If the theory of everything indicates a singularity-free and finite universe, will Poincare recurrence be relevant to the universe? If so, is there any interesting physical consequence, e.g. in superstring or quantum gravity?

  • $\begingroup$ A practical application of the theorem is described at physics.stackexchange.com/q/330139 . $\endgroup$
    – Edouard
    Apr 1, 2020 at 21:21
  • $\begingroup$ In my comments on Motl's answer to this question, I'd been overlooking the basic problem involved, which is that the sort of vacuum fluctuations that might seem to make Boltzmann Brains (or, perhaps, local universes) possible would all require passage thru that single direction of time whose ongoing passage we sense as present reality, whereas Poincare recurrence would be indistinguishable from a return to previous points in time which we currently sense as past reality: Each of those possibilities necessarily excludes the other. The recurrence possibility's stronger in spherical models. $\endgroup$
    – Edouard
    Dec 30, 2021 at 21:09
  • $\begingroup$ Spherical models (local universes such as our own "Universe", which includes both its horizon and the region of it that's visible to us) are currently appearing to be a more plausible possibility than flat ones, as seen at nature.com/articles/s41550-019-0906 . $\endgroup$
    – Edouard
    Dec 30, 2021 at 21:11
  • $\begingroup$ As this site caters to all education levels, I should point out that "flat" means lacking a net (overall) spatial curvature. $\endgroup$
    – Edouard
    Dec 30, 2021 at 21:22
  • $\begingroup$ Experimental proof of a quantum version of Poincare recurrence is described at arxiv.org/abs/1506.02938 . $\endgroup$
    – Edouard
    Jun 12, 2023 at 15:29

3 Answers 3


Yes, our Universe is approaching the empty de Sitter space – we are already pretty close to it, actually, because the cosmological constant dominates the vacuum energy (68% of it). It has a cosmic horizon (the boundary behind which we can't see) and the degrees of freedom are formally living on that surface.

Via the holographic principle (in a somewhat less tested context), one may claim that this means that de Sitter space has a finite entropy so a finite-dimensional Hilbert space is enough to describe everything that happens in it (including the matter in not-yet-empty mostly de Sitter space, like the present Universe). If that's so, the Universe we inhabit behaves much like any system with finitely many degrees of freedom, and it has Poincaré recurrences.

The Poincaré recurrence time is extremely long, something like $\exp(10^{120})$ billion years – it is because the entropy of the de Sitter horizon is $10^{120} k_B$ (the cosmological constant is $10^{-120}$ in Planck units or so). After this superlong time, approximately, events start to repeat themselves. At least in some sense, it is fair to say that the time is literally periodic.

This very long timescale is the ultimate "maximum duration" that may be discussed in physics. For example, if a calculation in field theory or string theory implies that the lifetime of a vacuum is longer than this Poincaré recurrence time, the decay is considered unphysical because "it cannot happen in time", anyway.

The timescale is vastly longer than anything we have a chance to experimentally test. Each 10 billion years or so, the linear distances between galaxies double and the density of normal matter decreases by an order of magnitude. In hundreds of billions of years, a vast majority of the currently active stars will be inactive and even "new generations" of the stars will already be gone or dying. In trillions or certainly quadrillion years, there will be nothing left to energize star-powered life as we know it, and similar "local, more modest" sources of useful energy will be diminishing in similar ways.

It is hard to imagine that there will be any intelligent beings in a quadrillion years. This is still vastly smaller than the Poincaré recurrence time. And this result of the comparison is no coincidence. Of course that things must have a chance to "destroy any pattern" before the chaos has a chance to reassemble itself into the patterns again.

  • $\begingroup$ The cosmic horizon depends on the observer, so does the finite-dimensional Hilbert space depend on the observer ? If yes, how are the Hilbert spaces related ? $\endgroup$
    – jjcale
    Jan 17, 2014 at 21:29
  • $\begingroup$ As the second question correctly assumes, it's really the same Hilbert space for all observers. It's just being interpreted - the observables are being chosen - in a way that depends on the observer. The key fact is about observables, not the Hilbert space itself. And the fact here is that the field operators that are behind the horizon of each other don't quite commute with one another. The fields within a single horizon are building blocks for a maximum commuting set of observables, kind of. Locality breaks further. $\endgroup$ Jan 18, 2014 at 6:47
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    $\begingroup$ Are you THAT Lubos Motl who discovered Matrix String theory ? $\endgroup$ Sep 24, 2016 at 22:19
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    $\begingroup$ Thanks for formulating the question in this flattering way. The correct original name is Screwing String Theory, however. $\endgroup$ Sep 27, 2016 at 10:27
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    $\begingroup$ Dear Edouard, the Poincare recurrence is a process that happens in principle, that may happen etc., The existence of this notion doesn't imply that some extreme examples of the recurrence will be likely or common. $\endgroup$ Nov 10, 2021 at 6:14

The Poincaré recurrence theorem will hold for the universe only if the following assumptions are true:

  1. All the particles in the universe are bound to a finite volume.
  2. The universe has a finite number of possible states.

If any of these assumptions is false, the Poincaré recurrence theorem will break down.

  • $\begingroup$ The only physicist I've heard to dispute either of these assumptions is Lee Smolin, and, of course, he's basing that dispute on his imagination (-not that there's anything wrong with that, but figments of the imagination generally seem more artistic than physical). $\endgroup$
    – Edouard
    Apr 1, 2020 at 20:53
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    $\begingroup$ @Edouard Could you reference the publication(s) where he disputes these assumptions? I'd like to review his arguments. $\endgroup$
    – Zenadix
    Apr 2, 2020 at 22:25
  • $\begingroup$ Its identifier had been changed, but the preprint of Smolin's paper's currently visible at arxiv.org/abs/1506.02938 . $\endgroup$
    – Edouard
    Jun 12, 2023 at 15:22
  • $\begingroup$ Gujrati has, at arxiv.org/pdf/0803.0983.pdfas, formulated a logical proof of the recurrence theorem's complete compatibility with the 2nd Law of Thermodynamics, as it arrives at the conclusion that "a Poincare recurrence can occur at any time". As he also points out, Boltzmann (who, in at least one of whatever approximations he had to himself, originated the 2nd Law) never disputed the recurrence theorem, and felt it to be compatible with that law. $\endgroup$
    – Edouard
    Aug 7, 2023 at 16:46

Yes, the recurrence time for this universe is about $10^{{10}^{{10}^{{10}^{2.08}}}}$. The unit (seconds, or years) isn't important.

  • $\begingroup$ Welcome on Physics SE and thank you for your answer =) Do you think you could sketch in your answer how exactly one arrives at this number? $\endgroup$
    – Sanya
    Dec 25, 2016 at 0:40
  • $\begingroup$ @Sanya : I guess it is mentioned youtube.com/watch?v=1GCf29FPM4k and here : arxiv.org/abs/hep-th/9411193 . But I have not verified it nor read the paper. $\endgroup$
    – dexterdev
    Nov 16, 2017 at 12:00

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