# Is spacetime discrete or continuous?

Is the spacetime continuous or discrete? Or better, is the 4-dimensional spacetime of general-relativity discrete or continuous? What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous?

Are there experimental evidences of continuity/discreteness?

When particles move inside space do they occupy spacetime by little chunks? What would imply if spacetime is discrete on continuous theories?

I've found little information on the web and books.

Probably my question is ill-posed and I apologize for this.

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Possible duplicate: physics.stackexchange.com/q/9720/2451 –  Qmechanic Aug 1 '12 at 15:42
This is bound to generate comments and answers where people say that discrete space-time can never be Lorentz-invariant. I have no expertise and no opinion on the matter, but I'd like to do my part to throw gasoline on the fire and point out that some people claim that a 'Poisson sprinkling' of space-time points is Lorentz invariant. en.wikipedia.org/wiki/Causal_sets –  DJBunk Aug 1 '12 at 16:00
@Regarding your question on experimental evidence, I just got a comment linking to the following article about lorentz invariance being confirmed by the Fermilab:motls.blogspot.gr/2009/08/… –  Dimensio1n0 Jun 6 '13 at 8:52
Thomas Campbell, former a physicist at NASA claims space time to be granular. So that time and space both are discrete. Atleast that is what I understood from his theories. I'm not sure which experiments or experience of his backs it up. But you can easily find him using Google. –  Mike de Klerk May 15 '14 at 19:23

is the 4-dimensional spacetime of general-relativity discrete or continuous?

In the usual definition of general relativity, spacetime is continuous. However, general relativity is a classical theory and does not take quantum effects into account. Such effects are expected to show up at very short distances, where your question is relevant.

Are there experimental evidences of continuity/discreteness?

All the experimental evidence points to continuous space, down to the shortest distances at which we have been able to measure. We don't know what happens at shorter distances. We also do not have any direct experimental evidence that gravity is a quantum theory, with the same caveat.

On the other hand, we are quite confident that a complete theory of nature must include quantum gravity and not just classical gravity. And, we have an educated guess of the distance scale at which quantum effects should become measurable: this is the Planck length, roughly $10^{-33}$ cm. This is much much shorter than the shortest distance at which we can carry out experiments, so at least we are not surprised that we did not see any such effects so far.

Before proceeding, one more caveat. There is an interesting and quite recent astrophysical experiment that showed that Lorentz symmetry holds even below the Planck length. If Lorentz symmetry is broken, it generally means that photons with different energies will travel at different velocities. At the experiment, they managed to detect a pair of photons that were created at almost the same time but had very different energies. They reached the detector almost simultaneously, which means their velocities were similar. Because the photons travelled an enormous distance before reaching us, they must have had almost the same velocity.

So we know that at least Lorentz symmetry holds at very short distances, and it seems difficult to reconcile this experimental fact with a discrete spacetime. So at least naively it seems that this is evidence against discreteness.

Is the spacetime continuous or discrete?

At long distances spacetime can certainly be thought of as continuous. At short distances, the short answer is: we don't know.

String theory is the only consistent theory of quantum gravity we know of, where we can actually compute things with some confidence. (You will probably hear some opinions that contradict this statement, mentioning loop quantum gravity, causal sets, etc., which are not related to string theory, but what I said is the common view in the community of high-energy theorists.) String theory is giving us some strong hints that perhaps spacetime at short distances is not continuous or discrete, but something else that we don't understand yet.

So the situation is that even theoretically, without talking about actual experiments that check the theory, we don't know what spacetime is like at short distances. Perhaps this is why you don't see this question mentioned a lot. My personal guess is that spacetime at short distances is neither continuous nor discrete, but has a different nature that may require new mathematical tools to describe.

Or better, What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous?

Adding extra dimensions does not change any of the above.

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+1 for the intellectual honesty. It seems that currently humanity has no mathematical tools to deal with this kind of answers. I've found an article on scientic american. The physicist Hogan claims to build an experiment that finally can prove the discreteness or continuity of space. What about this? scientificamerican.com/article.cfm?id=is-space-digital –  linello Aug 1 '12 at 20:32
I cannot access the full article because I do not have a scientific american subscription. I will say that statements like "If it works, it could rewrite the rules for 21st-century physics" are generally not indicative of interesting work. –  Guy Gur-Ari Aug 1 '12 at 23:06
What about the latest Nemiroff article on "Physical Review Letters"? Probably we should discard the concept of quantum foam, so it seems that spacetime still remains continuous under Planck lengths. –  linello Oct 12 '12 at 19:39

There is a beautiful theory of quantum gravity called "Canonical Quantum Graivty" which aims to quantized general relativity using typical canonical methods (canonical quantization/path integral formulation). This theory predicts a granular structure to spacetime while maintaining local lorentz invariance. The theory gives a spectrum of eigenvalues for quantized area and volume based on Penrose's spin-network graphs, except the theory consideres equivilence classes of spin-networks under diffeomorphisms. The Path-integral formulation of the theory consists in considering a sum-over-geometries which is entirely background independent, carried out in sum over 2-complexes, which are themselves graphs. Here is a small set of lectures that might interest you: http://arxiv.org/abs/1102.3660

Response to comment by OP: There are no experimental tests of quantum gravity that we know as of now, whether because we don't know how to interpret what we already have in front of us, or because we simply don't have the technical power/creativity yet, although there are a number of new papers that suggest experiments that may be done at the LHC for Canonical Quantum Gravity, which have to do with the evaporation of micro-black holes and their radiation spectra which differs the classical spectra predicted by QFT in curved spacetime. Canonical Quantum Gravity is also the only mainstream theory of QG on the table that gives falsifiable, numerical predictions that are novel; at least I have yet to see anything else on the forums and arxiv that does, so that doesn't mean much.

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People generally use the term "Loop Quantum Gravity" to describe this. –  Jerry Schirmer Aug 1 '12 at 17:45
What about experiments to prove/disprove it? –  linello Aug 1 '12 at 20:34

For the four dimensions space-time that we're used to, atoms of space-time is incompatible with special relativity. If we were to try claim a size of these grains of space time we would also have to say in what reference frame in which they have that size. So they introduce a preferred reference frame. From my understanding, supersymmetry introduces completely discrete dimensions of space-time, but these are radically different from the dimensions we are accustomed to. Here's a much more better discussion of the topic by one of the leading theorists in the world.

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I had a similar thought along these lines when I read the question. I wonder how they affect quantum fluctuations, which are only relevant at very small sizes. What would happen if you accelerated an object to a speed so close to c that it shrinks to quantum size to an outside observer? –  Daniel Allen Langdon Dec 1 '14 at 21:41

the idea of spacetime having a fundamental length does not necessarily translate in a discretized structure.

Let's think intuitively this in term of path integrals (lets assume one-dimensional paths and forget about stringy structure for now, is not relevant to the discussion). When we do path integrals, we usually take all kinematical paths of the system in configuration space (what is usually called off-shell states), assign an amplitude given by the dynamical action, and sum them all to obtain physical observable amplitudes (the on-shell states)

Now, the planck scale sets a natural cutoff for on-shell states, because paths that have energies above that scale must result in black holes in the path (or the quantum gravity equivalent of black holes, whatever those turn out to be). So in your amplitudes for on-shell states, you get systems that do not have observable structure beyond the planck scale, and in fact, increasing the energy makes it worse because it makes the resulting black holes bigger. But they live nonetheless in a Lorentz-invariant background

Now, all this is speculative, and likely not entirely correct picture, but my point is that a finite minimum physical scale does not contradict a continuous Lorentz-invariant background

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This article contains a mathematical argument that a discretized (or minimal unit of) time would spoil the uncertainty principle in context of Feynman path integrals. It seemed reasonable to me ... –  Dilaton Aug 1 '12 at 21:54