Hierarchy of the Incomputable: General and Scale Relativity

We often talk about the universe as a simulation, a grand computer program running the laws of physics. But what if the very fabric of reality is impossible to program? Physicists and logicians have long known that any simulation is just an approximation. However, a fascinating duel between two great theories of gravity—Einstein's General Relativity and Laurent Nottale's Scale Relativity—reveals that there are different levels of impossibility. Some theories are simply "more" non-simulable than others.

To understand this, let's forget about spacetime for a moment and think about a simpler task: simulating a coastline.

Level 1: Simulating a Smooth Beach (The General Relativity Problem)

Imagine a perfect, sweeping, sandy beach. Its curve is smooth and gentle. This is the world of Einstein's General Relativity, where spacetime is described as a smooth, continuous fabric.

Now, you want to create a computer model of this beach. Your computer can't handle a perfect curve; it can only handle a finite set of points. So, you do the obvious: you place a series of points along the beach and connect them with straight lines.

This is an approximation, of course. But it's a good one. If you want a better model, you just add more points. As you increase the resolution, your jagged, digital model gets closer and closer to the smooth reality. You can get arbitrarily close to a perfect representation because the underlying reality is differentiable—it's smooth. The physics between your data points is simple.

This is why we can simulate General Relativity. We create a grid of points in spacetime and calculate the physics on that grid. It’s an approximation of a continuous reality, but it’s a manageable one. The theory is non-simulable in principle (because of the infinite points in a continuum), but we can get as close as we need for any practical purpose.

Level 2: Simulating a Rocky Coastline (The Scale Relativity Problem)

Now, imagine a different kind of coast: the rugged, rocky coastline of Norway. This is the world of Scale Relativity, where spacetime is described as a fractal fabric.

You start the same way, by placing a series of points along the coast to map it out. But here, something strange happens. You decide to zoom in on a single straight-line segment between two of your points to see how good your approximation is.

You don't find a gently curving, nearly straight line. You find a whole new world of complexity: smaller bays, jagged rocks, and tiny inlets that were completely invisible from a distance. The complexity doesn't smooth out; it increases. If you zoom in again on a tiny piece of this new coastline, the same thing happens. This self-repeating complexity at every level of magnification is the definition of a fractal.

This presents a fundamentally deeper problem for any simulation. Your approximation isn't just a low-resolution version of reality; it's a completely different and far simpler object. The physics between your grid points is not simple—it's infinitely complex. Adding more points doesn't just refine the picture; it reveals entirely new universes of structure you didn't even know were there.

A Hierarchy of the Impossible

This is why Scale Relativity is "more" non-simulable than General Relativity. It contains two nested layers of impossible infinity:

1. The Infinity of Points: The classic problem of trying to model a continuous line with a finite number of dots. (Shared by both theories).

2. The Infinity of Structure: The radical problem that the path between any two of those dots is itself infinitely complex. (Unique to Scale Relativity).

Simulating General Relativity is a challenge of resolution. Simulating Scale Relativity is a challenge of infinite, nested complexity.

This isn't just an abstract mathematical game. It cuts to the heart of what reality might be. Is the universe, at its smallest scales, a smooth and simple place, as Einstein assumed? Or is it an infinitely intricate coastline, a fractal reality whose depths we can explore forever without ever reaching the end? The answer determines not just what our theories look like, but what is fundamentally knowable, what can be computed, and whether the universe can ever be fully captured in the ones and zeros of a simulation.