# How can an infinite universe expand?

I understand the expansion of the universe as actually an increase in the ratio of space to matter. Is this a correct understanding? Otherwise, I don't understand how an infinite structure can expand.

• Infinities come in different sizes: en.wikipedia.org/wiki/Aleph_number – Wayfaring Stranger Nov 21 '15 at 18:06
• @WayfaringStranger, thats true, but irrelevant in this context – James K Nov 21 '15 at 22:45
• Possible duplicate of How can the universe be infinite? – FJC Nov 23 '15 at 14:26
• universe not is infinite – RBoschini Dec 29 '15 at 18:47
• Observable universe is not infinite. Hubble's law gets things receding faster than light, unobservable, at 40 some billion light years right now. We tend to assume that an observer on that last planet receding at just under c sees a universe that looks just like ours, but with us receding at just under c. That assumption will get you close to an infinite universe. – Wayfaring Stranger Jul 5 '18 at 18:02

Expansion means that distances are increasing as a function of time. Say if the distance between two galactic clusters is $D$, then in an expanding Universe the distance is governed by some strictly increasing function of time $a(t)$ called the scale factor where

$$D=a(t)D_0$$

where $D_0$ is the distance at the present time and by definition $a(t_{0})=1$.

Cosmology assumes that the Universe is on large scales the same everywhere (homogeneous) and the same in all directions (isotropic) so the above applies to all distances above a certain scale. The scale factor $a(t)$ can be found from the Friedmann equations and initial conditions.

Expansion is possible in Universes of both finite and infinite spatial extent.

As the volume of a (large enough) region of space increases in proportion to $[a(t)]^3$, but the amount of matter remains constant, the matter density changes in proportion to $[a(t)]]^{-3}$. Expansion however also decreases the kinetic energy of of its contents, so the energy density decreases by a greater factor if the contents has kinetic energy.

• But if the universe is infinite, how can it expand? Saying it just can does not answer the question. – John Duffield Nov 22 '15 at 11:09
• Before I made that statement I explained how expansion is the increase of distances between (comoving) objects. Clearly there is no dependence on the space being finite for distances to increase. – John Davis Nov 22 '15 at 15:06
• @JohnDuffield mentalfloss.com/article/78583/infinite-hotel-paradox – Florin Andrei Jul 5 '18 at 17:43
• @Florin Andrei : IMHO it says nothing useful about the real universe. See my answer below for my own thoughts on the matter. – John Duffield Jul 5 '18 at 19:40
• "Expansion is possible in Universes of both finite and infinite spatial extent". This implies the universe to be held within a space. So, space is not part of the universe, but the universe is just one part of space? Seems quite debatable. – RodolfoAP Dec 7 '19 at 7:44

There is absolutely not contradiction between being infinite and able to expand (in contrast to what your question seems to suggest). This simple fact is not confined to the actual universe we are living in.

As an illustraction, take the infinite 'universe' of the natural numbers $i=0\dots\infty$. Now consider the sets $2i$ and $2i+1$, each equally infinite as the natural numbers, but stretched. Now combine those two sets to get an expanded 'universe' and you obtain the natural numbers again.

• a) If we make an analogy, this is equivalent to a Ponzi scheme. It works in theory. But considering nature's limitations, seems quite dubious. See Kant's first antinomy. The infinite attribute of the universe would just be a fallacy of perception. b) Extrapolating rules at different scales seems naive. What you are stating here is that galaxies are expanding, not the universe. c) AFAIK universe expansion implied "creating new space". Quite far from this response. d) This kind of universe expansion is equivalent to measurements contraction. – RodolfoAP Dec 7 '19 at 7:53
• I accept this explanation, I'd like to add an intuitive example. How many numbers divide by four? inf. How many numbers divide by 2? Also inf, but this inf is two times larger. So I guess it is true, the universe started infinitely large, and it keeps growing. – Yuval Harpaz Feb 13 at 17:50

How do you describe how far away two points are? You have to have some way of describing the concept of distance.

When we say that the universe is expanding, what we really mean is that distances inside it are increasing.

The idea that the expanding universe is some sort of 3D bubble or balloon that can be seen to expand from outside isn't meaningful, as there is no outside.

Perhaps a more helpful way to think of it is to say that the concept of distance is a property of the universe, and that property is changing over time.

• "The concept of distance is a property of the universe": such was the view before the XVII century. Modern philosophy considers space and time to be subjective features. See plato.stanford.edu/entries/kant-spacetime – RodolfoAP Dec 7 '19 at 8:00

If you he universe is finite in size, something(matter) would have to be present to to fill in the empty space surrounding the entire entire universe or else it would be just more empty space. Let me explain what I mean. A room is finite in size because when you get to a certain point that the walls, floor, and ceiling stop you. So if a universe is finite, it would have to have some equivalent to walls, to keep it the same size and finite and/or to constitute an end. If space ends at some point because nothing can go any further, like if we were inside of an incomprehensibley large hollow sphere and traveled to the end of space and found the end of the universe, something would still be on the other side of whatever the end of the was. Infinite is not a hard concept at all and is really the only possibility if you think a about. Infinite can't expand. It's like a child saying infinite plus one. Only now its college educated grown men now and all the logic in the world wont stop them from trying to make it make sense. The universe is definitely infinite, anyone can observe that by looking up at the night sky. We dont live inside of a cosmic domicile with limited space. We live in infinite empty space on a finite in size spherical shaped mass of a finite amount of matter. an Infinite universe of empty space that has stars and planets, moons and whatever else. But the universe itself is already infinite and therfore can't get any bigger. Its empty space, in every direction, forever(with stars and planets here and there). Why is that so hard for science to accept?

• "If you he universe is finite in size, something(matter) would have to be present to to fill in the empty space surrounding the entire entire universe or else it would be just more empty space." -- this statement is completely mistaken. So is your reasoning. There is absolutely no problem with a spatially finite universe that has no spatial boundary. – Stan Liou Dec 20 '15 at 19:03
• A hackneyed analogy perhaps, but you can compare the 3D volume of the Universe to the 2D surface of some arbitrarily-shaped body, e.g. a ball. The surface area of this object is finite, yet it doesn't have an edge (you're not allowed to fly up / dig yourself down, that would be cheating since you'd leave the 2D world). An infinite universe would be comparable to a 2D surface of an infinite object like an infinite table top, or an infinite saddle. – pela Dec 21 '15 at 14:13

Don't Forget This Part!! The other answers simply explain that an infinite universe can continuously expand because the increase in size is filled with space, not new matter. They confirm your explanation. However, the answer to "how can an infinite universe expand?" requires an explanation of "empty space" and mention of the fact that the universe's expansion is also accelerating.

Empty Space gives physicists headaches. Is there even such a thing as empty space? Or just space filled with matter we cannot yet detect? More on this below.

Accelerating Expansion: The universe is not only expanding, it's speeding up. That keeps some very smart people awake at night. "How can an infinite universe expand?" Well the big bang would explain it... if it were slowing down. It's not. So... it has to be pushed or pulled, right?

Why Does That Happen?

Einstein's theories first predicted that the gravity of the universe would cause the universe to collapse in on itself so he introduced something called the "cosmological constant" in his theory of relativity. That constant enabled the equations to predict a static universe.

Wiki excerpt:

[The cosmological constant] was originally introduced by Albert Einstein in 1917 as an addition to his theory of general relativity to "hold back gravity" and achieve a static universe, which was the accepted view at the time. Einstein abandoned the concept after Hubble's 1929 discovery that all galaxies... are moving away from each other, implying an overall expanding universe.

Modern Understanding

Although Einstein would later call that "cosmological constant" his "greatest blunder", it turned out to be correct. However, he also thought the value would be used to explain why things are static. It turned out the value would explain why things are continually expanding. But what is that constant, besides a number in an equation? Today most physicists seem to think it's dark matter or dark energy. It's a very new science but it is highly researched.

Wiki excerpt from 'Accelerating Universe':

Different theories of dark energy suggest different values of w, with w < -1/3 for cosmic acceleration... The simplest explanation for dark energy is that it is a cosmological constant... this leads to the Lambda-CDM model, which has generally been known as the Standard Model of Cosmology from 2003 through the present...

Wiki excerpt from Dark Energy:

around 70% of the mass–energy density of the universe can be attributed to dark energy. While dark energy is poorly understood at a fundamental level, the main required properties of dark energy are that it functions as a type of anti-gravity, it dilutes much more slowly than matter as the universe expands... The cosmological constant is the simplest possible form of dark energy since it is constant in both space and time

So I think the real answer to your question isn't that an infinite universe is possible simply because the amount of empty space increases. Actually, that might not even be possible. There may be no such thing as empty space. Empty space might just be a type of matter we can't see or detect yet.

Wiki:

Dark matter is a hypothetical kind of matter that cannot be seen with telescopes but accounts for most of the matter in the universe. The existence and properties of dark matter are inferred from its gravitational effects on visible matter, on radiation, and on the large-scale structure of the universe. Dark matter has not been detected directly, making it one of the greatest mysteries in modern astrophysics.

To understand the infinite expansion of the universe, ask first how such a thing could be possible. Ask why it expands infinitely instead of the mutual gravity of all the galaxies slowing each other down and causing a collapse.

• "Answering your question: the answer to your question isn't...". Synthesized answer: the universe is expanding, although we don't understand space expansion because we don't understand space. – RodolfoAP Dec 7 '19 at 8:15

Think of the universe as a birthday cake. If you had a cake that was a million light years across, you could fit at least a hundred thousand candles on it, right? Probably many more. But just think how many candles you could fit on a cake that was infinite in size - millions, if not billions. But if you put this cake back in the oven and bake a smaller cake onto the side of it, then you could say you've "expanded the cake", despite it being infinite in size. In this way, you could keep adding to your cake until it truly was infinite.

• How would you put an infinite cake in an oven? – pela Jul 5 '18 at 17:38
• Have to use a Big oven. – Wayfaring Stranger Jul 5 '18 at 17:57
• Actually, re-baking a cake extracts its humidity, and drastically reduces its size. So, the universe is a growing cake reducing its size. – RodolfoAP Dec 7 '19 at 8:17

I understand the expansion of the universe as actually an increase in the ratio of space to matter. Is this a correct understanding?

It isn't wrong. The ratio is increasing. But it isn't a "correct understanding". It's merely an observation of one of the results of the expansion of space.

If the universe is infinite how can it expand?

I don't know. Nor do I know how big bang cosmology can be reconciled to an infinite universe. If you look around on the internet, you can find articles like this which say this:

"The linear dimensions of the early universe increases during this period of a tiny fraction of a second by a factor of at least 10$^{26}$ to around 10 centimetres (about the size of a grapefruit)".

However in 2013 results from the WMAP mission appeared to confirm that space is flat. Then a non-sequitur crept in. See this article and pay careful attention to this:

"We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe."

That's a massive error. It absolutely doesn't suggest that the universe is infinite in extent. Or that the Universe is much larger than the volume we can directly observe. But this myth has legs, and people repeat it ad-infinitum, even though they can't explain how it fits in with Big Bang cosmology. What you tend to hear is that the observable universe was the size of a grapefruit, but it absolutely doesn't satisfy. Moreover there's a dreadful flaw lurking in the shadows. Take a look at the stress-energy-momentum tensor, and note the energy-pressure diagonal. A gravitational field is something like a spatial pressure gradient, and you can think of space as having an innate "pressure". So you can reason that the universe must expand. As to why Einstein didn't, I just don't know. But anyway, for an analogy, squeeze a stress-ball down in your fist, and let go. It expands because of the pressure. However if that material was infinite in extent, the pressure is counter-balanced at all locations. So it can't expand. In similar vein, in my opinion, an infinite universe can't expand.

People claim the universe must be infinite because of the cosmological principle. But this is merely an assumption. There's an assumption that the universe is homogeneous and isotropic, but this isn't fact. You cannot use it to make sweeping claims about an infinite universe that was always infinite. For all we know some observer 50 billion light years away might be looking up at the night sky wondering why half of it is black. Or a mirror-image of the other. Or some kind of edge.

It is said that in days gone by, people could not conceive of a world that was curved. They could only conceive of a world with an edge. Nowadays I rather fancy that there are some people who cannot conceive of a world that is not curved. They cannot conceive of a world with an edge.

Edit:

See The Foundation of the General Theory of Relativity: "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy". Energy is the source of the stress-energy tensor. Matter is only a source because of the energy-content. Also see Inhomogeneous and interacting vacuum energy which refers to spatial energy. An interesting read is the article Universe 156 billion light-years wide featuring Neil Cornish. This isn't entirely accurate, but the compound interest and the hall of mirrors concepts are of interest. As for the non-sequitur, see this interview with Joseph Silk:

"We do not know whether the Universe is finite or not."

I hope nobody will argue with that. Reading on:

"To give you an example, imagine the geometry of the Universe in two dimensions as a plane. It is flat, and a plane is normally infinite. But you can take a sheet of paper [an 'infinite' sheet of paper] and you can roll it up and make a cylinder, and you can roll the cylinder again and make a torus [like the shape of a doughnut]. The surface of the torus is also spatially flat, but it is finite".

This is akin to the old Asteroids game. But the Planck mission found no evidence of any torus. Reading on further:

"So you have two possibilities for a flat Universe: one infinite, like a plane, and one finite, like a torus, which is also flat."

I dispute that. There is a third possibility. A flat finite universe with no intrinsic curvature. If anybody can cite some reliable sources that support the assertion that a flat universe must be infinite, I'd like to see them.

• Pressure in the stress-energy tensor of matter actually decelerates the expansion. If you like, this is because the pressure is itself a source of gravity. General relativity does not deal with space, matter, pressure etc in the exactly same way as Newtonian physics, which is why your assumptions fail. The expansion of the Universe should be seen to be the result of initial conditions, though in later times negative-pressure dark energy is thought to have played a role in accelerating the expansion. – John Davis Nov 21 '15 at 18:21
• @John Davis : the stress-energy-momentum tensor "describes the density and flux of energy and momentum in spacetime". It's a mistake to say it's the stress-energy tensor of matter. Einstein described a gravitational field as space that was "neither homogeneous nor isotropic". Given that the FLRW assumption is correct and the universe is homogeneous on the large scale, he should have known there's no overall gravitational field. Why he didn't "ditch the dust", I shall never know. – John Duffield Nov 22 '15 at 11:07
• Matter is the source of the stress-energy tensor in this instance, so it is correct and common usage. That is not quite what Einstein said and it is abundantly clear that general relativity predicts expansion/contraction in isotropic and homogeneous spaces, except when either space is empty (Minkowski space) or, unstably, in the presence of a finely-tuned cosmological constant (Einstein static Universe). So there's no point quibbling about what Einstein said or what he meant, what is important is what GR says and what the observational evidence says. – John Davis Nov 22 '15 at 15:15
• @John Davis : see my notes above. Energy causes gravity. Also note the dark matter pie. Only circa 4% of the energy in the universe is thought to be (ordinary) matter. That doesn't justify your claim that matter is the source of the stress-energy tensor in this instance. – John Duffield Nov 23 '15 at 22:17
• I did not say the Universe was contracting, only that given the conditions above it must be contracting or expanding. The physical evidence shows it is expanding. The reason you gave for an infinite Universe to contradict expansion is from trying to apply Newtonian physics to general relativity and therefore is deeply flawed. – John Davis Nov 24 '15 at 1:53