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There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms.

There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete. From the article: > Importing the atomic philosophy of Democritus into modern physics might be essential for reconciling general relativity (which assumes a continuous reality) with quantum mechanics (which very much does not). As far as infinite vs finite space: because there is a fixed rate of information propagation, we only need to show that time is finite in extent, and it follows that space is also finite. We know that the visible universe is finite (by looking in the sky) and we can explain why: the big bang happened at a finite time in the past. We don't know if the universe will have a finite future, but it might be the case. One possible cause of a finite future could be accelerating expansion of the universe due to dark energy. [2] Of course there are other interpretations of "real space", but this one (causally connected) seems parsimonious to me. [1] https://news.ycombinator.com/item?id=13466254 https://news.ycombinator.com/item?id=13466254 [2] https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe

Jweb_Guru 10y ago
To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations. So again, your argument doesn't seem that compelling to me.
- cygx 10y ago
  And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model Just to clarify, FLRW is a class of models, which includes ones with finite as well as infinite space. We've been able to determine that our universe is close to spatially flat. However, that does not mean it's infinite: If spatial curvature is positive but small compared to the cosmic horizon (which is due to spatial expansion), we won't be able to tell. That said, even given finite space, spacetime can of course still be infinite if the universe does not recollapse.
- warrenpj 10y ago
  > To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. Your criticism is right and that is a bad argument. The real reasons they are hard to reconcile are technical (see for example [1]), and not mentioned in the source I referred to. The article only says that a discrete theory of spacetime "might be essential" for the reconciliation; the arguments for this are philosophical - it would be beautiful and historically completing. Read the article if you're interested in the arguments, which aren't included in my comments. > And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations. I agree with you that the universe could be infinite, but that doesn't contradict my previous comment. I explained that I was talking about local (causally connected to us) space-time, only. I admit this is a bit of a trick, but the parent comment used the term "real space" which I interpreted in such a way to favour an argument for finite space. > So again, your argument doesn't seem that compelling to me. I've given reasons but I think it might be helpful to restate what I am actually claiming. I admit I don't know that space and time are discrete. The source I gave is a poetic and philosophical argument that it might be. I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite. I think it's possible that the future is finite too. I don't know what FLRW is. My best guess is https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–W... [1] http://www.theory.caltech.edu/people/jhs/strings/str115.html http://www.theory.caltech.edu/people/jhs/strings/str115.html
  - cygx 10y ago
    > I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite. None of that is established, actually: Metric expansion of space comes with a cosmic event horizon where redshift goes to infinity. While the energy contained within that region is finite, there's no reason to assume that the universe outside the horizon looks any different than inside the horizon, and it might very well be spatially infinite. As to time being finite in the past, it is possible to generalize Einstein's equations so that they can be extended past the big bang singularity, and what you get is a mirror universe on the other side. The issue here is that as we approach Planck scale, quantum gravity becomes relevant, and we have no predictive theory of that.
cygx 10y ago
we only need to show that time is finite in extent, and it follows that space is also finite [...] That does not follow: As our universe is spatially flat (or close to it), it is consistent with the infinite-space Friedmann model. The cosmic horizon that bounds the observable universe would still exist due to the metric expansion of space.

Jweb_Guru 10y ago

To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations. So again, your argument doesn't seem that compelling to me.

cygx 10y ago
And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model Just to clarify, FLRW is a class of models, which includes ones with finite as well as infinite space. We've been able to determine that our universe is close to spatially flat. However, that does not mean it's infinite: If spatial curvature is positive but small compared to the cosmic horizon (which is due to spatial expansion), we won't be able to tell. That said, even given finite space, spacetime can of course still be infinite if the universe does not recollapse.
warrenpj 10y ago
> To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. Your criticism is right and that is a bad argument. The real reasons they are hard to reconcile are technical (see for example [1]), and not mentioned in the source I referred to. The article only says that a discrete theory of spacetime "might be essential" for the reconciliation; the arguments for this are philosophical - it would be beautiful and historically completing. Read the article if you're interested in the arguments, which aren't included in my comments. > And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations. I agree with you that the universe could be infinite, but that doesn't contradict my previous comment. I explained that I was talking about local (causally connected to us) space-time, only. I admit this is a bit of a trick, but the parent comment used the term "real space" which I interpreted in such a way to favour an argument for finite space. > So again, your argument doesn't seem that compelling to me. I've given reasons but I think it might be helpful to restate what I am actually claiming. I admit I don't know that space and time are discrete. The source I gave is a poetic and philosophical argument that it might be. I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite. I think it's possible that the future is finite too. I don't know what FLRW is. My best guess is https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–W... [1] http://www.theory.caltech.edu/people/jhs/strings/str115.html http://www.theory.caltech.edu/people/jhs/strings/str115.html
- cygx 10y ago
  > I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite. None of that is established, actually: Metric expansion of space comes with a cosmic event horizon where redshift goes to infinity. While the energy contained within that region is finite, there's no reason to assume that the universe outside the horizon looks any different than inside the horizon, and it might very well be spatially infinite. As to time being finite in the past, it is possible to generalize Einstein's equations so that they can be extended past the big bang singularity, and what you get is a mirror universe on the other side. The issue here is that as we approach Planck scale, quantum gravity becomes relevant, and we have no predictive theory of that.

cygx 10y ago

we only need to show that time is finite in extent, and it follows that space is also finite [...] That does not follow: As our universe is spatially flat (or close to it), it is consistent with the infinite-space Friedmann model. The cosmic horizon that bounds the observable universe would still exist due to the metric expansion of space.