Construction, Structure and Domains of Application

Construction of the Theory

The theory of scale relativity is based on the giving up of the hypothesis of manifold differentiability which is a key assumption of Einstein's general relativity. In the new theory, the coordinate transformations are continuous but can be differentiable (and therefore it includes general relativity) or nondifferentiable. The giving up of the assumption of differentiability implies several consequences [57], leading to the successive steps of the construction of the theory:

(1) It has been proved [57, 58, 59] that a continuous and nondifferentiable curve is
fractal in a general meaning, namely, its length is explicitly scale dependent and
goes to infinity when the scale interval ε goes to zero, i.e.when ε → 0. This result can be readily extended to a continuous and nondifferentiable manifold.

(2) The fractality of space-time [60, 54] involves the scale dependence of the reference frames. We therefore add to the usual variables defining the reference frames (position, orientation, motion), new variables ε characterizing their 'state of scale'. In particular, the coordinates themselves become functions of these scale variables, i.e. X = X(ε) (in the simplified case of only one variable). In an experimental situation, these scale variables are identified with the resolution scale of the measurement apparatus. In the case of a theoretical physics description, they are identified with the differential elements themselves, of which the coordinates become explicit functions,
i.e. X = X(dX).

(3) The scale variable ε can never be defined in an absolute way, but only in a relative way. Namely, only their ratio ρ = ε'/ε does have a physical meaning. This universal behavior leads to extend to scales the principle of relativity [33, 53, 57], in order to include in the possible changes of reference frames the new ones which are described by the transformations of these scale variables.

(4) Though the nondifferentiability manifests itself at the limit ε → 0, the use of differential equations is made possible by representing physical quantities ƒ by fractal functions ƒ[X(ε),ε] [57]. Even if the function ƒ(X,0) is nondifferentiable with respect to the variable X, the fractal function ƒ(X,ε) is differentiable for any ε ≠ 0 with respect to both X and ε. This allows us to complete the differential equations of standard physics by new differential equations of scale, which are constrained by the principle of scale relativity. The study of the scale laws derived from these differential equations has been developed according to various levels of relativistic transformations [53, 58, 61].

(5) The simplest possible scale differential equation is a first order equation,
∂X/∂lnε = β(X), which can be simplified again by Taylor expanding the unknown function β, so that it reads ∂X/∂lnε = a + bX + ... The solution of this equation is made of two terms, a scale-independent, differentiable, classical part and a powerlaw, nondifferentiable fractal part, which read where x = -a/b. When the coefficient b is constant, the second term is the standard expression for the length of a fractal curve of dimension D_F = 1 - b [38]. Moreover, the transformation law of this power-law term under a scale transformation
ln(λ/ε) → ln(λ/ε') takes the mathematical form of the Galileo group, and it therefore comes under the principle of relativity [53], as initially required (however, see Special and General Scale Relativity for more on this subject).

General structure of the theory

The theory of scale relativity is constructed by completing the standard laws of classical physics (motion in space / displacement in space-time) by new scale laws (in which the space-time resolutions are used as intrinsic variables, playing for scale transformations the same role as played by velocities for motion transformations). The hope is that such a stage of the theory is only provisional, and that motion and scale laws will be treated on the same footing in the final theory. However, before reaching such a goal, one must realize that the various possible combinations of scale laws and motion laws lead to a large number of sub-sets of the theory to be developed. Indeed, three domains of the theory are first to be considered:

(i) Pure scale-laws: description of the internal structures of a non-differential space-time at a given point / event;

(ii) Induced effects of scale laws on the equations of motion: generation of the quantum mechanics as mechanics on a non-differentiable space-time;

(iii) Scale-motion coupling: effects of dilations induced by displacements, that are tentatively interpreted as gauge fields.

Several levels of the description of scale laws (point i) can be considered. These levels are quite parallel to that of the historical development of the theory of motion:

(i1) Galilean scale-relativity: standard laws of dilation, that have the structure of a Galileo group (fractal power law with constant fractal dimension). When the fractal dimension of trajectories is D = 2, the induced motion laws are that of standard quantum mechanics.

(i2) Special scale-relativity: generalization of the laws of dilation to a Lorentzian form [15]. The fractal dimension itself becomes a variable, and plays the role of a fifth dimension, called 'djinn'. It is combined, not with the standard space-time coordinates, that keep their four-dimensional nature of signature (+,-,-,-), but with the four fractal fluctuations. Two impassable length-time scales, invariant under dilations, appear in the theory; they replace the zero (and the infinite), and play for scale laws the same role as played by the speed of light for motion. The minimal horizon scale is identified with the Planck length-scale, and the maximal one with the scale of the cosmological constant. Such a proposal has several implications for high energy physics and for cosmology, which allow to make new theoretical predictions and to put the theory to the test with success.

(i3) Non-linear scale laws and scale-dynamics: while the first two cases correspond to "scale freedom", one can also consider distorsion from strict self-similary, as described by second-order differential equations of scale transformations. This generalisation includes log-periodic corrections to scale invariance. Still more general distorsions from self-similarity can also be described in terms of a 'scale-dynamics', i.e. of the effect of a "scale-force" (that is a mere Newton-like way to describe geometric effects in the scale space).

(i4) General scale-relativity: in analogy with the field of gravitation being ultimately attributed to the geometry of space-time, a more profound description of the scale-field can be done in terms of geometry of the scale 'space-djinn' and its couplings with the standard classical space-time. The account of scale-motion couplings, that leads to a new interpretation of gauge fields (third step hereabove), is a part of such a general theory of scale-relativity.

(i5) Quantum scale-relativity: the above cases assume differentiability of the scale transformations. If one assumes them to be continuous but, as we have assumed for space-time, non-differentiable, one is confronted for scale laws to the same conditions that lead to quantum mechanics in space-time. One may therefore conjecture that quantum mechanical scale laws could be constructed.

The possible complication of the theory becomes apparent when one realizes that these various levels of the description of scale laws will lead to different levels of induced dynamics (point ii) and scale-motion coupling (iii), and that other sublevels are to be considered, depending on the status of motion laws (non-relativistic, special-relativistic, general-relativistic).

Domains of application of the theory

A new complication comes from the fact that three domains of application of the theory can be considered:

(a) Microphysics: new scale-relativistic effects are expected in the realm of elementary particle physics at high energy. We expect hbar to have an effective value that varies with scale beyond the top quark energy. Moreover, new experiments can also be considered: in analogy with the fact that the free fall reference system for gravitation is an accelerating system, one may conjecture that some particle-fields could be absorbed in coordinate systems characterized by a "scale-acceleration". Recall indeed that a resolution is identified, in the scale-relativity theory, with a scale-velocity, which is a derived quantity; one can go one step further with the concept of scale-field and introduce a second-derived quantity, the scale-acceleration. In such a new experiment, the resolution of the measurement apparatus should be variable in space-time, while the scale dependance should be no longer self-similar.

(b) Cosmology: another natural domain of application of scale-relativity is very large scales. We have proposed that Lorentzian scale laws could also be valid toward the large scales, implying the existence of a finite, maximal resolution, impassable, invariant under dilations, owning the physical properties of the infinite, which has been identified with the scale of the cosmological constant. In this new framework, new solutions can be brought to the problems of the vacuum energy density, of Mach principle and of the large number hypothesis.

(c) Complex systems: it has been suggested that a general theory of structure formation should take a quantum mechanical-like form. Indeed, several structures, in particular in the biological domain, are density waves rather than solid objects. For describing such a system, one is no longer interested in the individual trajectories, but instead in the overall structure, that the individual particles only cross. Our proposal is to go even further and, basing ourselves on the fractal character of individual trajectories and on the fundamental irreversibility of such a process at the infinitesimal level, to describe complex structures as density waves in term of probability amplitudes, themselves being solutions of a generalized Schrödinger equation. An application of this approach to the problem of the formation and evolution of gravitational structures has given several results in the recent years, at scales ranging from planetary systems to large scale structures of the universe: solar system, extra-solar planetary systems, planets around pulsars, obliquities and inclinations in the solar system, satellites of giant planets, binary stars, binary galaxies, large scale distribution of galaxies.

Let us finally remark that large parts of the theory (i2, iii) are still uncomplete, since they are up to now expressed only in the framework of a reduced number of dimensions. Only a very small fraction of the theory has been developed up to now, and far more remains to be done, in particular concerning the levels (ii)+(i2-i5) and (iii).
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Quoted from:

Nottale, L., 1999, Chaos, Solitons and Fractals, 10, 459
"The Scale-Relativity Program"

Nottale L., Célérier M.N. & Lehner T., 2006, J. Math. Phys. 47, 032303
"Non Abelian gauge field theory in scale relativity"