# B How can the Universe be infinite and yet have a finite age?

1. Aug 27, 2016

### RingNebula57

How can the universe be infinite and yet have a finite age?

2. Aug 27, 2016

### phinds

It could have started off infinite. We don't know whether it is finite or infinite but whichever it is, that's how it started off.

3. Aug 27, 2016

### bapowell

The observable universe has a finite age, and it has a finite size. We know nothing of the universe outside what we can observe.

4. Aug 28, 2016

### RingNebula57

Ok ,thank you! I will do more research on this topic.

5. Feb 28, 2018

### laymanB

I thought I would resurrect ones of these infinite universe threads to make some points and hear feedback.

Is it not the case in physics that when we get infinities as a solution to our equations, it usually indicates a problem? For instance, we say that GR cannot be the whole story because the equations give us singularities with infinite densities at the beginning of the universe and in the center of black holes. Why do we think that this indicates a problem with the theory and that we need new physics, as opposed to saying that there is indeed infinite density there?

Let me take this idea into the question of whether the universe is spatially infinite or finite. We have confirmed that the local geometry is flat with small error bars. We don't observe anything that would lead us to believe that the local topology is anything besides Euclidean space. So, we plug these observations into a FRW model and the solution that we get out is a spatially infinite universe.

Now my question is: why don't we think this points to something wrong with the model, instead of embracing this infinity and trying to get rid of the other infinities (like infinite densities). It seems to me in the history of physics, that trying to get rid of infinities has led to meaningful progress.

6. Feb 28, 2018

### phinds

Usually, yes

Because infinite density makes no physical sense.
Because infinite extent does not seem to be physically impossible.

7. Mar 1, 2018

### PeroK

You are confusing two different concepts of infinity. Let's look at the mathematics. There is nothing inherently problematic with an "infinite" set - or, more precisely, an "unbounded" set. The simplest example is:

$\mathbb{N} = \{1, 2, 3 \dots \}$

In mathematical terms, that set not only has an infinite number of elements, but the distance between any pair of elements is unbounded.

Note that when it comes to cosmology, physicists tend to use "infinite" or "of infinite extent" to mean "unbounded" in the mathematical sense.

Infinitity, however, turns up again in mathematics in other cases. E.g.:

$\lim_{t \rightarrow 0^+} \frac{1}{t} = +\infty$

This is problematic if you want to assign a value at $t = 0$. And, if that limit came from a model of a physical process, then there is a "singularity" at the point $t = 0$. In the sense that you cannot assign a finite value at that point.

It is these "singularities" that indicate a problem with the model.

8. Mar 1, 2018

### Chronos

Infinite density creates an issue because the volume of a body with a non-zero finite can only be zero. That strikes most scientists as unphysical and just plain wrong. Since all known black hole have a finite mass, it is difficult to avoid concluding they must also occupy a non-zero volume. It's not just black holes that have this problem, it also pops up all the time in particle physics. As soon as you wedge a finite amount of anything into zero volume you get an infinite density of whatever it is you just tried to stuff into a size zero sack, be it bull droppings or the charge of an electron. Strictly speaking mathematically the rule is the result of division by zero is undefined [i.e., nonsense]. So the best answer for the density of matter in a black hole singularity is "undefined". Undefined is not a problem until you try to impose zero as the only allowable option for volume. For a black hole the obvious choice for volume is defined by its event horizon - considering, by definition, a black hole is any region of space where escape velocity equals or exceeds c. The same thing applies to the volume of the universe. We do not kinow if it truly goes on forever. Fortunately, that's only a guess [its hard to imagine what would constitute a spatial boundary on the universe] The easy way out of this mess is to assume it only extends as far as we can see - and we already know that cannot exceed the distance light could have traveled since the universe began. Answers are easier when you poae the right questions.

9. Mar 1, 2018

### laymanB

This is part of what I'm trying to make sense of. The place where mathematical abstraction maps onto physical realities. Let's say that we consider a square with sides of length 1. We know that the length of the diagonals are $\sqrt{2}$. We know that $\sqrt{2}$ is an irrational number and that the decimal places never end nor repeat. But don't we conclude that the length of the diagonal is a discrete length with end points in reality when we build this square out of wood or some other material?

We can ask a mathematician to show us different types of infinities, discuss the Hilbert Hotel, and talk about the continuum hypothesis, but do any of these mathematics give us justification for discussing infinities in physics? I'm thinking along the lines of Zeno's paradox, where only traveling half the distance between two points never gets us to the end point. While it is true if you can only travel by halving distances, real world particles are under no such constraint. They simply traverse the whole distance.

10. Mar 1, 2018

### Chris Miller

I was going to begin a new thread, but seeing this, I think I'll just post my questions here:

According to Sylvia Nasar's biography of John Nash ("A Beautiful Mind"), Nash once presented Einstein with some mathematics theorizing a flat, non-expanding universe. Einstein seemed to patronize him, suggesting he "study physics." Now, from what I gather here, Nash's theory was closer to current cosmological thinking than Einstein's at the time. Am I mistaken in my understanding that the universe is now believed to have begun as an infinitely large, dense mass? That matter has fragmented and is separating according to Hubble's constant, but that space itself has always been and remains infinite? That only our infinitesimal observable portion of the universe is actually growing?

11. Mar 1, 2018

### PeroK

You have infinite sets at the heart of physics, because calculus relies on the real numbers. You can't do (normal) calculus on finite sets of points. And, it is problematic to try to model spacetime as a discrete set (of points a non-zero distance apart).

I don't see the relevance of Zeno's Paradox. Which is fairly feeble in any case, IMO.

12. Mar 1, 2018

### laymanB

Good point.

Sorry, this was non-linear rambling on my part. Let me try again.

If we are talking about the escape velocity of an object, what is the physical meaning of the object slowing as it approaches infinity. In other words, does infinity have to exist to launch something with escape velocity? Sorry if this is still muddled.

13. Mar 1, 2018

### phinds

As far as that sort of thing is concerned, infinity is just a mathematical fiction and does not need to exist to do the computation.

14. Mar 1, 2018

### PeroK

This is a good example, IMO, of how turning the question round to use finite quantities resolves the matter.

If you launch a rocket at a speed $v$ and it returns to Earth in a finite time, then $v$ is less than the escape velocity. Otherwise, $v$ is greater than or equal to the escape velocity.

Put another way, you can set up an equation for the time the rocket returns. If that equation has no solution, then the velocity is greater than or equal to the escape velocity, which then is the lowest velocity for which the equation has no solution.

No need to mention the "i" word!

15. Mar 1, 2018

### Staff: Mentor

Not explicitly, but it's still there implicitly, in the derivation of the equation you set up. That equation implicitly assumes that space is infinite. If you drop that assumption and substitute the assumption that space is a 3-sphere with some finite total 3-volume, you can find solutions with the same initial conditions where you couldn't before.

16. Mar 1, 2018

### laymanB

What, if any, are the implications of a spatially infinite universe at t > 0? Let me elaborate.

Am I correct in saying that the energy density, the stress-energy tensor, and therefore the spacetime curvature was extreme in the first second of the Big Bang?

Assuming this is right, then what are the implications if that energy density is extended into infinity versus being finite? Does it put constraints on the scalar field for inflation? Does it make it much more likely that the spacetime curvature will result in an immediate collapse?

17. Mar 1, 2018

### Staff: Mentor

Why do you think this? Current cosmological thinking is that the universe is spatially flat and expanding, not spatially flat and non-expanding.

18. Mar 1, 2018

### Staff: Mentor

Yes.

AFAIK, no; any constraints on the inflaton field are local, not global; they don't "care" whether the universe is globally finite or infinite, spatially.

No. The equations governing this are also local, not global.

19. Mar 1, 2018

### PeroK

If you are already on Chapter 22 of Hartle, you must have bashed through Chapters 1-21 fairly quickly!

20. Mar 1, 2018

### laymanB

Ha! I am much lazier than you think. I'm only about a quarter of the way through, and haven't touched it for the last month or two. I'm trying to get more big picture ideas while trying to learn the theories. Instead of going through the standard protocol of learning the math and foundations first, before moving on to more complex subjects. I would not advise it for a college curriculum.