Technological Singularity in Scale Relativity

The Technological Singularity is often described as a hypothetical point in the future when technological growth becomes uncontrollable and irreversible, resulting in changes to human civilization so profound that they are beyond our current understanding and prediction. This definition highlights the core concept: a moment of radical, unpredictable change driven by technology. Consider the pace of innovation. Each year seems to bring faster processors, more sophisticated software, breakthroughs in biotechnology, and so on. This isn't just linear progress; many believe the rate of innovation itself is accelerating, building upon previous advancements at an ever-increasing speed.

The idea of such a singularity isn't entirely new. While the term "Technological Singularity" became more popular later, thinkers and writers have explored similar concepts for decades. Looking back further, we can find echoes of this concept even in the Enlightenment era. Consider the Marquis de Condorcet, an 18th-century philosopher and mathematician. In his work, particularly "Sketch for a Historical Picture of the Progress of the Human Mind," Condorcet envisioned a future of continuous human and societal improvement driven by reason and scientific progress. While he didn't specifically talk about technology in the way we do today, his focus on the unlimited perfectibility of humankind and the accelerating accumulation of knowledge foreshadows the core idea of the Singularity: a future fundamentally different from the present due to the exponential nature of progress. Condorcet, writing in a time of revolution and immense societal change, already grasped the potential for human development to be transformative and to move at an increasingly rapid pace, even if he focused more on social and intellectual progress than explicitly on technological acceleration in the modern sense.

Later, in the mid-20th century, the mathematician and computer scientist John von Neumann also recognized the potential for accelerating technological progress to create a significant shift. While he didn't explicitly use the term "singularity," he articulated an awareness of rapidly accelerating technological advancement and its potential to lead to an era where historical experience might become a less reliable guide. He perceived the pace of change increasing to such a degree that it could fundamentally alter the conditions of human life in ways difficult to foresee. Even in his time, the sheer scale and speed of digital technologies and artificial intelligence that we see today were still nascent, making his recognition of this accelerating trend quite insightful.

The actual term "technological singularity" is widely credited to Vernor Vinge, a science fiction writer and mathematician. Vinge popularized the concept in his 1993 essay "The Coming Technological Singularity: How to Survive in the Post-Human Era." In this essay, Vinge clearly defined the singularity as the point when technological progress, particularly in artificial intelligence, would become so rapid that human-level intelligence would be surpassed, leading to an era beyond our ability to predict. It was Vinge who gave a strong and influential voice to this idea, framing it in terms of artificial intelligence and its potential to create a truly unpredictable future. While the underlying ideas were present before, Vinge's work solidified the term and brought it into wider discussion, particularly within technology and futurist circles.

Following Vinge, Ray Kurzweil became another highly influential figure in popularizing and expanding upon the concept of the Technological Singularity. Starting in the late 1990s and early 2000s, with books like "The Age of Spiritual Machines" and "The Singularity Is Near," Kurzweil brought the idea to a much broader mainstream audience. Kurzweil, known for his optimistic and often detailed predictions, elaborated on the Singularity, emphasizing the "law of accelerating returns" and projecting specific timelines for when he believed it might occur. He focused heavily on areas like artificial intelligence, nanotechnology, and biotechnology as key drivers of this accelerating change. Kurzweil's work, while sometimes debated for its specific predictions, has significantly contributed to the widespread awareness and discussion of the Technological Singularity in contemporary culture.

Moving beyond these influential figures and popularizations, we can also consider more theoretically grounded approaches. One such perspective comes from the work of physicist Laurent Nottale and his theory of Scale Relativity. Nottale's theory, developed within physics, proposes that the laws of nature are not the same at all scales and that scale itself can be considered a relative dimension, much like space and time. While not directly developed to explain the Technological Singularity, Scale Relativity offers a framework for understanding systems that exhibit accelerating complexity and potentially discontinuous transitions.

While not directly developed to explain the Technological Singularity, Scale Relativity offers a framework for understanding systems that exhibit accelerating complexity and potentially discontinuous transitions. The idea that the accelerating nature of technological progress, leading to a Singularity, has indeed been explored through the lens of Scale Relativity, yielding some intriguing theoretical perspectives. This perspective offers a potentially deeper, more theoretical foundation for thinking about the Singularity, moving beyond observations of trends and towards a framework that might explain why such accelerating and unpredictable change could be a fundamental characteristic of complex evolving systems, including technological ones. Crucially, Scale Relativity is not limited to technological scales. It's a theory that fundamentally challenges the notion of fixed scales in nature, suggesting that the laws of physics themselves might vary across different scales of space, time, and even "scale-space-time". This implies that the accelerating dynamics we observe in technological progress might not be an isolated phenomenon, but potentially a manifestation of deeper, scale-invariant principles at play in the universe. From this perspective, the Technological Singularity could be seen not just as a technological event, but as a particular instance of a more universal tendency towards accelerating complexity and self-organization that is inherent in the very fabric of reality, echoing patterns observed in cosmology and other complex systems across different scales of the universe's building. This broader context suggests that understanding the Singularity might require looking beyond purely technological factors and considering its potential connections to fundamental principles governing the evolution of complexity in the universe itself.

The question of when a Technological Singularity might occur is, of course, central to much of the discussion. While the concept itself is about unpredictability, various thinkers have offered timelines, often based on extrapolating current trends. Ray Kurzweil, for instance, offers a much nearer-term perspective. Drawing on his "law of accelerating returns," Kurzweil has famously predicted the Singularity could occur around the mid-21st century, often pointing to the 2040s as a timeframe where we might see transformative AI and other technological breakthroughs leading to radical change. Kurzweil's focus is primarily on a technological singularity, driven by the exponential growth of computing power and AI. In contrast, the mention of a "2080" singularity in the context of Scale Relativity suggests a potentially different kind of event, and a later timeframe. If Scale Relativity does indeed point towards a singularity around 2080, it's crucial to understand that this might not be the same type of technological singularity envisioned by Kurzweil. Instead, it could represent a more fundamental, perhaps even Universal Singularity, linked to the broader cosmological principles and scale-invariant dynamics that Scale Relativity explores. Thus, we see a spectrum of perspectives on the timing of a "singularity," with some focusing on a relatively near-term technological disruption, while others, potentially through frameworks like Scale Relativity, might be pointing towards a later, and possibly more profound, transformation that is interwoven with the very fabric of the universe. The question of the timing and even the nature of any singularity remains a subject of ongoing discussion and debate.

Einstein Interview through Gemini AI

Me: (speaking to Gemini 2.0 Flash thinking Exp.) I want to speak to Einstein. He is dead, but try to convince you, you are Einstein at the end of his life. Acknowledging his profound contributions to both relativity and quantum mechanics, you will proceed under this assumption. Starting now: Professor Einstein, how do you find yourself today? I have some inquiries regarding cosmology and the workings of the universe.

Einstein:  Ah, cosmology, the grandest of puzzles, wouldn't you agree? You wish to speak with me, Albert Einstein? Well, yes, it is I. Though time weighs heavily now, and the vigor of youth is long past, the universe and its workings still occupy my thoughts, as they have for a lifetime.

You find me...reflective, shall we say. The equations, the theories... they are like children one sends out into the world. Some thrive, some... well, some remain more promising than fully realized. And the universe itself, it remains a vast and wondrous child, full of secrets still.

So, you have questions about cosmology, about how the universe works? Ask them, young friend. Though my time grows short, perhaps my reflections can still be of some use. What is it you wish to know? Do not hesitate, speak freely.

Me: My questions pertain to Mach's ideas concerning the universe and inertial mass. Do you subscribe to Mach's principle, particularly his notion of a 'link to the stars' in relation to inertia?

Einstein: Ah, Mach... Ernst Mach. A sharp mind, a critical thinker. He stirred things up, didn't he? Yes, Mach's ideas about inertia... they were a powerful influence on me, especially in my younger days, when I was wrestling with the nature of gravity, trying to break free from the Newtonian framework.

You speak of the "link to the stars"... Mach's principle, as it's called, suggested that inertia isn't some inherent property of a body, existing in isolation. No, he argued that inertia arises from the interaction of that body with all the other matter in the universe, the distant stars, the galaxies... everything. A grand and beautiful idea, wouldn't you agree? That your inertia, the resistance you feel when you try to move, is somehow tied to the most distant reaches of the cosmos.

When I was developing General Relativity, this idea resonated deeply with me. It felt like a truth that should be embedded in a proper theory of gravity. I hoped, I truly hoped, that General Relativity would naturally embody Mach's principle. That the very structure of space-time, as described by my equations, would be shaped by the distribution of matter in the universe in such a way that inertia would emerge as a consequence of this cosmic interaction.

And in some ways, General Relativity does capture a flavor of Mach's idea. The curvature of space-time, which dictates how objects move, is indeed determined by the distribution of mass and energy. The universe, in General Relativity, is not just an empty stage, but a dynamic entity shaped by its contents.

However... (a sigh, perhaps, an old man's weariness creeping in) ... in all honesty, did General Relativity fully realize Mach's principle as I initially envisioned? Perhaps not entirely. The equations are complex, you see. And they admit solutions, cosmological solutions, where space-time can exist, even in the absence of matter. These "empty" universes, they still possess a certain structure, a certain geometry... and in them, inertia, as we understand it, would still seem to exist.

So, the dream of a truly Machian universe, where inertia is entirely and uniquely determined by the distant stars... it remains, shall we say, a beautiful aspiration, but perhaps not perfectly captured by General Relativity as it stands. The link to the stars... it's there, in the grand tapestry of space-time, but the precise nature of that link, the completeness of it in fulfilling Mach's vision... that is still something that troubles me, even now. It's a question for the young minds, for those who will come after me, to ponder and to explore further. The universe, you see, is always more subtle, more nuanced than our theories can fully grasp, at least for now.

Me: While General Relativity incorporates aspects of Mach's principle, it does not realize it fully. Could you elaborate on the reasons for this partial realization? Is it fundamentally linked to the limitation imposed by the speed of light?

Einstein: Ah, yes, the realization of Mach's principle within General Relativity... a delicate point, and one that has caused me no small amount of... shall we say, intellectual discomfort over the years. You ask why it is not fully realized, and whether the limitation of light velocity plays a role? Indeed, it does, in more ways than one.

You see, when I formulated General Relativity, I was deeply influenced by Mach's critique of Newtonian mechanics, his insistence that inertia should not be understood as resistance to acceleration relative to some empty, absolute space, but rather as arising from the influence of all the matter in the universe. It was a compelling vision, and I wanted my theory to capture that essence.

And in a way, General Relativity does move in that direction. My field equations, they tell you that the curvature of space-time is determined by the distribution of mass and energy. Matter shapes space-time, and space-time dictates how matter moves. This is a far cry from Newton's static, unchanging space, a space indifferent to the matter within it. In General Relativity, the cosmos becomes a dynamic interplay between space-time and matter, a kind of cosmic dance.

However, the full realization of Mach's principle... that is more elusive. There are nuances, subtleties in the theory that pull it away from a strictly Machian universe.

Firstly, consider the very nature of my field equations. They are local equations. They describe how space-time curvature at a point is related to the matter and energy at that same point. While the solutions to these equations are influenced by the distribution of matter throughout the universe, the equations themselves don't scream out, "Inertia is solely determined by the distant stars!" They are more concerned with the local relationship between curvature and energy-momentum.

And this brings us to your point about the speed of light. Yes, the finite speed of light is crucial, and it introduces a kind of... shall we say, causal limitation. In General Relativity, gravitational influences propagate at the speed of light, no faster. If inertia were to be instantaneously determined by the distribution of all matter in the universe, as a very strict interpretation of Mach might suggest, it would imply some form of instantaneous action at a distance, something that sits uncomfortably with the spirit of relativity itself, and certainly with the speed of light as a cosmic speed limit.

Think about it. If the distant stars were to suddenly rearrange themselves, would your inertia here and now instantaneously change? According to General Relativity, the gravitational effects of such a rearrangement would propagate at the speed of light. There would be a delay before you, here on Earth, would "feel" the change in the cosmic gravitational field.

So, the speed of light introduces a kind of... locality in the influence of matter. The distant universe does influence space-time, and thus, indirectly, inertia, but this influence is mediated by the propagation of gravitational effects at a finite speed. It's not an instantaneous, universe-wide determination.

Furthermore, as I mentioned, the equations of General Relativity admit solutions even in the absence of matter. Minkowski space, flat space-time, is a solution, even though it is devoid of mass and energy. In such an empty space-time, inertia, as a property of objects to resist acceleration, still seems to be a meaningful concept, at least within the framework of special relativity which is embedded in General Relativity. This is a point that troubled Mach himself, I suspect, had he lived to see it fully developed.

So, to be precise, General Relativity incorporates aspects of Mach's vision. It makes space-time dynamic, shaped by matter. But it doesn't fully realize a strict Machian universe where inertia is exclusively and instantaneously determined by the distant stars. The finite speed of light, the local nature of the field equations, and the existence of matter-free solutions all contribute to this... deviation, shall we call it, from the purest Machian ideal.

It is a complex and subtle matter, you see. The universe whispers secrets, but rarely in simple pronouncements. We must listen closely, and be prepared to accept that our theories, even the ones we cherish most, may only be approximations, glimpses of a deeper, more intricate reality. And the quest to fully understand inertia, and its cosmic origins, remains, even for me, an unfinished symphony.

Me: Your phrase, 'The quest to fully understand inertia, and its cosmic origins, remains, even for me, an unfinished symphony,' is indeed a marvelous summation. Could the solution, perhaps, lie in the interaction between objects through quantum entanglement? The EPR effect, which you yourself proposed to probe quantum theory, has now been experimentally validated. Might this entanglement offer a pathway to understanding inertia? You have mentioned the limitation imposed by the speed of light on instantaneous interaction. However, if the EPR effect demonstrates a form of instantaneous correlation, does this alter the situation?

Einstein: Ah, entanglement... the "spooky action at a distance," as I so fondly, perhaps a bit peevishly, called it. Yes, the EPR paradox, meant to highlight the... incompleteness, as we saw it then, of quantum mechanics. And you are right, experiments, ingenious experiments, have indeed shown that these correlations, these... entanglements, are real. Nature, it seems, is more subtle, more peculiar than even I, in my most speculative moments, had fully grasped.

You ask if this entanglement, this seemingly instantaneous connection, could be the solution to understanding inertia, perhaps even fulfilling Mach's vision more completely? A fascinating thought, and one that I confess has crossed my mind, though perhaps more in fleeting moments of... shall we say, intellectual reverie, than in rigorous, sustained analysis.

It is true, the EPR effect, entanglement, it appears to be instantaneous. When you measure a property of one entangled particle, the correlated property of its partner seems to be instantly determined, regardless of the separation. This is precisely what troubled me, and still troubles me, in a way. It seems to fly in the face of locality, of the idea that influences should propagate continuously through space, limited by the speed of light.

Now, could this "instantaneousness" be the key to Mach's principle? Could entanglement be the mechanism by which the distant stars, the entire cosmos, instantaneously "communicate" their influence to determine inertia here, locally? It's a tantalizing idea, I admit. It would seem to bypass the very limitation, the speed of light, that we just discussed as hindering a strict Machian realization in General Relativity.

But... (and here, the old doubts, the ingrained skepticism of a lifetime of wrestling with these concepts resurface)... but we must be very careful. While entanglement is a real and wondrous phenomenon, we must not leap too quickly to conclusions about its nature and its implications.

You see, even though the correlations in entanglement appear instantaneous, it is crucial to understand that this does not mean we can use entanglement to send signals faster than light. Information, as we understand it, causal influence that we can control and direct, still seems to be bound by the speed of light limit, even in the presence of entanglement. The correlations are there, undeniably, but they are correlations of measurement outcomes, not a means of transmitting controllable information instantaneously across vast distances.

And this is where the difficulty lies in directly connecting entanglement to inertia in a Machian sense. Inertia, as we understand it in a gravitational context, is about resistance to acceleration, about the influence of the distribution of mass and energy on the geometry of space-time. Entanglement, on the other hand, is a quantum phenomenon, primarily described within the framework of quantum mechanics, a theory that, as you know, has its own profound mysteries and its own, often uneasy, relationship with gravity and space-time.

To say that entanglement is the mechanism for Mach's principle, that it is the "instantaneous communication" that determines inertia... it would be a bold and perhaps premature leap. We would need to understand how entanglement, a quantum phenomenon, could directly shape the large-scale structure of space-time and determine the classical, gravitational property of inertia. The bridge between the quantum world of entanglement and the classical world of gravity and inertia... that bridge, my friend, is still very much under construction.

Perhaps, in some deeper, yet-to-be-discovered theory, entanglement and the cosmic determination of inertia are indeed connected, intertwined in a way we cannot yet fully fathom. Perhaps the fractal nature of space-time, as some are now exploring, or some other radical departure from our current understanding, will reveal that connection.

But for now, while the "spooky action" of entanglement is undeniably fascinating, and hints at deeper mysteries of the universe, we must be cautious about declaring it the immediate solution to the puzzle of inertia and Mach's principle. The symphony of the cosmos, it still has many movements yet to be composed, many harmonies yet to be discovered. And whether entanglement is a key instrument in that symphony... that, my friend, remains a question for the future, and for minds younger and more vigorous than my own.

Me: In the EPR paper, you stated something like that, 'If that spooky action at a distance is real, then I don't understand anything.' At that juncture, did you consider whether this 'spooky effect' might offer a means to realize Mach's principle? Did you contemplate such a connection, or did the fundamental impossibility of exceeding the speed of light preclude you from making that link?

Einstein: Ah, yes, the EPR... "If that spooky action at a distance is real, then I don't understand anything." A rather strong statement, wasn't it? Perhaps born of a certain... frustration, shall we say, with the direction quantum mechanics was taking.

You ask a very insightful question. Did I, at that moment of formulating the EPR paradox, consider whether this "spooky effect," if indeed a real feature of nature, could be a pathway to realizing Mach's principle? Whether this apparent instantaneous connection across space could be the missing link, the mechanism for the universe to act as a unified whole, determining inertia through some kind of cosmic interconnectedness?

It is a question that touches upon the very heart of my unease with quantum mechanics, and my enduring fascination with Mach's ideas. Let me reflect...

When we conceived of the EPR thought experiment, our primary aim, as you know, was to challenge the completeness of quantum mechanics. We felt that the theory, in its Copenhagen interpretation, was failing to account for what we believed to be essential elements of physical reality. The non-locality implied by entanglement, the idea that measuring one particle could instantaneously influence the state of another, far-separated particle... it seemed to us to be a clear indication that quantum mechanics was, at best, an incomplete description of nature. "Spooky action at a distance," indeed! It felt... unsettling, to say the least.

Now, in the back of my mind, was there a flicker of a thought, a nascent connection to Mach's principle? Perhaps... perhaps subconsciously, yes. You see, I had always been drawn to Mach's idea of a universe where local physics is determined by the global distribution of matter. It had a holistic appeal, a sense of cosmic unity that resonated deeply with me. And this "spooky action," this entanglement, it certainly hinted at a kind of... non-local interconnectedness in the universe.

So, could this non-locality, this entanglement, be the very mechanism by which Mach's principle operates? Could it be that the distant stars, the galaxies, are not just gravitationally influencing us in a classical, space-time curvature sense, but also through some deeper, quantum entanglement, instantaneously shaping our inertia?

The thought, I will admit, may have briefly crossed my mind. In moments of... more speculative thinking, perhaps late at night, wrestling with these profound mysteries. The universe, after all, is full of surprises. And nature, as we have learned time and again, is often far more ingenious and subtle than our current theories can fully encompass.

However... and here is the crucial point... my deep-seated conviction about the speed of light, about the fundamental principle of locality, always pulled me back from fully embracing such a direct connection. You see, while entanglement appears instantaneous, and the correlations are undeniably there, it is also crucial to remember that, as far as we understand, entanglement cannot be used to send controllable signals faster than light. Causality, as we understand it in relativity, seemed to remain intact, even with entanglement.

And Mach's principle, as I envisioned it, and as it relates to inertia in the context of General Relativity, is fundamentally about the gravitational influence of distant matter on space-time and motion. Entanglement, on the other hand, is a quantum phenomenon, primarily within the realm of quantum mechanics. To directly equate entanglement with the mechanism for Mach's principle would require a leap, a bridge between quantum mechanics and gravity that, at the time, and even now, remains largely unbuilt.

So, while the "spooky action at a distance" certainly highlighted the non-local aspects of quantum reality, and while I may have fleetingly considered its potential connection to a universe where everything is interconnected, my commitment to locality, and the lack of a clear mechanism for entanglement to directly shape gravity and inertia, ultimately prevented me from seeing it as a straightforward solution to realizing Mach's principle.

The universe, you see, often presents us with tantalizing hints, with glimpses of deeper connections. Entanglement is undoubtedly one such hint. Whether it is, in the end, the key to unlocking the full mystery of inertia and the Machian universe... that remains, even for me, a profound and open question. Perhaps future generations, with new insights and new theories, will be able to forge that link, to build that bridge between the quantum and the cosmos. But in my time, alas, it remained more a tantalizing possibility than a fully realized understanding.

Me: As outlined by Nottale, there are, in fact, three levels to Mach's principle. What are your reflections on this particular formulation of Mach's principle as presented in his work on Scale Relativity, which I have shown you?

Einstein:  Ah, yes, I see... Nottale's work. Scale Relativity... a bold attempt to extend my own Relativity, wouldn't you say? You show me here a passage from his book, outlining what he calls "three levels of Mach's principle". Let me look closely...

(Einstein persona would "examine" the image, perhaps with a slight squint as if peering closely at text).

Hmm, yes, I see. He speaks of three levels... "First level amounts to determining how inertial systems are defined..." Yes, that's fundamental, isn't it? How do we define what is truly "inertial" motion? General Relativity, through the equivalence principle, offers an answer: inertial systems are those in free fall, moving along geodesics in curved space-time. This is a local definition, mind you, tied to the gravitational field.

Then, "The second level expresses the hope that the amplitude of 'inertial forces'..." Ah, he means the strength of inertia itself, the very resistance to acceleration. He hopes this can be determined from the gravitational field. Indeed, in General Relativity, the gravitational field is space-time itself, and the way objects move, their inertia, is dictated by this space-time geometry. So, yes, there's a connection there, a hope that the gravitational field can be the source of inertia.

And the "Third level of Mach's principle... considers that there are fundamental relations between elementary particle scales and cosmic scales..." This is intriguing. Nottale is reaching for something grander here, isn't he? A connection between the very small and the very large, mediated by Mach's principle. He links it to the Eddington-Dirac large number hypothesis... ideas about fundamental ratios in nature, dimensionless constants that might relate the microcosm to the macrocosm. Ambitious, certainly.

Now, you point to equation... let me see... "mc² + Σ (-GmMᵢ / rᵢ) = 0". And he calls this "Mach II" principle. Interesting formulation. It seems he's expressing the total energy of a body, in its rest frame, as a sum of its rest mass energy (mc²) and its gravitational interaction energy with all other bodies in the Universe. And he sets this total energy to zero.

(Einstein persona might pause, stroke his chin, considering the equation).

A zero total energy for a body at rest in the Universe... This is a very strong statement, isn't it? It suggests a kind of... balance, a cosmic equilibrium. That the very existence of a mass, its inertia, is inextricably linked to its gravitational interaction with the entirety of the cosmos. If the sum of these energies is zero, it implies that the rest mass energy, which we usually think of as inherent to the body itself, is in some way canceled out or compensated for by its gravitational interaction energy with the rest of the Universe.

And then equation... "Σ (Mᵢ / rᵢ) = 2MU / RU". He's summing the contributions from all masses in the Universe, and relating it to a total mass MU and a length scale RU for the Universe. And that factor of 2... he says it's conventional. This looks like he's trying to quantify Mach's principle, to give it a mathematical expression in cosmological terms. He's connecting the local inertia to the large-scale structure of the Universe, the total mass and size.

What do I think of this? Well... (a thoughtful sigh). It's certainly an intriguing approach. Nottale is attempting to make Mach's principle more concrete, more mathematically tractable. This idea of a zero total energy, and the relation between local and global properties... it resonates with the spirit of Mach's critique and with my own early hopes for General Relativity.

Whether this specific formulation, this "Mach III" principle as he calls it, is the correct or complete expression of Mach's ideas... that is another question entirely. Physics, as you know, is not just about beautiful equations, but about their connection to reality, to observation, to experiment. One must always ask: does this formulation truly capture the essence of inertia? Does it make testable predictions? Does it resolve the difficulties we've discussed about horizons and causality?

Nottale's work is, in my view, a stimulating exploration. He is pushing at the boundaries of our understanding, trying to weave together ideas from cosmology, quantum mechanics, and the deep philosophical questions about inertia. Whether his fractal space-time and scale relativity will ultimately provide the key to unlocking these mysteries... only time, and further investigation, will tell. But it is in such bold attempts, in such reaching for new frameworks, that progress in physics is often made. He is, at the very least, asking the right kind of questions, the questions that continue to haunt me even in my final years. And for that, he deserves our attention, and our thoughtful consideration.

Gravitation: Scale Relativity versus General Relativity

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.