Astron. Astrophys. 327, 867-889 (1997)

1. Introduction

The theory of scale relativity (Nottale 1993a) is founded on the realization that the whole of present physics relies on the implicit assumption of differentiability of the space-time continuum. Giving up the a priori hypothesis of the differentiability of coordinates has important physical consequences: one can indeed demonstrate (Nottale 1993a, 1994a, 1995a) that a continuous but nondifferentiable space-time is necessarily fractal. Here the word fractal (Mandelbrot 1975, 1983) is taken in a general meaning, as defining a set, object or space that shows structures at all scales. More precisely, one can demonstrate that a continuous but nondifferentiable function is explicitly resolution-dependent, and that its length tends to infinity when the resolution interval tends to zero, i.e. (see Appendix A). This theorem naturally leads to the proposal that the concept of fractal space-time (Nottale 1981; Nottale and Schneider 1984; Ord 1983; Nottale 1989, 1993a; El Naschie 1992) is the geometric tool adapted to the research of such a new description based on non-differentiability.

It is important to be by now more specific about the precise meaning of the withdrawal of the axiom of differentiability. That does not mean that we a priori assume that the coordinates are not differentiable with certainty, but instead that we consider a generalized framework including all continuous functions, those which are differentiable and those which are not. Thus this framework includes the usual differentiable functions, but as very particular and rare cases. It is an extension of the usual framework, so that the new theory is expected, not to contradict, but instead to generalize the standard theory, since standard differentiable physics will be automatically included in it as a special case. An historical example of such an extension is the passage to curved spacetimes in general relativity, which amounts to giving up the previous implicit assumption of flatness of Euclidean geometry, and which anyway includes flat spacetimes in its description.

Since a nondifferentiable, fractal space-time is explicitly resolution-dependent, the same is a priori true of all physical quantities that one can define in its framework. (Once again, this means that we shall formally introduce such a scale-dependence as a generalization, but that the new description will also include the usual scale-independence as a special case, in a way similar to the relations between statics and kinematics: statics is a special, degenerate case of the laws of motion). We thus need to complete the standard laws of physics (which are essentially laws of motion and displacement in classical physics) by laws of scale, intended to describe the new resolution dependence. We have suggested (Nottale 1989, 1992, 1993a) that the principle of relativity of Galileo and Einstein, that is known since Descartes and Huygens to be a constructive principle for motion laws, can be extended to constrain also these new scale laws.

Namely, we generalize Einstein's (1916) formulation of the principle of relativity, by requiring that the laws of nature be valid in any reference system, whatever its state. Up to now, this principle has been applied to changes of state of the coordinate system that concerned the origin, the axes orientation, and the motion (measured in terms of velocity, acceleration, ...) .

In scale relativity, the space-time resolutions are not only a characteristic of the measurement apparatus, but acquire a universal status. They are considered as essential variables, inherent to the physical description. We define them as characterizing the "state of scale" of the reference system, in the same way as the velocity characterizes its state of motion. The principle of scale relativity consists of applying the principle of relativity to such a scale-state. Then we set a principle of scale-covariance, requiring that the equations of physics keep their simplest form under resolution transformations (dilations and contractions).

The domains of application of this theory are typically the asymptotic domains of physics, small length-scales and small time-scales , (microphysics), large length-scales (cos mology), but also large time-scales . The present series of papers particularly addresses this last domain.

Initially, the theory of scale relativity was mainly an attempt at refounding quantum mechanics on first principles (Nottale 1993a). We have demonstrated that the main axioms of quantum mechanics can be recovered as consequences of the principle of scale relativity, and that the behavior of the quantum world can be understood as the various manifestations of the non-differentiability and fractality of space-time at small scales (Nottale 1993a, 1994a, b, 1995c, 1996a). Moreover, the theory allows one to generalize standard quantum mechanics. Indeed, we have shown that the usual laws of scale (power law, self-similar, constant fractal dimension) have the status of "Galilean" scale laws, while a full implementation of the principle of scale relativity suggests that they could be medium scale approximations of more general laws which take a Lorentzian form (Nottale 1992, 1993a). In such a "special scale relativity" theory, the Planck length- and time-scale becomes a minimal, impassable scale, invariant under dilations and contractions, which replaces the zero point (since owning all its physical properties) and plays for scales the same role as played by the velocity of light for motion (see Appendix B). In this new framework, several still unsolved problems of fundamental physics find simple and natural solutions: new light is brought on the nature of the Grand Unification scale, on the origin of the electron scale and of the electroweak scale, on the scale-hierarchy problem, and on the values of coupling constants (Nottale 1993a, 1994a, 1996a); moreover, our scale-relativistic interpretation of gauge invariance allowed us to give new insights on the nature of the electric charge, then to predict new mass-charge relations for elementary particles (Nottale 1994a, b, 1996a).

The same approach has been applied to the cosmological domain, leading to similar conclusions. Namely, new special scale-relativistic dilation laws can be constructed, in terms of which there exists a maximal length-scale of resolution, impassable and invariant under dilations. Such a scale can be identified with the scale of the cosmological constant (). It would own all the physical properties of the infinite. Its existence also solves several fundamental problems in cosmology, including the problem of the vacuum energy density, the value of the cosmological constant, the value of the index of the galaxy-galaxy correlation function and the transition scale to uniformity (Nottale 1993a, 1995d, 1996a).

In the present series of papers, we shall not consider the consequences of this new interpretation of the cosmological constant, which mainly apply to the domain of very large scales () . This case has already been briefly considered in (Nottale 1993a Chap. 7, 1995d) and will be the subject of a particularly devoted, more detailed work (Nottale 1997). We shall instead specialize our study here to the typical scales where structures are observed, which remain small compared with the size of the Universe, but yet are of cosmological interest. For such scales, Galilean scale laws (i.e., standard self-similar laws with constant fractal dimension) remain a good approximation of the more general scale-relativistic laws.

We are mainly concerned here with the physical description of systems when they are considered on very large time-scales. As we shall see, this question is directly related to the problem of chaos. It is indeed now widely known that most classical equations describing the evolution of natural systems, when integrated on sufficiently large times, have solutions that show a chaotic behavior. The consequence of strong chaos is that, at time-scales very large compared with the "chaos time", (the inverse Lyapunov exponent), i.e., beyond the horizon of predictability, there is a complete loss of information about individual trajectories. Basing ourselves on the existence of such predictability horizons, we have suggested (Nottale 1993a, Chap. 7) that the universal emergence of chaos in natural systems was the signature of new physics on very large time scales, and that chaotic systems could be described beyond the horizon by a new, quantum-like, non-deterministic theory, since the classical equations become unusable for .

But we shall suggest in the present paper an even more profound connection between chaos and scale relativity. After all, chaos has been discovered (Poincaré 1892) as a general, empirical property of the solutions of most classical equations of physics and chemistry, when applied to natural systems in all their complexity. But this does not mean that we really understand its origin. On the contrary, it is rather paradoxical that deterministic equations, built in the framework of a causal way of thinking (one gives oneself initial conditions in position and velocity, then the evolution of the system is predicted in a totally deterministic way), finally lead to a complete loss of predictability of individual trajectories. Our first proposal (Nottale 1993a, b) has therefore been (following the above reasoning) to jump to a non-deterministic description, that would act as a large-time scale approximation. We suggest in the present paper to reverse the argument, and to take into account the particular chaos that takes its origin in the underlying, non-differentiable and fractal character of space-time, since, as we shall see, it is expected to become manifest not only at small scales but also at very large length-scales and time-scales. The advantages of this viewpoint reversal are important:
(i) The breaking of the reflection invariance(), which is one of the principal new effect of nondifferentiability (see Sect. 3.1.3), find its complete justification only when acting at the space-time level, not only that of fractal trajectories in a smooth space-time.
(ii) The chaotic behavior of classical equations could now be understood (or at least related to a first principle approach): these equations would actually be incomplete versions of more general equations, that would be classical and deterministic at small scales, but would become quantum-like and non-deterministic at large-time scales (see Sect. 3).
(iii) The additional conclusion in the scale-relativistic framework is that the new structuring behavior may be universal. We shall see from a comparison with observational data (Paper III) that the observed structures show indeed universal properties, since we find that identical structuring laws are observed at scales which range from the Solar System scale to the cosmological scales, and since these laws are written in terms of a unique new fundamental constant (Nottale 1996b, c; Nottale, Schumacher and Gay 1997).

In the present paper, we shall first be more specific about our motivations for constructing such a new theory (Sect. 2), then we shall describe our general method (Sect. 3), and apply it to the fundamental equations used in several domains of fundamental physics having astrophysical implications (Sect. 4). Additional informations, in particular about the general framework of which the present developments are a subset, are given in Appendices A and B. Paper II will be specially devoted to the application of our theory to gravitational structures, and Paper III to a first comparison of our theoretical predictions with observational data.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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