I have transcribed the text of the interview in french language with Laurent Nottale, conducted in 2024 by Olivier Bordes of Radio France.
English Translation:
Welcome to the second part of the interview with Laurent Nottale on relativity, vacuum, and fractals.
To begin, Laurent Nottale, could you first explain what fractals are, and then talk about fractality?
Fractals are a concept that Benoît Mandelbrot formulated. He realized that there was a tradition of describing objects, systems, mathematical, physical or biological, that had common properties. In 1974-75, he called this "fractals." Initially, fractals are objects, but the concept goes beyond the notion of objects, which are explicitly dependent on scales. They are structured in scales. That's the essence of fractals.
To know if a curve, a surface, a volume, or something else is fractal, we cannot simply look at it at our scale because we can be mistaken. You can have extremely structured objects, and a zoom can show us that they are not fractal. The key to fractals is that you have to zoom in. We cannot know if something is fractal at a single scale. You have to change the scale. Fractality is characterized by the fact that we see structures at a given scale, we zoom and there are still structures, and so on. It's really structure within structure, within structure. When it's not fractal, even a complex curve smooths out as you zoom in. Fractals are not differentiable; we cannot define a slope. At each scale, the slope varies again.
We must distinguish between ultimate mathematical fractals, where the structuring in successive scales is limitless, and fractals in nature, which are structured in scales over a certain range. Fractal objects are of this nature, with a minimum and a maximum scale.
For example, at our scale, it is around the meter. We can see the millimeter. For the micron (thousandth of a millimeter), we need a microscope. Ordinary light is not enough. We need to use X-rays, then tunneling microscopes to see atoms (nanometers, billionth of a meter). We can manage to see the inside of atoms, but to go further, we need particle colliders. In fact, when we look at something with light, we bombard the object with photons. We can do this with light, X-rays, etc. We always need a system of interaction to see something. We use photons that are more and more energetic, electrons, and recently, we measured the radius of the proton with neutrinos. It's extraordinary because neutrinos interact very little. We send interacting particles to measure the characteristics of a given object.
We are in full fractality. We move from one scale to another, and we see new things each time, different structures. The essence of fractality is that it is not just self-similarity (where you see the same structure again). In nature, we see structures within structures, but they are different. There are common properties despite these different structures.
For this reason, in the theory of scale relativity, I was inspired by the relativity of position. We wondered what happens here, then a little further away. The great idea is that these are the same laws, but they are expressed in different coordinate systems. I applied this idea to the 4th quantity characterizing the state of the coordinate system which is the scale. To make a measurement, we must characterize the relative position, the relative orientation, the relative movement, and the relative resolution. A coordinate system, as Einstein pointed out, is a set of rulers and clocks. We must orient, build a system with rules X, Y, Z, and a clock to measure time, with which we measure an event (position and time). We think we can work with that, but there is a bug. We say that we will make measurements with a ruler, but a ruler without markings does not measure anything! In official physics, we do not mention that coordinate systems must include the resolution of the measurement, that is, the marks on the ruler. Resolution has an essential role in understanding the result. All physics is organized like that: if we do not give the uncertainty of measurement, it has no meaning. We are obliged to define the maximum and minimum scale of the ruler. The key point is the resolution. When we change the resolution, we really change the nature of what we are measuring. We use another instrument to go from a resolution of 1 mm to 1 micron. The big mistake is to think that by looking at something on a smaller scale, we see the same thing more precisely. But it is false.
In biology, we used to think that the fetus was a small man, but it's more like a small fish. In defining fractality, we talk about fractal spacetime, do you have any further information? Yes, to explain fractals, I was led to describe what things are really like. We have to change the scale towards the small, as towards the large.
At the global level, the scale ratios reach around 10^60 between the smallest conceivable scales (Planck scale) and the largest (cosmological constant). The Planck scale is made from the fundamental constants: speed of light, gravitational constant, and Planck's constant. These constants are beyond measurements; they are ratios between mass and energy on one side, and spacetime on the other. We combine this for a length, a time, and a Planck mass.
The scale of the cosmological constant is around 3 gigaparsecs. We take the ratio of the two, and we get 10^60.
Mandelbrot showed that fractals were everywhere and that it was our view that was deficient. Our view of the world was built on the idea of a steam engine builder. Nature is not smooth. It is fractal. Fractal entities are not smooth and are very dynamic. The non-smooth aspect does not mean it is fractal. You have to zoom in. A tree has a trunk, main branches, then smaller branches. It is fractal at the limit of its canopy, its leaves. Before that, it is a "pre-fractal", the making of a fractal by successive scales.
Is there a continuity in fractals? This is indeed an important point. Fractals are often organized as iterated systems. These systems are discrete, not continuous. The Von Koch curve is an example. At each step, it is not yet a fractal but a fractal generator. We reproduce a structure at each step, and this fractal dimension is not an integer; it's the log of 4 over the log of 3. It's an object between a curve and a surface, a plane seen as a curve. There are a lot of paradoxes with fractals. We see the link between fractal spacetime and relativity of vacuum.
So, what is the place of the fractals of fractal spacetime in the concept of vacuum relativity? In wanting to apply the concept of fractal not only to an object but to space and time, I was forced to describe them in a different way, not by iterated systems, but by differential equations, because space and time must be continuous. The method is the same as for the description of a physical system: we take the state of the system at a given point and look at another point very close to it. That's the great idea of Descartes: a complex system is broken down into simpler parts. We use differential equations to describe fractals in spacetime, which are equations of motion in space, and of dilation or contraction in the space of scales.
Laurent Nottale, what is the link with Einstein's generalized relativity? The theory of scale relativity applies the principle of relativity to the change of scale, in addition to the changes in position, orientation, and movement. Scale relativity adds this scale transformation and is described naturally by fractal spaces. General relativity is described by curved spaces. A fractal space is a space that is explicitly dependent on the scale. Physical quantities diverge when the resolution interval tends towards zero. Fractals and non-differentiability are linked. Non-differentiability says we cannot define a slope. In physics, derivatives are velocities and accelerations. In classical mechanics, we need derivatives, but in quantum mechanics, we also use derivatives in the Schrödinger equation, even when talking about the Heisenberg relation. Non-differentiability is not assuming that a system has a velocity and acceleration. A continuous function is continuous once differentiable or twice differentiable. Quantum mechanics supposes at least a C2 function. Scale relativity relaxes this hypothesis. Non-differentiability does not mean that we cannot define a difference, but that we cannot define the ratio of these differences.
A continuous and non-differentiable spacetime is fractal. Fractality is a tool for describing a non-differentiable space, but this is just one property. A non-differentiable spacetime has other essential properties that do not come from fractality. The measurement quantities (length, time, etc.) are explicitly dependent on the scale and are divergent when the interval tends towards zero. Length is no longer a number, but a function of the scale.
Where did the idea of scale relativity come from? It was an intuition about the foundations of quantum mechanics. In its official version, it is not founded; it is a set of rules without origin. A great physicist like Feynman even said that these rules were absurd. Most physicists gave up hope of founding quantum mechanics on something deeper. Yet, this was Einstein's big question, which has been completely misunderstood. When he said that quantum mechanics was incomplete, he meant incomplete at the level of its principles, of its foundation. Yet, physicists in quantum mechanics interpreted it as “incomplete within itself” when it works perfectly and there is no need to add to it. It must be completed to make sense, but it doesn't mean adding something inside, but that a new construction and a translation is needed between them. I thought that if a spacetime existed, it should create quantum mechanics and then, I asked what is the relativity of quantum mechanics, whereas, paradoxically, the spacetime of quantum mechanics is flat (Minkowski).
For gravitation, Einstein said that an ellipse is just a manifestation of the geodesic of a curved space. I wanted the trajectories of particles to be geodesics of a spacetime. It must be such that the quantum manifests itself through its structure. Just as the universal property of spacetime describes gravitation (curvature), there should be a spacetime with a universal property that describes the quantum. Quantum measurements say that if I make measurements at smaller and smaller spatial resolutions, the impulse becomes larger and larger and even diverges. I told myself that the key was that the measurement results depend on the interval of resolution and tend towards infinity when that interval tends to zero. This is what I told you about a non-differentiable and fractal spacetime. Spacetime that describes quantum mechanics must then be explicitly dependent on the scale, and the nature of that dependence is a fractal one. A colleague, Thibault Moulin, told me it was Mandelbrot's fractal geometry. That’s why I called it a fractal, but my initial idea was a spacetime that we could call "scaling spacetime". I thought I would have had fewer problems calling it like that. "Fractal spacetime," in the 80s, was considered totally ridiculous by theoretical physicists.
I had an incredible number of referee opinions who rejected my articles, considering the idea of a fractal spacetime totally ridiculous. I really had opinions like that.
As for Benoît Mandelbrot himself, I think his problem was that he had a mind focused on objects and did not know about general relativity. The concept of spacetime was not clear to him. When I tried to talk to him about fractal spacetime, he did not understand. Later he began to understand, but it took a very long time. It was not his fractals. Mandelbrot's fractal is an object viewed from the outside. Spacetime must be described from the inside. This is the great discovery of curved spacetimes, made by Gauss and Lobachevsky: we can characterize the fact that a space is not Euclidean only through measurements inside the space.
I would like to talk about your research on galaxies. I'm working on the problem of dark matter. It has been known for almost a century that galaxies are much more massive than we thought, from studying velocity dispersion. There needed to be a factor of 100 between the mass and the luminous mass. It was seen that this was true even at the level of an individual galaxy. Normally, the velocity should decrease when looking beyond the radius of a galaxy. The velocity remains constant: these are the flat rotation curves.
This abnormal dynamics suggests an abnormal potential energy, or missing one. Scientists have hypothesized dark matter. Another hypothesis proposes to modify the Newtonian dynamics (MOND). The dark matter hypothesis works well for galaxy clusters, but badly at the scale of individual galaxies. We can study this dynamics with objects around galaxies, but also with pairs of galaxies. In Newtonian dynamics, a two-body system can be transformed into a one-body system plus a small body. I have created a catalog of 13,000 galaxies and have found the characteristic rate of dark matter and I have compared that with the planetary system.
The rotation velocity of the galaxies goes to a probability peak at 150 km/s, and it is exactly the same probability peak that we have for exoplanets. Exoplanets orbit their star at 150 km/s. A third of the ones we know orbit at that speed. In the solar system, we find a law of structuring of velocities, with integer ratios. There is a fundamental value and then the velocities are fractional multiples of this value.
All of this is predicted by the theory of scale relativity. In 1990, I demonstrated the axioms of quantum mechanics and the Schrödinger equation. It, which is a postulated equation in quantum mechanics, becomes demonstrated by scale relativity. The trajectories are the geodesics of a fractal space, and they are not unique, but a beam. This beam of trajectories is a geodesic fluid that follows a probability distribution which has a peak. In my equations, there is a constant, but nothing says it has to be Planck's constant. These equations can therefore apply to any chaotic system.
The solar system is a chaotic system, and protoplanetary nebulae too. I wrote the Schrödinger equation for the solar system, and I found that it organizes itself according to a fundamental speed, and fractional multiples of that speed. Mercury is on orbital 3, Venus 4, Earth 5, and Mars 6. I then predict that there must be two deeper levels. At 0.17 astronomical units, I predicted a ring of asteroids (which turns out to be the source of the Earth-crossing asteroids). I predict an orbital at 0.045 astronomical units per solar mass, which corresponds to the speed of 145 km/s. All of this was published in 1992.
Exoplanets were discovered three years later. The speed of 150 km/s is that of exoplanets, pairs of galaxies, and now satellites of giant galaxies. We find ourselves in the same situation as between a star and its planets. This theory quantifies planetary orbits in stellar systems.
Quantum mechanics says that momentum is mass multiplied by velocity. But in relativity, mass disappears due to the equivalence principle. Therefore, it will be the velocity that will be quantified, and not the energy or momentum. I predict that all gravitational structures (stars, galaxies, clusters, etc.) organize themselves in the velocity space, which explains the existence of the same characteristic velocity at 150 km/s that I mentioned before.
Thank you Laurent Nottale.
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