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I still don't understand how the expansion of the universe works. If the universe is made up of an infinite number of points that make up space-time, then how can space expand or stretch. Common sense dictates that in order for space to be stretched, the length between points must increase, the minimum length. But when we have an infinite number of points, changing the length between them simply does not make any sense. It wouldn't affect anything.

So maybe space-time has a finite volume with some fundamental length?

I understand that the minimum length can be so small (below the Planck length) that we cannot notice the graininess of the spacetime and think that it is a continuum-like. But this breeds contradictions.

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  • $\begingroup$ The Planck length is described (in Wikipedia, etc.) as "the smallest meaningful length", but that doesn't mean that it is the smallest length: It only means that it's the smallest length that we can, without the use of magnification energies that would collapse whatever object (or portion thereof) might occupy it, observe. $\endgroup$
    – Edouard
    Apr 25, 2022 at 15:40

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But when we have an infinite number of points, changing the length between them simply does not make any sense.

It absolutely does make sense. Infinite sets need care. Many of the ideas that work for finite sets are changed for infinite sets.

Consider the set of real numbers from 0 to 1 and the set of real numbers from 1 to 3. Both sets have an infinite number of reals, and in fact they have the same number of points. Each one in the first can be uniquely mapped to one in the second by multiplying it by 2 and adding 1.

Furthermore, the difference between any two elements in the second set is twice the difference between the mapped points in the first set.

So, yes, it does make sense to have an infinite number of points and to change the length between them.

in order for space to be stretched, the length between points must increase, the minimum length

For space to stretch the length between points indeed must increase. However, that in no way implies that there is a minimum length.

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  • $\begingroup$ This answer seems to be the one most consistent with Nobel laureate John Mather's observation that "the universe is expanding into itself". $\endgroup$
    – Edouard
    Apr 25, 2022 at 15:41
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An infinite number of points is not a definite number. Don't think of it like a normal number.

No matter how small a distance you make between two any points there is always an infinite number of points between them. There's no cutoff for that. So when you imagine two points spreading apart there are an infinite number of points between those and always wil be. There will never be a gap because there are an infinite number of points between any finite gap.

Put another way, there is no way to select two points which are not themselves separated by an infinite numebr of points.

Common sense dictates that in order for space to be stretched, the length between points must increase, the minimum length.

Common sense is fine for everyday things in normal human experience. When you start think about infinite anything you need to move beyond common sense because human experience (our rules of thumb we call common sense) don't ever have to deal with infinite. We do have a tool for managing infinities and it's called math.

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  • $\begingroup$ So why is space expanding then? $\endgroup$
    – Peter
    Apr 24, 2022 at 19:45
  • $\begingroup$ the space in between any two points is increasing. kind of like $f:x \to 2x$ $\endgroup$ Apr 24, 2022 at 19:51
  • $\begingroup$ But you said before that there is no gap between any points. $\endgroup$
    – Peter
    Apr 24, 2022 at 19:54
  • $\begingroup$ @Peter The expansion of the universe is due to the geometry of spacetime and requires some knowledge of general relativity to grasp. I'd suggest you read about the FLRW metric as a starting point. $\endgroup$ Apr 24, 2022 at 23:15

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