This post is also available in: Spanish

The very existence of space is a real puzzle. Almost everybody has a crude, working notion of space which is usually “that place where objects live”. However, once one starts to dig deeper into the concept, the apparent clarity of our everyday notion vanishes.

The first problem we face is what physicists call “scale invariance”. When we say something is scale invariant, we mean it looks the same, no matter at which scale you look at it: for example, a white, infinite screen with infinite resolution would be scale invariant.

Physicists look at space as a set of numbers: these numbers are needed in order to specify the position of an object in that space. Imagine, for example, a line: in order to specify a point on it, we only need one number, namely, the distance from the tip. Thus, we say a line is a 1-dimensional object. On a board, we need two numbers, one for the distance to each of the borders; we then say a board is a 2-dimensional object. Similarly, for our ordinary space we need 3 numbers, so we say we live in a 3-dimensional space. If we include time as one of the coordinates, we get the famous Einsteinian space-time of Special Relativity.

The numbers we use to specify positions in space are Real numbers. Real numbers include all rational numbers (those which can be expressed as a quotient between two integers) as well as irrational ones. Irrational numbers have a random-looking order in their decimal expression and cannot be expressed as a quotient between integers. However and even though they may seem much stranger than rational ones, they are far more numerous. Between any two real numbers there is an infinite number of irrational ones.

Real numbers have a truly astonishing property: one can make a one-to-one correspondence between any two intervals of them. For example, one can assign a different real number between 0 and 1 to each and every one of the real numbers between 0 and 100, or 0 and 100000 or 0 and infinity. That is, the quantity of numbers at each interval is exactly the same, notwithstanding the length of that interval. Let’s rephrase that: real numbers are scale invariant. One cannot distinguish between any interval, no matter how big or small.

That, of course, leaves us with a big problem. Because the Universe we see around us is definitely not scale invariant: one does not see galaxy-sized diapers or electron-sized computers. Our Universe is very, very variant. But the numbers we use to describe it are not.

This is not a mathematical difficulty, but a conceptual one. People have been using Real numbers as a way of describing space for hundreds of years, just by arbitrarily setting the measurement standards or by setting cutoffs, as in QFT. However, the underlying conceptual problem remains: why is the Universe not scale invariant, when the space it lives in is?

A possible answer is that, in fact, space is not scale invariant at all; that describing it in terms of Real numbers is practical, but not fundamental. In this view, space, when probed at small enough distances, should look rough, made of tiny tiles of a fundamental nature. Space would be what is usually called an emergent property, something which arises from the interaction of these fundamental blocks and which, at our scale, looks pretty much like a smooth continuum. This is the view advocated by Quantum Loop Theorists, for example.

This view has its problems, though. And it does because it rises a conflict with Special Relativity, in a way that makes it very difficult to solve. Actually, Quantum Loop Theorists expect to see violations of Special Relativity at high enough energies.

In Special Relativity, space and time are merged in a 4-dimensional monster called space-time. They are not just glued together: in fact, they are interchangeable. For some observer, some piece of space-time may be seen as time whereas, for some other, that same piece may be seen as space. This is an unavoidable consequence of the constancy of the speed of light. This in turn implies that notions such as the length of an object are dependent on the observer. It has been shown that an object looks shorter for an observer who’s moving closer to the speed of light.

That is why the idea of a space built from basic blocks conflicts with Relativity. If we take these basic blocks to have length, say, *l*, they will automatically have a different length for an observer who’s moving at a different speed. Therefore, the fundamental length *l* becomes not so fundamental, but dependent on the observer. On top of that, in order to define it we would need a privileged observer, who would be the one to establish the fundamental length to be *l* and not some other quantity. That goes against everything Special Relativity has taught us, that is: every observer should see the same laws of Physics. Only time will tell if they’re right.

Is there another solution, then? Yes, there are. On one hand, there’s the solution from string theory, which addresses the minimum length problem with a duality between position and momentum space, something which is far too complex to address it in this article.

A different suggestion which has been raised in the last twenty years is fractal space. A fractal is an approximately self-similar object which exists at every scale level. They can be built on a mother space but, conversely, a space can be built from them. One starts with a simple recurrence relationship which generates a fractal; the fractal, in turn, has the properties one would expect from a Real space.

Furthermore, some fractals have a very interesting property: even though they’re infinitely finely grained, they exhibit slightly different behaviors at every possible scale. Thus, these particular fractals are not scale invariant. This would then solve our two problems: the Universe would not be scale invariant and, since fractals’ building blocks are infinitely small, there is no conflict with Special Relativity.

So far, there’s no evidence to favor any of the three possible solutions. Probably, no piece of evidence will be found in the next decade, even though the problem of space permeates every attempt to build a unified, coherent theory. In the meanwhile, we have no choice but to keep inhabiting this Space which, deep down, we still don’t understand.

See more answers to this question.

Benoit Mandelbrot is the father of the fractal concept. In the talk below, he explains what they are in an accessible yet fascinating way.

## Leave a Reply