## 1. IntroductionThe 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 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 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 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, 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 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 |