Our quest to understand the universe has spawned revolutionary theories, with relativity holding a central position amongst them. From Galileo’s insights to the models of Einstein and Nottale, we've discovered that space, time, and physical phenomena aren't absolutes, but realities that depend on the observer's viewpoint, their resolution, measurement instruments, and their positioning in space and time. This document doesn't seek to pit the two theories of relativity against each other, but rather to comprehend their differences, particularly concerning gravitation, exploring what Scale Relativity (SR) brings to the table and how it fits in with General Relativity (GR).
General Relativity (GR), established by Albert Einstein at the beginning of the 20th century, revolutionised our perception of gravitation and demonstrated that it's intimately linked to the very structure of spacetime. His genius lay in his ability to conceptualise gravitation not as a force acting at a distance, but as a manifestation of the curvature of spacetime caused by the presence of mass/energy. This vision, underpinned by Riemannian geometry, not only enabled the description with unprecedented precision of the movements of celestial bodies at cosmic scales, but also introduced a link between the distribution of mass and how time and space evolve. In this sense, it introduced the notion that local time is directly linked to the presence of mass, which manifests itself as a slowing down of time. Indeed, in general relativity, time flows more slowly in an intense gravitational field. Mass no longer acts as a source of force at a distance (as in Newton’s theory), but creates a deformation of spacetime, which results in a slowing down of time in regions of strong gravity. However, whilst GR has enabled exceptional advances, it hasn't succeeded in integrating the effects of the quantum world and, moreover, it doesn't fully explain the rapid evolution of galaxies in the primordial universe. On the other hand, it isn’t established that GR is incapable of explaining the rapid evolution of galaxies in the primordial universe, it hasn't done so to date and this would require a more precise description of the interactions at play. Moreover, recent research has used GR applied to galaxies and a non-uniform universe, demonstrating that dark mass can be explained using simplified versions of GR (2022) and similarly for the origin of dark energy (2024).
Scale Relativity (SR), developed by Laurent Nottale in the 1980s-1990s, also presents a description with a variation of time from one point to another, but it demonstrates that mass is an emergent property of the fractal nature of spacetime, and that it’s this geometry that causes these variations in time. For SR, physics isn’t just a question of motion, but also a question of resolution. Nottale demonstrates that the very concept of dimension is relative, that length has no meaning except by comparison, and that space "fills up" depending on the resolution with which we observe it, that space has characteristics that depend on how we look at it, indeed, the scale used. Thus, in SR, the influence of mass on time is no longer an effect of the curvature of spacetime, but rather a consequence of the local increase in distance caused by its fractal structure. In other words, the more mass is present, the more space is "filled" with fractal paths that increase the effective distance between two points, and therefore an impression of a slowing down of time.
In this perspective, SR thus supersedes GR when it comes to dealing with the quantum world (something that GR fails to do) by its approach, which no longer relies on the variation of time imposed (we don't know how) by mass, but on the idea that mass is an emergent property of the fractal geometry of spacetime. For SR, time is only a global measure that is slowed down locally by the fractality of space in contact with mass. The idea is therefore to recognise that time evolves with the dimension and the resolution that is explored and that all the variations that we observe in the effect of gravitation are in fact due to variations in distance or time and, consequently, to a modification of the geometry of space/time.
Remarkably, SR, in addition to mass, also has profound implications on our understanding of the notion of charge and spin, which it proposes no longer to regard as imposed fundamental quantities, but rather as consequences of the exploration of geodesic paths and transformations of scales. SR can thus define a particle, not as a unique point, but as a fractal and dynamic exploration of a spacetime that is itself fractal.
Finally, GR and SR explain why the speed of light is constant regardless of the reference frame by considering it as a limitation which cannot be reached by a mass. Inertial mass increases until it reaches infinity were it achievable at the speed of light, so not possible. SR adds two other limitations, the zoom limitations, on one side to the resolution of the Planck scale, on the other the visualisation of the entire universe from a single point. These limits cannot be reached because they would require an infinite amount of energy (the entire universe). To get an idea of this zoom limitation, imagine what energy was needed to reach the resolution of the Higgs Boson. That is how all the energy of the universe would be required to see even smaller, to reach the Planck resolution, it is a zoom impossible to achieve. Thus, SR proposes a conception of spacetime where three spatial dimensions, time and one dimension linked to resolution (what Nottale called the "djinn") are sufficient to account for all physical phenomena, from the smallest particles to the largest cosmic structures. It's a description that highlights the properties of spacetime that must be described, and that they are consistent across both high and low energy regimes, and that time evolves with the scale properties and the geometry of spacetime.
Through this approach, the idea is no longer to assume different descriptions for the theories of the quantum world or of large cosmic structures, but to create a single description, with fractal mathematics as the keys to understanding these two extremes, and to make space an entity that underlies all things.
This perspective on relativity has a wealth of potential and has already demonstrated this across the entire spectrum of the universe’s scales. In fact, SR is more than just a physical theory: it represents a shift in perspective and an extraordinary way of approaching the challenges that nature presents to us, with the intuition that simplicity is hidden in a description based on a fractal geometry of space, which is at the origin of all that exists in the universe.
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