# Is the hypothesis “at some level, spacetime becomes discrete” falsifiable?

Suppose I conjectured that, at some length scale, spacetime was discretized into "cells", Minecraft-style. For simplicity, I guess let's say they're cubes with side length $n$.

Presumably we can put an upper bound on $n$ from observation. For instance, myself and the table occupy the same $10 \text{m}^2$ cube of space, and we are two different objects, so $n < 10 \text{m}^2$. (Is this conclusion correct? Is my reasoning correct?)

Is there any experiment that would disprove this hypothesis for all $n$? My intuition is no, and I was using this as an example of a non-falsifiable hypothesis earlier today, but I was seized by doubt, as I have a math & computer science background without much physics.

Is my conjecture falsifiable?

• One place to look for trouble in constructing such a theory is with the conservation of angular momentum. In the Noetherian view this principle arises from the isotropy of space. – dmckee Dec 31 '15 at 5:51
• @dmckee: So, this could in principle be falsifiable? Do you and Timaeus disagree? – Eli Rose Dec 31 '15 at 6:03
• People are trying to do that right now by looking at the dispersion of high energy gammas of cosmic origin. Naive models (where the effect is supposed to be first order) can rule out the existence of a Planck scale by many orders of magnitude, already. The problem is that these models are really naive and they are probably not the correct way of understanding spacetime at the quantum level. – CuriousOne Dec 31 '15 at 6:04
• @CuriousOne: Cool! So suppose I'm being a jerk and, no matter what experiments you show me, I keep saying "nope, $n$ is smaller than that". Is there any way to finally prove me wrong? – Eli Rose Dec 31 '15 at 6:08
• Can we rule out an arbitrarily small effect? No. We can only set limits that say "We are certain with this probability that it is not larger than this...". In mathematical terms this means that all physical theories are really equivalence classes of theories that have elements that can differ by as much in their predictions as the most precise null experiments. Physicists are fine with that. The challenge is to make the limits as small as possible. It's like a 100m sprint, if you like. Will anybody ever run it in 0 seconds? No. They are still trying to be world champion in 100m sprint! – CuriousOne Dec 31 '15 at 6:11

If you want a falsifiable theory you must make a prediction about specific mutually exclusive ways the universe could be versus not be and then argue that one the groups must happen or cannot happen. Then when you investigate and find which one you get, you have falsified or not falsified the theory.

So ask yourself what you predictions are. Your predictions are a bit vague, but they sound like the predictions of a continuous theory. Why do I say that. It's because of the "at some level" part. It sounds like it means that at scales much larger than that hypothetical level every prediction agrees with the continuous theory.

So you could make the same predictions as a continuous theory and then whenever data is collected, no matter what data we see, the data was collected at some scale and you could just say that if only the scale was smaller things would have turned out differently.

But however small the scale is, you can pull the same trick. You never even need to bother making discrete predictions because whatever data fits the continuous theory also allows the discrete one to exist at the much smaller resolution without being exposed.

This freedom to wiggle out of any data is the hallmark of an unfalsifiable theory.

But if you claimed there was a discrete theory at a fixed level where experiments with a particular nonzero scale were small enough to require different predictions for the discrete theory. Now you have made a prediction that can be tested and thus your theory is falsifiable.

So you could have a whole family of theories, each predicting deviations at a different scale. And each one would be falsifiable. But the meta claim that at least one of them is correct, that meta claim is not falsifiable.

• Right, that agrees with my intuition. – Eli Rose Dec 31 '15 at 5:40

Apologies for the late response. Also, I mean to post this as a comment to Timaeus's answer since this doesn't really answer your question, and I do not have 50 rep to comment.

A user in Phys.SE wrote this answer to similar question.

Seems like a rather simple thought experiment, with a conclusion that " no matter how much you reduce the scale, there's always a smaller distance possible". So, the universe has to prohibit movement along certain directions to preserve the smallest unit of displacement. Is this what actually the proponents of discrete spacetime believe?

Now, how would one claiming discretized spacetime get around this?

• Here's another problem that arises: if time and space are both quantized, then we might say that the speed of light is such that each t the wave propogates 1 unit of space. But how to explain things moving less than the speed of light? Something travelling at 1/10th of the speed of light would only propogate 1 unit every 10t. How does it "know" when 10t are up and it is time to move? – Will Graham Sep 25 '16 at 1:53
• Furthermore, if an object is travelling at 0.8 times speed of light, it would need 1.25t to travel 1 unit of space. Since 0.25 unit of time is disallowed, it would be travelling 1 unit of space in 2t – Will Graham Sep 25 '16 at 1:59

I will state that specific models, regardless of scale as discussed in the other answers, can be falsified by not being able to incorporate the standard model of particle physics in its mathematics. The standard model represents innumerable measurements/data and any theory has to be able to demonstrate that the model emerges naturally from its formalism.

In addition, any model must obey Lorenz transformations, which also is validated by innumerable measurements. These have shown that the interactions of the particles in the standard model are local, and this has to be seen to emerge from the model naturally for it to have a connection to real data and not be a mathematical game.

There exist other gauges than scale , and these can invalidated the model.