in 1993
was:
ΩΛ h2 = 0.36Depending on model choices,
the most recent predictions give the following
range:
0.311 < ΩΛ h2 (predicted) < 0.356,while
the measured cosmological constant from the
Planck satellite is:
ΩΛ h2 (measured) = 0.318 ±0.012.Given the
improvements of the empirical measures from
1993 until 2011, Nottale commented:
The convergence of the observational values
towards the theoretical estimate, despite
an improvement of the precision by a factor
of more than 20, is striking.
Dark energy can be considered as a measurement
of the cosmological constant. In scale relativity,
dark energy would come from a potential energy
manifested by the fractal geometry of the
universe at large scales, in the same way
as the Newtonian potential is a manifestation
of its curved geometry in general relativity.
=== Horizon problem ===
Scale relativity offers a new perspective
on the old horizon problem in cosmology. The
problem states that different regions of the
universe have not had contact with each other's
because of the great distances between them,
but nevertheless they have the same temperature
and other physical properties. This should
not be possible, given that the transfer of
information (or energy, heat, etc.) can occur,
at most, at the speed of light.
Nottale writes that special scale relativity
"naturally solves the problem because of the
new behaviour it implies for light cones.
Though there is no inflation in the usual
sense, since the scale factor time dependence
is unchanged with respect to standard cosmology,
there is an inflation of the light cone as
t → Λ/c″, where Λ is the Planck length
scale (ħG/c3)1/2. This inflation of the light
cones makes them flare and cross themselves,
thereby allowing a causal connection between
any two points, and solving the horizon problem
(see also).
== Applications to other fields ==
Although scale relativity started as a spacetime
theory, its methods and concepts can and have
been used in other fields. For example, quantum-classical
kinds of transitions can be at play at intermediate
scales, provided that there exists a fractal
medium which is locally nondifferentiable.
Such a fractal medium then plays a role similar
to that played by fractal spacetime for particles.
Objects and particles embedded in such a medium
will acquire macroquantum properties. As examples,
we can mention gravitational structuring in
astrophysics (see section 5), turbulence,
superconductivity at laboratory scales (see
section 7.1), and also modeling in geography
(section 7.4).
What follows are not strict applications of
scale relativity, but rather models constructed
with the general idea of relativity of scales.
Fractal models, and in particular self-similar
fractal laws have been applied to describe
numerous biological systems such as trees,
blood networks, or plants. It is thus to be
expected that the mathematical tools developed
through a fractal space-time theory can have
a wider variety of applications to describe
fractal systems.
=== Superconductivity and macroquantum phenomena
===
The generalized Schrödinger equation, under
certain conditions, can apply to macroscopic
scales. This leads to the proposal that quantum-like
phenomena need not to be only at quantum scales.
In a recent paper, Turner and Nottale proposed
new ways to explore the origins of macroscopic
quantum coherence in high-temperature superconductivity.
=== Morphogenesis ===
If we assume that morphologies come from a
growth process, we can model this growth as
an infinite family of virtual, fractal, and
locally irreversible trajectories. This allows
to write a growth equation in a form which
can be integrated into a Schrödinger-like
equation.
The structuring implied by such a generalized
Schrödinger equation provides a new basis
to study, with a purely energetic approach,
the issues of formation, duplication, bifurcation
and hierarchical organization of structures.An
inspiring example is the solution describing
growth from a center, which bears similarities
with the problem of particle scattering in
quantum mechanics. Searching for some of the
simplest solutions (with a central potential
and a spherical symmetry), a solution leads
to a flower shape, the common Platycodon flower
(see Fig. 2). In honor to Erwin Schrödinger,
Nottale, Chaline and Grou named their book
"Flowers for Schrödinger" (Des fleurs pour
Schrödinger).
=== Biology ===
In a short paper, researchers inspired by
scale relativity proposed a log-periodic law
for the development of the human embryo, which
fits pretty well with the steps of the human
embryo development.
With scale-relativistic models, Nottale and
Auffray did tackle the issue of multiple-scale
integration in systems biology.Other studies
suggest that many living systems processes,
because embedded in a fractal medium, are
expected to display wave-like and quantized
structuration.
=== Geography ===
The mathematical tools of scale relativity
have also been applied to geographical problems.
=== Singularity and evolutionary trees ===
In their review of approaches to technological
singularities, Magee and Devezas included
the work of Nottale, Chaline and Grou:
Utilizing the fractal mathematics due to Mandlebrot
(1983) these authors develop a model based
upon a fractal tree of the time sequences
of major evolutionary leaps at various scales
(log-periodic law of acceleration – deceleration).
The application of the model to the evolution
of western civilization shows evidence of
an acceleration in the succession (pattern)
of economic crisis/non-crisis, which point
to a next crisis in the period 2015–2020,
with a critical point Tc = 2080. The meaning
of Tc in this approach is the limit of the
evolutionary capacity of the analyzed group
and is biologically analogous with the end
of a species and emergence of a new species.
The interpretation of this emergence of a
new species remains open to debate, whether
it will take the form of the emergence of
transhumans, cyborgs, superintelligent AI,
or a global brain.
== Reception and critique ==
=== 
Scale relativity and other approaches ===
It may help to understand scale relativity
by comparing it to various other approaches
to unifying quantum and classical theories.
There are two main roads to try to unify quantum
mechanics and relativity: to start from quantum
mechanics, or to start from relativity. Quantum
gravity theories explore the former, scale
relativity the latter. Quantum gravity theories
try to make a quantum theory of spacetime,
whereas scale relativity is a spacetime theory
of quantum theory.
==== String theory ====
Although string theory and scale relativity
start from different assumptions to tackle
the issue of reconciling quantum mechanics
and relativity theory, the two approaches
need not to be opposed. Indeed, Castro suggested
to combine string theory with the principle
of scale relativity:
It was emphasized by Nottale in his book that
a full motion plus scale relativity including
all spacetime components, angles and rotations
remains to be constructed. In particular the
general theory of scale relativity. Our aim
is to show that string theory provides an
important step in that direction and vice
versa: the scale relativity principle must
be operating in string theory.
==== Quantum gravity ====
Scale relativity is based on a geometrical
approach, and thereby recovers the quantum
laws, instead of assuming them. This distinguishes
it from other quantum gravity approaches.
Nottale comments:
The main difference is that these quantum
gravity studies assume the quantum laws to
be set as fundamental laws. In such a framework,
the fractal geometry of space-time at the
Planck scale is a consequence of the quantum
nature of physical laws, so that the fractality
and the quantum nature co-exist as two different
things.
In the scale relativity theory, there are
not two things (in analogy with Einstein's
general relativity theory in which gravitation
is a manifestation of the curvature of space-time):
the quantum laws are considered as manifestations
of the fractality and nondifferentiability
of space-time, so that they do not have to
be added to the geometric description.
==== Loop quantum gravity ====
They have in common to start from relativity
theory and principles, and to fulfill the
condition of background independence.
==== El Naschie's E-Infinity theory ====
El Naschie has developed a similar, yet different
fractal space-time theory, because he gives
up differentiability and continuity. El Naschie
thus uses a "Cantorian" space-time, and uses
mostly number theory (see Nottale 2011, p.
7). This is to be contrasted with scale relativity,
which keeps the hypothesis of continuity,
and thus works preferentially with mathematical
analysis and fractals.
==== Causal dynamical triangulation ====
Through computer simulations of causal dynamical
triangulation theory, a fractal to nonfractal
transition was found from quantum scales to
larger scales. This result seems to be compatible
with quantum-classical transition deduced
in another way, from the theoretical framework
of scale relativity.
==== Noncommutative geometry ====
For both scale relativity and non-commutative
geometries, particles are geometric properties
of space-time. The intersection of both theories
seems fruitful and still to be explored. In
particular, Nottale further generalized this
non-commutativity, saying that it "is now
at the level of the fractal space-time itself,
which therefore fundamentally comes under
Connes's noncommutative geometry. Moreover,
this noncommutativity might be considered
as a key for a future better understanding
of the parity and CP violations, which will
not be developed here."
==== Doubly special relativity ====
Both theories have identified the Planck length
as a fundamental minimum scale. However, as
Nottale comments:
the main difference between the "Doubly-Special-Relativity"
approach and the scale relativity one is that
we have identified the question of defining
an invariant length-scale as coming under
a relativity of scales. Therefore the new
group to be constructed is a multiplicative
group that becomes additive only when working
with the logarithms of scale ratios, which
are definitely the physically relevant scale
variables, as we have shown by applying the
Gell-Mann–Levy method to the construction
of the dilation operator (see Sec. 4.2.1).
==== Nelson stochastic mechanics ====
At first sight, scale relativity and Nelson's
stochastic mechanics share features, such
as the derivation of the Schrödinger equation.
Some authors point out problems of Nelson's
mechanics in multi-time correlations in repeated
measurements.
On the other hand, perceived problems could
be resolved.
By contrast, scale relativity is not founded
on a stochastic approach. As Nottale writes:
Here, the fractality of the space-time continuum
is derived from its nondifferentiability,
it is constrained by the principle of scale
relativity and the Dirac equation is derived
as an integral of the geodesic equation. This
is therefore not a stochastic approach in
its essence, even though stochastic variables
must be introduced as a consequence of the
new geometry, so it does not come under the
contradictions encountered by stochastic mechanics.
==== Bohm's mechanics ====
Bohm's mechanics is a hidden variables theory,
which is not the case of scale relativity.
In this way, they are quite different. This
point is explained by Nottale (2011, p. 360):
In the scale relativity description, there
is no longer any separation between a "microscopic"
description and an emergent "macroscopic"
description (at the level of the wave function),
since both are accounted for in the double
scale space and position space representation.
=== Cognitive aspects ===
Special and general relativity theory are
notoriously hard to understand for non-specialists.
This is partly because our psychological and
sociological use of the concepts of space
and time are not the same as the one in physics.
Yet, the relativity of scales is still harder
to apprehend than other relativity theories.
Indeed, humans can change their positions
and velocities but have virtually no experience
of shrinking or dilating themselves.
Such transformations appear in fiction however,
such as in Alice's Adventures in Wonderland
or in the movie Honey, I Shrunk the Kids.
=== Sociological analysis ===
Sociologists Bontems and Gingras did a detailed
bibliometrical analysis of scale relativity
and showed the difficulty for such a theory
with a different theoretical starting point
to compete with well-established paradigms
such as string theory.Back in 2007, they considered
the theory to be neither mainstream, that
is, there are not many people working on it
compared to other paradigms; but also neither
controversial, as there is very little informed
and academic discussion around the theory.
The two sociologists thus qualified the theory
as "marginal", in the sense that the theory
is developed inside academics, but is not
controversial.
They also show that Nottale has a double career.
First, a classical one, working on gravitational
lensing, and a second one, about scale relativity.
Nottale first secured his scientific reputation
with important publications about gravitational
lensing, then obtained a stable academic position,
giving him more freedom to explore the foundations
of spacetime and quantum mechanics.
A possible obstacle to the growth in popularity
of scale relativity is that fractal geometries
necessary to deal with special and general
scale relativity are less well known and developed
mathematically than the simple and well-known
self-similar fractals. This technical difficulty
may make the advanced concepts of the theory
harder to learn. Physicists interested in
scale relativity need to invest some time
into understanding fractal geometries. The
situation is similar to the need to learn
non-euclidian geometries in order to work
with Einstein's general relativity. Similarly,
the generality and transdiciplinary nature
of the theory also made Auffray and Noble
comment: "The scale relativity theory and
tools extend the scope of current domain-specific
theories, which are naturally recovered, not
replaced, in the new framework. This may explain
why the community of physicists has been slow
to recognize its potential and even to challenge
it."
Nottale's popular book, written in French,
has been compared with Einstein's popular
book Relativity: The Special and the General
Theory. A future translation of this book
from French into English might help the popularization
of the theory.
=== Reactions ===
The reactions from scientists to scale relativity
are generally positive. For example, Baryshev
and Teerikorpi write:
Though Nottale's theory is still developing
and not yet a generally accepted part of physics,
there are already many exciting views and
predictions surfacing from the new formalism.
It is concerned in particular with the frontier
domains of modern physics, i.e. small length-
and time-scales (microworld, elementary particles),
large length-scales (cosmology), and long
time-scales.
Regarding the predictions of planetary spacings,
Potter and Jargodzki commented:
In the 1990s, applying chaos theory to gravitationally
bound systems, L. Nottale found that statistical
fits indicate that the planet orbital distances,
including that of Pluto, and the major satellites
of the Jovian planets, follow a numerical
scheme with their orbital radii proportional
to the squares of integers n2 extremely well!
Auffray and Noble gave an overview:
Scale relativity has implications for every
aspect of physics, from elementary particle
physics to astrophysics and cosmology. It
provides numerous examples of theoretical
predictions of standard model parameters,
a theoretical expectation for the Higgs boson
mass which will be potentially assessed in
the coming years by the Large Hadron Collider,
and a prediction of the cosmological constant
which remains within the range of increasingly
refined observational data. Strikingly, many
predictions in astrophysics have already been
validated through observations such as the
distribution of exoplanets or the formation
of extragalactic structures.
Although many applications have led to validated
predictions (see above), Patrick Peter criticized
a provisionally estimated value of the Higgs
boson in Nottale (2011):
a prediction for the Higgs boson that should
have been observed at mH ≃113.7GeV...it
would appear, according to the book itself,
that the theory it describes would be already
ruled out by LHC data!
However, this prediction was initially made
at a time when the Higgs boson mass was totally
unknown. Additionally, the prediction does
not rely on scale relativity itself, but on
a new suggested form of the electroweak theory.
The final LHC result is mH = 125.6 ± 0.3
GeV, and lies therefore at about 110% of this
early estimate.
Particle physicist and skeptic Victor Stenger
also noticed that the theory "predicts a nonzero
value of the cosmological constant in the
right ballpark". He also acknowledged that
the theory "makes a number of other remarkable
predictions".
== See also ==
Fractal cosmology
General relativity
Special relativity
Galilean invariance
Multifractal system
