Scale relativity and non-differentiable fractal space-time

The theory of scale relativity [14] is an attempt to study the consequences of giving up the hypothesis of space-time differentiability. One can show [14] [15] that a continuous but nondifferentiable space-time is necessarily fractal. Here the word fractal [12] is taken in a general meaning, as defining a set, object or space that shows structures at all scales, or on a wide range of scales. More precisely, one can demonstrate [17] that a continuous but nondifferentiable function is explicitly resolution-dependent, and that its length L tends to infinity when the resolution interval tends to zero, i.e. L = L(ε)ε→0 → ∞. This theorem and other properties of non-differentiable curves have been recently analysed in detail by Ben Adda and Cresson [4]. It naturally leads to the proposal that the concept of fractal spacetime [21] [25] [14] [7] is the geometric tool adapted to the research of such a new description. In such a generalized framework including all continuous functions, the usual differentiable functions remain included, but as very particular and rare cases. 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. We thus need to complete the standard laws of physics (which are essentially laws of motion in classical physics) by laws of scale, intended to describe the new resolution dependence. We have suggested [13] that the principle of relativity can be extended to constrain also these new scale laws. Namely, we generalize Einstein’s 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 and acceleration). In scale relativity, we assume that the spacetime resolutions are not only a characteristic of the measurement apparatus, but