Scale-relativity, Fractal Space-time and Gravitational Structures

The theory of scale relativity extends Einstein's principle of relativity to scale transformations of resolutions. It is based on the giving up of the axiom of differentiability of the space-time continuum. Three consequences arise from this withdrawal: (i) The geometry of space-time must be fractal, i.e., explicitly resolutiondependent. (ii) The geodesics of the non-differentiable space-time are themselves fractal and in infinite number. (iii) Time reversibility is broken at the infinitesimal level. These three effects are finally combined in terms of a new tool, the scale-covariant derivative, which transforms classical mechanics into a generalized, quantum-like mechanics. We describe virtual families of geodesics in terms of probability densities, interpreted as a tendency for the system to make structures. In the present contribution, we apply our scale-covariant procedure to the equations of motion of test-particles in a gravitational potential. A generalized Newton-Schrödinger equation is obtained, and its solutions are studied. Our theoretical predictions are then successfully checked by a comparison with observational data at various scales, ranging from planetary systems to large scale structures. A possible interpretation of these results is that the underlying fractal geometry of space-time plays the role of a universal structuring "field" that leads to selforganization and morphogenesis of matter in the Universe.