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I read an article about possibility of existence of multiverse and came up with a conflicting view with one of the sentences written in the article which goes as follows:

"If space-time goes on forever, then it must start repeating at some point, because there are a finite number of ways particles can be arranged in space and time."

What does this mean? What does it mean to say when it hints that it must start repeating at some point? Also, where are these multiverses enclosed within? If they are expanding at such a rate, is the container in which they are enclosed also expanding?

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"If space-time goes on forever, then it must start repeating at some point..." No. This is an error in logic. –  DarenW Apr 23 '13 at 8:35
    
What's error over here? It would be too much helpful if you could rectify the error. –  AaKASH Apr 23 '13 at 11:13
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Which article are you reading? –  Qmechanic Apr 23 '13 at 12:42
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If the universe is infinite there is obviously an infinite number of ways of arranging the matter within it, so there is no requirement for the universe to repeat at large scales. What the article is suggesting is more subtle than this.

Suppose we take a finite volume. This could be as small as you, or as large as the observable universe, but in both cases there is a finite number of ways of arranging the matter in a finite volume. The reason there is only a finite number of ways is that we assume separations smaller than the Planck length can't be distinguished. So our finite volume is made up of a large but finite number of Planck volumes, and there is only a finite way of arranging the matter between this finite number of Planck volumes. Depending on how you do the calculation the number of ways of arranging the matter in the observable universe is around $2^{10^{118}}$.

So if you assume the universe is completely random then the probability of a randomly selected volume the size of the observable universe looking just like ours is 1 in $2^{10^{118}}$. This is obviously very unlikely, but in an infinite universe there are an infinite number of observable universe sized volumes, so somewhere there will be an exact replica of our observable universe. In fact there will be an infinite number of such exact replicas.

None of this has anything to do mith multiple universes. The argument above is just that there must be repeating regions within our universe.

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Can't these arrangements be unique? –  AaKASH Apr 23 '13 at 10:39
    
@John, why can't we distinguish separations smaller than the Plank length? –  Mew Apr 23 '13 at 10:56
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@Chris: the argument is that to resolve a distance $d$ takes (at least) the energy of a photon with wavelength $d$. So resolving smaller distances takes more energy. However to resolve a Planck distance takes so much energy that the system would immediately form a black hole with the event horizon radius roughly equal to the Planck length. Since we can't see inside the event horizon the conclusion is that there is no physical process that can resolve a length shorter than a Planck length i.e. anything shorter than the Planck length can't affect the universe differently to a Planck length. –  John Rennie Apr 23 '13 at 11:02
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The Planck length is the length at which the structure of spacetime becomes dominated by quantum effects, and it would become impossible to determine the difference between two locations less than one Planck length apart and because of that it is the shortest measurable length and no improvements in measurement instruments could change that. –  AaKASH Apr 23 '13 at 11:10
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@Chris when people say "The Planck length is the length at which the structure of spacetime becomes dominated by quantum effects" they mean an order of magnitude statement. Since we lack a complete theory of quantum gravity and "becomes dominated by" is a vague statement in itself we can't really do better than that. But at the order of magnitude level the argument is pretty strong. –  Michael Brown Apr 23 '13 at 13:14
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