## Appendix A: continuity and nondifferentiability implies scale-divergenceOne can demonstrate
(Nottale 1993a,
1996a) that the length of a
continuous and nowhere differentiable curve is dependent on resolution
, and, further, that
when , i.e. that this curve is fractal (we used
the word "fractal" in this extended meaning throughout this paper).
The scale divergence of continuous and almost nowhere differentiable
curves is a direct consequence of Lebesgue's theorem, which states
that Consider indeed a continuous but nondifferentiable function
between two points A
and A in the Euclidean plane. Since Therefore, if What about the reverse proposition: Is a continuous function whose
length is scale-divergent between any couple of points (such that
finite), i.e., when
, nondifferentiable? The answer is as follows:
This can be demonstrated by looking at the way the length increases and the slope changes under successive zooms of a constant factor . There are two ways by which the divergence can occur: either by a regular increase of the length (due to the regular appearance of new structures at all scales that continuously change the slope), or by the existence of jumps (in this case, whatever the scale, there will always exist a smaller scale at which the slope will change). The power law corresponds to a continuous length increase, , then to a continuous and regular change of slope when : therefore the function is nondifferentiable in this case. Divergences slower than power laws may correspond to a regular length increase, but with a factor which becomes itself scale-dependent: with when . In this case, some functions can be differentiable, if they are such that new structures indeed appear at all scales (and could then be named "fractal" under the general definition initially given by Mandelbrot 1975 to this word), but these structures become smaller and smaller with decreasing scale, so that a slope can finally be defined in the limit . Some other functions diverging slower than power laws are not differentiable, e.g. if there always exists a scale smaller than any given scale such that an important change of slope occurs: in this case, the slope limit may not exist in the end. ## Appendix B: special and generalized scale-relativity## B.1. Special scale relativityIt is well known that the Galileo group of motion is only a degeneration of the more general Lorentz group. The same is true for scale laws. Indeed, if one looks for the general linear scale laws that come under the principle of scale relativity, one finds that they have the structure of the Lorentz group (Nottale 1992). Therefore, in special scale relativity, we have suggested to substitute to the Galilean laws of dilation the more general Lorentzian law (Nottale 1992, 1993a): This expression is yet uncomplete, since under this form the scale
relativity symmetry remains unbroken. Such a law corresponds, at the
present epoch, only to the null mass limit. It is expected to apply in
a universal way during the very first instants of the Universe. This
law assumes that, at very high energy, no static scale and no space or
time unit can be defined, so that only pure contractions and dilations
have physical meaning. The corresponding physics is a physics of pure
numbers. In Eq. (B1), there appears a universal, purely numerical
constant , which plays the role of a maximal
possible However, the effect of the spontaneous scale symmetry breaking
which arises at some scale is to yield a new
law in which the invariant is no longer a dilation
, but becomes a A fractal curvilinear coordinate becomes now scale-dependent in a covariant way, namely . One of the main new feature of special scale relativity with respect to the previous fractal or scale-invariant approaches is that the scale-dimension , which was previously constant, is now explicitly varying with scale (see Fig. 5) and even diverges when resolution tends to the new invariant scales. In the microphysical domain, the invariant length-scale is naturally identified with the Planck scale, , that now becomes impassable and plays the physical role that was previously devoted to the zero point (Nottale 1992, 1993a). The same is true in the cosmological domain, with once again an inversion of the scale laws. We have identified the invariant maximal scale with the scale of the cosmological constant, . The consequences of this new interpretation of the cosmological constant have been considered in (Nottale 1993a, 1995d, 1996a) and will be developed further in a forthcoming work (Nottale 1997).
Note that special scale-relativistic laws (Nottale 1992) have also recently been considered by Dubrulle (1994) and Dubrulle and Graner (1996) for the description of turbulence, with a different interpretation of the variables. It is also noticeable that recent developments in string theories (Witten 1996) have reached conclusions that are extraordinarily similar to those of scale relativity. One finds that there is a smallest circle in string theory (whose radius is about the Planck length), and that strings are characterized by duality symmetries. Two of these dualities are especially relevant to our approach, since they make already part of it in a natural way. The first is the quantum / classical duality, which we recover in terms of our scale / motion duality. The second is a microscopic / macroscopic duality: it has been found that strings do not distinguish small spacetime scales from large ones, relating them through an inversion. But scale inversion is a transformation which is naturally included in the scale-relativistic framework (see Sect. 2.5), since this is nothing but the symmetric element of the scale group ( in the Galilean case). Therefore it has recently been claimed by Castro (1996) that scale relativity is the right framework in which the newly discovered string structures will take their full physical meaning. The string duality between the small and large scales adds a new argument to our main conclusion: namely, that the laws of physics take again a quantum-like form at very large spacetime scales. ## B.2. From scale dynamics to general scale relativityThe whole of our previous discussion indicates to us that the scale
invariant behavior corresponds to freedom in the framework of a scale
physics. However, in the same way as there exists forces in nature
that imply departure from inertial, rectilinear uniform motion, we
expect most natural fractal systems to also present distorsions in
their scale behavior respectively to pure scale invariance. Such
distorsions may be, as a first step, attributed to the effect of a
scale "dynamics", i.e. to "scale-forces". (Caution: this is only an
where is a "scale-mass", which measures the way the system resists to the scale-force. ## B.2.1. Constant scale-forceLet us first consider the case of a constant scale-force. Eq. (B4) writes where constant. It is easily integrated in terms of the usual parabolic solution (where : However the physical meaning of this result is not clear under this form. This is due to the fact that, while in the case of motion laws we search for the evolution of the system with time, in the case of scale laws we search for the dependence of the system on resolution, which is the directly measured observable. We find, after redefinition of the integration constants: The scale dimension becomes a linear function of resolution (the same being then true of the fractal dimension ), and the relation is now parabolic rather than linear as in the standard power-law case. There are several physical situations where, after careful examination of the data, the power-law models were clearly rejected since no constant slope could be defined in the plane. In the several cases where a clear curvature appears in this plane (e.g., turbulence, sand piles,...), the physics could come under such a "scale-dynamical" description. In these cases it might be of the highest interest to identify and study the scale-force responsible for the scale distorsion (i.e., for the deviation to standard scaling). ## B.2.2. Harmonic oscillatorAnother interesting case of scale-potential is that of a repulsive harmonic oscillator. It is solved as For it gives the standard Galilean case , but its large-scale behavior is particularly interesting, since it does not permit the existence of resolutions larger than a scale . Such a behavior could provide a model of confinement in QCD (Nottale 1997). More generally, we shall be led to look for the general non-linear scale laws that satisfy the principle of scale relativity (see also Dubrulle and Graner 1997). As remarked in (Nottale 1994b, 1996a), such a generalized framework implies working in a five-dimensional fractal space-time. The development of such a "general scale-relativity" lies outside the scope of the present paper and will be considered elsewhere (Nottale 1997). ## B.3. Scale-motion coupling and gauge invarianceThe theory of scale relativity also allows to get new insights
about the physical meaning of gauge invariance
(Nottale 1994b,
1996a).
In the previous scale laws, only scale transformations at a given
point were considered. But we must also wonder about what happens to
the structures in scale of a scale-dependent object when it is
displaced. Consider anyone of these structures, lying at some
(relative) resolution (such that
, where is the
fractal/nonfractal transition) for a given position of the particle.
In a displacement of the object, the relativity of scales implies that
the resolution at which this given structure appears in the new
position will a priori be different from the initial one. In other
words, is now a function of the space-time
coordinates, , and we expect the occurrence of
where a four-vector must be introduced since is itself a four-vector and a scalar (in the case of a global dilation). This behavior can be expressed in terms of a new scale-covariant derivative: However, if one wants such a "field" to be physical, it must be defined whatever the initial scale from which we started. Starting from another scale (we consider only Galilean scale-relativity here, see Nottale 1994b, 1996a for the additional implications of special scale-relativity), we get which depends on the relative "state of scale", . However, if one now considers translation along two different coordinates (or, in an equivalent way, displacement on a closed loop), one may write a commutator relation: This relation defines a tensor field which,
contrarily to , is independent of the initial
scale. One recognizes in the analog of an
electromagnetic field, in , that of an
electromagnetic potential, in © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |