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When physicists talk about the universe being infinite, or wondering whether it is or not, what do these two options actually mean? I am not interested whether the universe is infinite or not, I am interested in what are the two options actually looking like.

Does an "infinite universe" mean infinite space, but when you go far enough there isn't any matter anymore, just more of infinite empty space in that direction? Or does an "infinite universe" mean infinite matter, where it doesn't matter how far you go, you will find infinite stars all along the way?

In short, is the "infiniteness" meant to apply to space or to matter (which then would also include space of course)?

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    $\begingroup$ Continuing my personal understanding in a comment, since it's not really part of the question: The idea of a finite universe always seemed silly to me, since I can't image how the "border" of the universe would look like; surely you can always go further. So I'm inclined to believe the concept of an "infinite universe" means infinite matter, but I was never sure. $\endgroup$
    – Nohus
    Mar 8, 2021 at 21:55
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    $\begingroup$ we don't know.${}$ $\endgroup$ Mar 8, 2021 at 21:55
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    $\begingroup$ @AccidentalFourierTransform I am asking for the meaning of a common concept in physics, not for the state of reality. Surely when physicists talk about the concept, they know what they mean? The meaning is the only thing I am looking to learn. $\endgroup$
    – Nohus
    Mar 8, 2021 at 21:56
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    $\begingroup$ @Nohus It is possible to have a finite universe that is borderless. In a simple case, if you “keep going forward” you could eventually just come back to where you started. This kind of space would be a “3-sphere”. $\endgroup$
    – G. Smith
    Mar 8, 2021 at 22:23
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    $\begingroup$ @G.Smith There are other topologies than a 3-sphere with this property, e.g. a 3-torus. $\endgroup$
    – wizzwizz4
    Mar 9, 2021 at 21:33

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Generally when physicists talk about the universe being finite, they are talking about the existence of an upper bound $R$ on the distance between any two points in space. Such an upper bound could arise in several ways - perhaps the universe has an edge - a boundary which cannot be crossed - or perhaps the universe has the topology of a 3-sphere, and so if one travels sufficiently far in any direction they would eventually return to their starting point. A spatially infinite universe is one which does not have this feature - given any real number $M$, there exist two points in the universe which are separated by a distance which is greater than $M$.

A finite universe of course can presumably only host a finite amount of matter. An infinite universe could in principle host either a finite or infinite amount of matter. Mainstream cosmology generally assumes the cosmological principle, which states that on sufficiently large scales the distribution of matter in the universe is homogeneous; in such cases, an infinite universe would have an infinite amount of matter in it, but of course this principle may not be accurate.

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    $\begingroup$ @Nohus Right. Observations indicate that the universe is, on very large scales, quite homogeneous; that being said, we are restricted to observations of the visible universe, and it's possible that on scales vastly larger than that, the matter distribution is not homogeneous at all. In the absence of any evidence to the contrary, most physicists would probably guess that we do not occupy such a special position in the universe. But who knows? $\endgroup$
    – J. Murray
    Mar 8, 2021 at 22:35
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    $\begingroup$ @Nohus: It's (apparently) also considered unlikely that our universe has the topology of a 3-sphere, because space appears to have no curvature at large scales. However that doesn't preclude a finite unbounded universe with pacman topology (no curvature, but opposite ends of the universe still wrap around) $\endgroup$ Mar 9, 2021 at 9:40
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    $\begingroup$ @BlueRaja-DannyPflughoeft I love that "pacman topology" is something that you can seriously mention in a physics discussion, and its likely the most accurate description of what you mean. $\endgroup$ Mar 9, 2021 at 11:58
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    $\begingroup$ @aschepler That is a common misconception. The big bang model does not say that all matter in the universe came from a single event. The big bang happened everywhere. $\endgroup$
    – J. Murray
    Mar 9, 2021 at 19:45
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    $\begingroup$ @aschepler Alright, for a less pop-sci description see e.g. the cosmology chapter of any intro GR text. The idea is that there is no single point in space where the big bang occurred. Imagine a 3D coordinate grid with spacing $\delta$ which extends to infinity in every direction, with an inertial observer sitting at every grid point. Now imagine taking $\delta\rightarrow 0$. At every moment in time, the grid is infinitely large, but the spacing between observers is just getting smaller. That's the big bang (in a spatially infinite universe) in reverse. $\endgroup$
    – J. Murray
    Mar 9, 2021 at 20:08
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When cosmologists talk about an infinite Friedmann universe, they mean one with infinite spatial volume, finite and uniform matter density (averaged over cosmological scales), and thus infinite matter. There would be galaxies as far as you care to go, even if you could go far beyond what we call the “cosmological horizon”.

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    $\begingroup$ That answers the "infinite" side, thank you. But seeing that the question is unresolved (correct me if I'm wrong), what would it mean for the universe to be finite? Would it mean the space is finite, or matter is finite in infinite space, or are both options considered (and thus the terminology is unclear)? $\endgroup$
    – Nohus
    Mar 8, 2021 at 22:13
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    $\begingroup$ @Nohus That warrants a separate question. In this question you asked what an infinite universe means. $\endgroup$
    – G. Smith
    Mar 8, 2021 at 22:15
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    $\begingroup$ @Nohus But I will answer it anyway in a comment: A finite Friedmann universe has finite spatial volume, finite and uniform matter density, and thus finite matter. $\endgroup$
    – G. Smith
    Mar 8, 2021 at 22:17
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    $\begingroup$ In my first sentence I asked "what do these two options actually mean?", which I thought takes precedence on how I titled my actual question in the post. But thank you for answering in the comment! $\endgroup$
    – Nohus
    Mar 8, 2021 at 22:18
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    $\begingroup$ The word “finite” does not even appear in your question. $\endgroup$
    – G. Smith
    Mar 8, 2021 at 22:19
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Maybe it is best to mention the three major possibilities of "universal shapes" that exist today.

Let me start by saying that cosmologists envision the universe as a four-dimensional spacetime continuum that can exist in a variety of shapes as long as the shape conforms to the field equations of Einstein's general theory of relativity.

Hornet-shaped spacetime.
And let me start with a very strangely shaped universe. The funnel formed spacetime. Look here or here. This is a hornet-shaped (hrnet=curved) universe. That is if we look at two space dimensions because we obviously cannot see a curved three-dimensional space. This universe allegedly explains observations made on the universe. This is a finite universe. Here's a picture (note that the points of space where the ship turns back inward are problematic):

enter image description here

A spherical spacetime.
The most simple and most common form. This is the "balloon universe", due to the fact that the two-dimensional space part can be viewed as a balloon shape, which can be viewed to represent a curved form of a two-dimensional space.
This universe can be finite or infinite in spatial extent. And these spacetimes can have zero, negative or positive curvature.
Zero curvature corresponds to a spacetime that will continue to expand forever and this expansion will stop "after" infinite time, while its size is infinite. The balloon will stop expanding at infinity and have an infinite size.
The positive curvature represents a universe that expands and after a finite time, this expansion will stop. This universe obviously has a finite size. The balloon grows in size and will contract after some time (spacetime generally cannot stay motionless).
The negative curvature represents a spacetime that will expand at an ever-increasing rate. The ballon starts expanding and will grow in size faster and faster. It looks as if our own universe is (in these days) growing with this last feature.
In our universe, inflation has been assigned to a newly formed balloon spacetime. Initially, the universe expanded at an incredible rate and basically, the whole universe was formed during inflation. Since the end of inflation, the universe has only grown "only" three times as large as in the beginning. So what we see today is just a very tiny fraction of the whole. Behind the horizon the is a vast space.
The three universes are also referred to as flat, closed, or open:

enter image description here

The torus-shaped universe.
In 1984 this model was proposed as the [three-torus model]. 5This universe has a two-dimensional space equivalent of a torus. It has positive as well as negative curvature and can be finite or infinite (the infinite size will be reached after infinite time).

enter image description here

One can invent more shapes, of course, but they have to pass the tests of observation. I'm not aware that another form as the three mentioned is present though.

It is generally accepted (i.e., by most cosmologists) that our universe has a balloon=like topology. The amount of matter in these universes can be finite or infinite. In closed universes, the amount of matter will be finite while this amount is infinite in (asymptotically) flat or open universes.

What about time? Is spacetime eternal? That depends of course if there was something before the big bang (when the spacetimes started evolving very small size). For example, according to Smolin, there was a contracting universe present before the big bang (likewise he proposes that upon entering a black hole there will follow an exit to another spacetime, where there is a white hole present which ejects all that has fallen in from our own spacetime). It is generally claimed though that spacetime had a beginning in time and dependent on how much matter is in it and on the expansion speed, the universe will be there forever or not. It is thought (based on observations) that our universe is nearly flat and will expand forever.

So, what does the concept of an infinite universe mean? In the context of spacetimes that might be clear now: a universe is infinite if the spacetime has an infinite extent. Infinite space goes hand in hand with infinite time. The amount of matter contained in these universes can be finite or infinite.

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    $\begingroup$ The origin of the horn universe differs in the standard model. A finite distance back in time it is pinched down and cut off in a rounded end, known as the Big Bang. Time counts forward from zero at that beginning. $\endgroup$ Mar 10, 2021 at 9:22
  • $\begingroup$ @GuyInchbald I don't see what you mean. $\endgroup$ Mar 10, 2021 at 13:13
  • $\begingroup$ The tip of the horn does not extend infinitely into the past, as shown in the illustration. It has a blunt, more or less rounded end which is the inflationary period after the Big Bang, with the central point being the beginning of the Big bang and also the beginning of Time. $\endgroup$ Mar 10, 2021 at 13:27
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    $\begingroup$ The idea that "spacetime had a beginning in time" is definitely not "generally accepted": 1/2 of the Nobel Prize award for 2020 went to Roger Penrose, whose "conformal cyclic cosmology" is eternal both to the past and to the future, with observational evidence of past aeons in several "anomalous spots of significantly higher temperature" that are visible in the CMB. (I believe the main factor in the award was Penrose's contribution to the singularity theorems he formulated with Hawking in the 1960's, but 2010's CCC is compatible with them.) $\endgroup$
    – Edouard
    Apr 5, 2021 at 12:21
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    $\begingroup$ @Edouard It's the first time I hear that he favors a cyclic model. Like always, he somehow contra-establishment. Which I like. I'm a fan of a cyclic universe myself. No beginning, no end. Maybe I return after the next big bang. $\endgroup$ Apr 6, 2021 at 6:44
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When you ask about an aspect of the universe as talked about by most cosmologists, you need to take into account the common general assumptions about the universe. One of the most common assumptions is that the universe is approximately the same everywhere taking an average of such things as average matter density or radiation density in a sufficiently large volume. The is called homogeneity. A second assumption is that the universe looks the same with respect to a sufficeient large scale in all directions. This is called isotropy. With the assumption of homogeneity, an infinite universe is expected to have in a sufficiently large volume the same stuff and dynamics everywhere, just like a finite universe does. Different homogeneous universe models have different conditions at different times. It is a bit tricky do decide how to determine what it means for two different volumes in the universe to be at the same time. The simplest way to think about it is that if they have the same densities, then they are at the same time. Some of the other answers talk about rather strange shapes, like a hyper-torus. So far I have never been able to locate a description of how a torus can be homogeneous and isotropic.

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  • $\begingroup$ A hyper-torus in this context is a cube, or other parallelepiped, with opposite faces identified. It has a continuous translation symmetry, but no continuous rotation symmetry, so it breaks isotropy but not homogeneity. Drawings of a torus as a donut shape are misleading since they suggest the surface is curved, which it isn't (as used in cosmology). $\endgroup$
    – benrg
    Mar 11, 2021 at 1:04
  • $\begingroup$ I understand benrg's description of a torus. It is not isotropic becasue it is possible to have a straight lines from a chosen origin that returns to itself with different angles relative the six parallel surfaces, AND such different paths will NOT be the same length. This violates isotropy. $\endgroup$
    – Buzz
    Mar 11, 2021 at 16:23
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It doesn't make any sense to talk about spatial scale/distance without referring to some objects within space. Or have you ever used a ruler to measure the distance of two space points in empty space? This pretty much answers your question.

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