Scale relativity and non-differentiable fractal space-time

Abstract

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→o →∞. 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.