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I have been reading up on some approaches to quantum gravity apart from string theory. The popular conception of loop quantum gravity is that it says that space is actually physically discrete at some sufficiently small scale, where the lattice spacings are presumably given by the Planck length.

It seems that at least in the case of loop quantum gravity, that this is a misconception and the discreteness is just referring to certain eigenvalues of area in a way which is somewhat familiar from elementary quantum mechanics (see here for example).

Is there some approach where space is actually a graph or where spacetimes are actually chopped up into lattices such that for example the Universe is a cosmological spacetime which can be chopped up into a lattice?

(I don't mean just doing quantum gravity on a flat lattice by analogy with lattice Yang-Mills theory).

Edit: After some more searching, it looks like this kind of theory tends to go by the name of 'induced gravity' or 'emergent gravity'. The details of how it all works are about as confused as they are in LQG unfortunately.

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  • $\begingroup$ The phrase "space atoms" brings up 144,000 results, with the top one being from some Oxford site, but, as I'm not a fan of the concept (which might, nevertheless, simplify all sorts of calculations) for some aesthetic or psychological reason that I haven't been able to isolate, I won't be posting an answer. $\endgroup$
    – Edouard
    Commented Jul 8, 2022 at 19:34
  • $\begingroup$ Rovelli has a recent pop-sci titled "Helgoland" that's at my level (identical, reputation-wise, to your own), and discusses lattice formulation early on. $\endgroup$
    – Edouard
    Commented Jul 8, 2022 at 19:44
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    $\begingroup$ One problem with a discete spacetime lattice is making it compatible with Lorentz invariance. $\endgroup$
    – PM 2Ring
    Commented Jul 9, 2022 at 6:10
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    $\begingroup$ @PM2Ring This is a good point, it looks to be that violations of Lorentz invariance even at smallest scales have been essentially ruled out by experiment, unless I am missing something. $\endgroup$
    – Tom
    Commented Jul 9, 2022 at 12:45
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    $\begingroup$ @mikestone Yes I know....that's not what I was asking about in my question. $\endgroup$
    – Tom
    Commented Jul 9, 2022 at 13:35

3 Answers 3

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Empty space or vacuum cannot be a simple lattice with Planck length spacing, because such a lattice contradicts special relativity. The idea of a lattice is that it has a smallest length. But due to special relativity, lengths Lorentz-contract, and can get as small as desired for rapid observers, much smaller than the Planck length, thus invalidating the idea of a smallest length.

So the naive idea that empty space is a lattice must be rejected. Therefore, all quantum gravity approaches have some kind of fuzzy space at the Planck scale: something happens there that keeps the smallest length the same for all observers, also rapid ones. Finding a model that maintains minimum length for all observers is one reason that quantum gravity is hard to develop.

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  • $\begingroup$ As a fan (currently) of "inifinite divisibility" (described in a Wikipedia item by that name), I'd like it if you're right, but, as your conclusion renders a 2020 remark by Rovelli either wrong or ambiguous, could you cite a reference? $\endgroup$
    – Edouard
    Commented Jul 11, 2022 at 4:47
  • $\begingroup$ No, I do not have a reference, but I cannot imagine Rovelli contradicting this. I once read a similar argument by Hossenfelder (on her blog?). In any case, there is no space-crystal any more in research, since a long time. $\endgroup$
    – KlausK
    Commented Jul 11, 2022 at 5:11
  • $\begingroup$ I agree with KlausK, the last time I could find a really unambiguous suggestion that the Universe can be modelled as a crystal is the 1980s. $\endgroup$
    – Tom
    Commented Jul 11, 2022 at 5:22
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    $\begingroup$ H. Kleinert, ''Gravity as Theory of Defects in a Crystal with Only Second-Gradient Elasticity'' if anyone is curious. $\endgroup$
    – Tom
    Commented Jul 11, 2022 at 6:31
  • $\begingroup$ The Kleinert title seems to date from 2005, but, anyway, Rovelli's exact remark, on p.108 of his 2020 book titled "Helgoland", was "There is no infinite in going toward the small: things cannot get infinitely smaller". I'm only to its p.134, but, so far, he hasn't mentioned any crystal. My earlier comment on his book, re the OP's question, was about its description of the writing of lattices in their relation to Heisenberg's Uncertainty Principle (non-commutativity, etc.). $\endgroup$
    – Edouard
    Commented Jul 11, 2022 at 16:45
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Tom,

I just found your question about descret modes of space. H have a look here, it describes "mass and space" as a network, based on soliton wave solutions

https://doi.org/10.1515/zna-2016-0270

https://doi.org/10.1515/zna-2019-0343

Ingo

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    Commented Oct 6, 2023 at 13:40
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    – Community Bot
    Commented Oct 6, 2023 at 16:11
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Tom, I have come up with a highly speculative theory of gravity based on loop quantum gravity that you can download here: An Examination of Gluon Field Excitation as the Initiating Cause of Time Dilation, Gravity and Dark Matter

The basis of the theory is that it is the size of the particulate space of LQG that defines the speed of time, and that adding energy will cause a dilation of the particles and thus a dilation of time. This provides a clear definition of "spacetime". When looking at the underlying cause of gravity surrounding a planet, gluon field excitation (analogous to the magnetic field surrounding a bar magnet) leads to a dilation of the particulate space of LQG and thus the warping of spacetime and gravity.

As I said above, the theory is highly speculative and based on a number of unproven concepts, but perhaps it will get you thinking.

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