Scale relativity

Scale relativity is a geometrical and fractal space-time physical theory.the laws of physics must be such that they apply to coordinate systems whatever their state of scale.the structure of space has both a smooth (differentiable) component at the macro-scale and a chaotic, fractal (non-differentiable) component at the micro-scale, the transition taking place at the de Broglie length scale.The theory offers a new interpretation of gauge transformations and gauge fields (both Abelian and non-Abelian), which are manifestations of the fractality of space-time, in the same way that gravitation is derived from its curvature.With his equation for the probability density of planetary orbits around a star, Nottale has seemingly come close to the old analogy which saw a similarity between our solar system and an atom in which electrons orbit the nucleus. But now the analogy is deeper and mathematically and physically supported: it comes from the suggestion that chaotic planetary orbits on very long time scales have preferred sizes, the roots of which go to fractal space-time and generalized Newtonian equation of motion which assumes the form of the quantum Schrödinger equation.The suggestion to use the formalism of quantum mechanics for the treatment of macroscopic problems, in particular for understanding structures in the solar system, dates back to the beginnings of the quantum theoryThe Titius-Bode 'law' of planetary distance is of the form a + b × c n, with a = 0.4 AU, b = 0.3 AU and c = 2 in its original version. It is partly inconsistent — Mercury corresponds to n = −∞, Venus to n = 0, the Earth to n = 1, etc. It therefore 'predicts' an infinity of orbits between Mercury and Venus and fails for the main asteroid belt and beyond Saturn. It has been shown by Herrmann (1997) that its agreement with the observed distances is not statistically significant. ... n the scale relativity framework, the predicted law of distance is not a Titius-Bode-like power law but a more constrained and statistically significant quadratic law of the form an = a0n2.In the same way as there are well-established structures in the position space (stars, clusters of stars, galaxies, groups of galaxies, clusters of galaxies, large scale structures), the velocity probability peaks are simply the manifestation of structuration in the velocity space. In other words, as it is already well-known in classical mechanics, a full view of the structuring can be obtained in phase space.The convergence of the observational values towards the theoretical estimate, despite an improvement of the precision by a factor of more than 20, is striking.Utilizing the fractal mathematics due to Mandlebrot (1983) these authors develop a model based upon a fractal tree of the time sequences of major evolutionary leaps at various scales (log-periodic law of acceleration – deceleration). The application of the model to the evolution of western civilization shows evidence of an acceleration in the succession (pattern) of economic crisis/non-crisis, which point to a next crisis in the period 2015–2020, with a critical point Tc = 2080. The meaning of Tc in this approach is the limit of the evolutionary capacity of the analyzed group and is biologically analogous with the end of a species and emergence of a new species.It was emphasized by Nottale in his book that a full motion plus scale relativity including all spacetime components, angles and rotations remains to be constructed. In particular the general theory of scale relativity. Our aim is to show that string theory provides an important step in that direction and vice versa: the scale relativity principle must be operating in string theory.The main difference is that these quantum gravity studies assume the quantum laws to be set as fundamental laws. In such a framework, the fractal geometry of space-time at the Planck scale is a consequence of the quantum nature of physical laws, so that the fractality and the quantum nature co-exist as two different things.the main difference between the 'Doubly-Special-Relativity' approach and the scale relativity one is that we have identified the question of defining an invariant length-scale as coming under a relativity of scales. Therefore the new group to be constructed is a multiplicative group that becomes additive only when working with the logarithms of scale ratios, which are definitely the physically relevant scale variables, as we have shown by applying the Gell-Mann–Levy method to the construction of the dilation operator (see Sec. 4.2.1).Here, the fractality of the space-time continuum is derived from its nondifferentiability, it is constrained by the principle of scale relativity and the Dirac equation is derived as an integral of the geodesic equation. This is therefore not a stochastic approach in its essence, even though stochastic variables must be introduced as a consequence of the new geometry, so it does not come under the contradictions encountered by stochastic mechanics.In the scale relativity description, there is no longer any separation between a 'microscopic' description and an emergent 'macroscopic' description (at the level of the wave function), since both are accounted for in the double scale space and position space representation.Though Nottale's theory is still developing and not yet a generally accepted part of physics, there are already many exciting views and predictions surfacing from the new formalism. It is concerned in particular with the frontier domains of modern physics, i.e. small length- and time-scales (microworld, elementary particles), large length-scales (cosmology), and long time-scales.In the 1990s, applying chaos theory to gravitationally bound systems, L. Nottale found that statistical fits indicate that the planet orbital distances, including that of Pluto, and the major satellites of the Jovian planets, follow a numerical scheme with their orbital radii proportional to the squares of integers n2 extremely well!Scale relativity has implications for every aspect of physics, from elementary particle physics to astrophysics and cosmology. It provides numerous examples of theoretical predictions of standard model parameters, a theoretical expectation for the Higgs boson mass which will be potentially assessed in the coming years by the Large Hadron Collider, and a prediction of the cosmological constant which remains within the range of increasingly refined observational data. Strikingly, many predictions in astrophysics have already been validated through observations such as the distribution of exoplanets or the formation of extragalactic structures.a prediction for the Higgs boson that should have been observed at mH ≃113.7GeV...it would appear, according to the book itself, that the theory it describes would be already ruled out by LHC data! Scale relativity is a geometrical and fractal space-time physical theory. Relativity theories (special relativity and general relativity) are based on the notion that position, orientation, movement and acceleration cannot be defined in an absolute way, but only relative to a system of reference. The scale relativity theory proposes to extend the concept of relativity to physical scales (time, length, energy, or momentum scales), by introducing an explicit 'state of scale' in coordinate systems. This extension of the relativity principle was originally introduced by Laurent Nottale, based on the idea of a fractal space-time theory first introduced by Garnet Ord, and by Nottale and Jean Schneider. To describe scale transformations requires the use of fractal geometries, which are typically concerned with scale changes. Scale relativity is thus an extension of relativity theory to the concept of scale, using fractal geometries to study scale transformations.