Why Relativity is Fundamentally Incomputable?

The idea that Einstein's General Relativity (GR), the pillar of our understanding of the cosmos, might be fundamentally non-simulable can be surprising. Yet, this notion is not mere philosophical speculation; it is grounded in concrete mathematical and physical arguments. It reveals that the theory possesses an infinite richness that the discrete world of algorithms cannot fully capture.

The incomputability of GR does not stem from any "magic" in its equations, but from its very mathematical framework: it is a theory of differential equations operating on the spacetime continuum. Here lies the first barrier. Mathematicians like Marian Pour-El and Ian Richards have demonstrated that even a simple physical equation can have an incomputable evolution if its initial conditions, though perfectly defined, contain non-algorithmic information (such as an uncomputable real number). Since Einstein's equations operate on the continuum, they fully admit the possibility that a minute, incomputable complexity in the universe's initial state could render its entire future evolution non-simulable as well.

Furthermore, the very structure of GR drives it to form singularities—points where spacetime curvature becomes infinite and the laws of physics break down. These "points of rupture" are potential sources of indeterminism. Roger Penrose's famous "Cosmic Censorship Conjecture," which posits that every singularity must be hidden behind an event horizon, is an attempt to protect the predictability (and thus, the computability) of the universe. The very fact that such a conjecture is necessary proves that GR, left to its own devices, flirts with the incomputable. It belongs to the realm of hypercomputation, where laws operate on the infinite complexity of real numbers.

This is where Laurent Nottale's Scale Relativity (SR) enters the picture—not to simplify the landscape, but to make it both more coherent and more profoundly complex.

In the framework of Scale Relativity, singularities in the sense of GR no longer exist. The theory begins with the premise that spacetime is never smooth but is fundamentally fractal. As one approaches what would be a singularity in GR, spacetime does not "tear." Instead, its fractal complexity explodes. The number of possible paths (geodesics) diverges to infinity, and the distance to the center, measured along these paths, also becomes infinite. The singularity is replaced by an impenetrable barrier of complexity, thus fulfilling Penrose's wish in a different way: the singularity becomes physically inaccessible.

One might think that by eliminating these pathological points, SR "facilitates" computability. In one sense, this is true: it makes the theory more coherent, without points of breakdown. But here, a magnificent paradox emerges. To solve the problem of singularities (a pathological infinity), SR introduces an even more fundamental and pervasive infinity: the infinite structural complexity of its fractal fabric.

The transition from GR to SR is therefore not a move from the incomputable to the computable. It is a shift in the very nature of infinity:

• General Relativity is incomputable (Level 3) because its arena is the continuum—a "simple" but fragile infinity, prone to rupture.

• Scale Relativity is "even more" incomputable (Level 4) because, in addition to the infinity of the continuum, every segment of that space possesses an infinite internal structure.

In conclusion, Scale Relativity, by resolving the problem of GR's singularities, does not make the universe simpler or easier to simulate. On the contrary, it reveals that the reason singularities do not exist is because reality is, at a fundamental level, infinitely more complex and richer than the smooth geometry of Einstein could ever have imagined. It is a solution of extraordinary elegance and depth.