# Expanding from and eventually to a singularity?

• B
• Bran

#### Bran

Gold Member
I know this thread, about why the Universe can't expand inward, is fairly old; but I stumbled across it today and there was something mentioned here that sparked a question I feel like people here would be qualified to answer. What was mentioned, was that a singularity is a point at which our mathematics breaks down and can no longer describe what takes place (Chronos' comment). My question is, is there a point in the other direction where we can say the same thing? Would this 'point' be infinity? Or is there another word to describe it? If our Universe exploded from a singularity and is expanding in an infinite or unbounded manner, would that not mean that at some point, what we expand to is as impossible for our mathematics to describe as what we expand from? If that is the case, what would be the difference between a singularity and this other state? Could that possibly describe something like what the OP in the original thread suggested, a seemingly inward expansion, whereby the Universe expands toward the same mathematical breakdown / state that it exploded from? I'm not a physicist, and I'm not a mathematician - I'm simply interested in this sort of thing, my apologies if the answer to the question is too simple :-)

I don't think so, but I'm not sure. I think a singularity is similar to having a formula similar to 1/x. As x gets extremely small, the formula gets extremely large. We say that it goes to infinity as x approaches zero. But if you try to plug zero into the formula... well you can't. You cannot get a result from that. But you can plug any number other than 0 in for x. As x gets huge (goes to infinity) the formula approaches zero. Which is just fine. See here for more info: https://en.wikipedia.org/wiki/Singularity_(mathematics)

• Bran
I know this thread, about why the Universe can't expand inward, is fairly old; but I stumbled across it today and there was something mentioned here that sparked a question I feel like people here would be qualified to answer. What was mentioned, was that a singularity is a point at which our mathematics breaks down and can no longer describe what takes place (Chronos' comment). My question is, is there a point in the other direction where we can say the same thing? Would this 'point' be infinity? Or is there another word to describe it? If our Universe exploded from a singularity and is expanding in an infinite or unbounded manner, would that not mean that at some point, what we expand to is as impossible for our mathematics to describe as what we expand from? If that is the case, what would be the difference between a singularity and this other state? Could that possibly describe something like what the OP in the original thread suggested, a seemingly inward expansion, whereby the Universe expands toward the same mathematical breakdown / state that it exploded from? I'm not a physicist, and I'm not a mathematician - I'm simply interested in this sort of thing, my apologies if the answer to the question is too simple :-)
You can write down equations that work this way, but there's very little chance that those equations describe reality.

This is known as the "big rip" scenario. It happens if the dark energy increases in density as the universe expands. There are a number of theoretical reasons to believe that this can't happen, but if it did happen, then it takes a finite amount of time before the expansion rate of the universe reaches infinity.

Edit: in all of the realistic models of the future fate of the universe, there are no singularities in the future, aside from the unlikely possibility that the universe recollapses.

• Bran
I don't think so, but I'm not sure. I think a singularity is similar to having a formula similar to 1/x. As x gets extremely small, the formula gets extremely large. We say that it goes to infinity as x approaches zero. But if you try to plug zero into the formula... well you can't. You cannot get a result from that. But you can plug any number other than 0 in for x. As x gets huge (goes to infinity) the formula approaches zero. Which is just fine. See here for more info: https://en.wikipedia.org/wiki/Singularity_(mathematics)
Thank you for not only the response, but the pointer toward Wikipedia! I'll admit, there were some parts of that entry that made my head spin (accelerated by the caffeine in my system as I mainline coffee, I'm certain); but there was something there that caught my attention (the smallest section, of course) about Coordinate Singularities. This seemed to describe what I had in mind with my question, as I was thinking along the lines of the Big Bang and subsequent expansion of the Universe, and thinking of it as a closed and circular system. This would have expansion from singularity and expansion to singularity represented by the jumping from longitude 0 to longitude 180 and then again from 180 to 0. This led to something else fascinating ('further reading' sections are dangerously seductive), called Chronometric singularities ... again, somewhat similar to what I had in mind, as when there is no Universe (either before or after), there is no measurable time. The problem, of course, is that I can also see from these definitions that what I have in mind doesn't seem very realistic. Still, thank you for your explanation - it gave me the chance to read over a few things I hadn't before encountered!

You can write down equations that work this way, but there's very little chance that those equations describe reality.

This is known as the "big rip" scenario. It happens if the dark energy increases in density as the universe expands. There are a number of theoretical reasons to believe that this can't happen, but if it did happen, then it takes a finite amount of time before the expansion rate of the universe reaches infinity.

Edit: in all of the realistic models of the future fate of the universe, there are no singularities in the future, aside from the unlikely possibility that the universe recollapses.

Thank you for your reply - this also helps me to understand why the idea of expanding toward a singularity doesn't work too well. I hadn't thought about the Big Rip ... I was honestly thinking something along the cyclical nature implied by the donut-shaped Universe, where the Big Bang eventually pushes everything back in on itself before exploding all over again. Along those lines, but different in that I really like Hubble's description of the shape of the Universe being such that wherever you stand in it, it seems as though you are in the center. I thought somehow, in a situation where this is possible, that the return to singularity that the donut shape would imply could happen at a point of infinite expansion. Ludicrous, now that I think further about it; but I appreciate that people here took the time to explain why this couldn't work without derision or laughter!

Remember that 'the singularity' is not a physical object.
It means that we have a situation which is undefinable mathematically (by the best theories we currently have).
Just like dividing by zero, it's logical nonsense, it's not a description of something.

• weirdoguy and Bran
Remember that 'the singularity' is not a physical object.
It means that we have a situation which is undefinable mathematically (by the best theories we currently have).
Just like dividing by zero, it's logical nonsense, it's not a description of something.

You raise a really good point here - and this is honestly an area where I find myself a bit confused. When I hear or read 'singularity,' what comes to mind is the origin of the Universe - in the General Relativity / Big Bang model, a gravitational singularity. As I understand it, this singularity, when it exploded, did so with all the energy / matter that would comprise our Universe. It is difficult for me to reconcile the source of all existence as being a non-physical object - even if that source, when named as a singularity, is describing an event in mathematics more than it is a physical object. Somehow, the logical nonsense of dividing by zero starts to seem inviting, lol.

As I understand it, this singularity, when it exploded, did so with all the energy / matter that would comprise our Universe.

This is not correct. The "initial singularity" is a mathematical artifact of a particular idealized model of the universe. Nobody believes it is actually physically real.

The "explosion" you refer to, according to our best current model, happened at the end of inflation, when the energy that had been stored in the "inflaton" field (the field that drove inflation) got transferred to the matter and radiation fields included in the Standard Model (quarks, leptons, photons, etc.). That created a very hot, very dense, rapidly expanding (hence the often-used term "explosion") state of the universe that is the correct referent of the term "Big Bang".

• Bran
This is not correct. The "initial singularity" is a mathematical artifact of a particular idealized model of the universe. Nobody believes it is actually physically real.
Not exactly. General Relativity predicts a singularity in the finite past under very general conditions. Just the fact that we have an expanding universe is enough.

So it's not so much that the singularity is part of any idealized model, but rather than the singularity in our models underscores the fact that General Relativity cannot accurately describe gravity under situations of extreme density.

• Bran
General Relativity predicts a singularity in the finite past under very general conditions.

Yes, and inflationary cosmology violates those conditions; specifically, the inflaton field violates at least one of the energy conditions that has to be assumed for the singularity theorems to hold.

the singularity in our models underscores the fact that General Relativity cannot accurately describe gravity under situations of extreme density.

This is true, but it's not necessarily the same issue as the one I described. Classical GR can describe inflation quite well; you just assume a scalar field with a large constant energy density everywhere. So from a classical GR perspective, an inflationary cosmological model is just an inflation region of spacetime, with a scalar field as the only field with a nonzero stress-energy tensor, joined to a spacetime region where ordinary matter and radiation (and a very small dark energy density) are the only nonzero contributions to the stress-energy tensor. The 3-surface at the junction of the two regions is an idealized "reheating" surface, where the energy density gets transferred from the inflaton field to the matter and radiation fields.

None of this requires the energy density or spacetime curvature to get large enough anywhere for classical GR to break down for "singularity" reasons (i.e., because, heuristically, the radius of curvature of spacetime is at the Planck scale or smaller). So the latter is, conceptually, a separate issue.

• Bran
Yes, and inflationary cosmology violates those conditions; specifically, the inflaton field violates at least one of the energy conditions that has to be assumed for the singularity theorems to hold.
There's still a past singularity in inflation models:
http://arxiv.org/abs/gr-qc/0110012

• Bran
"The Universe,would never come into existence. It would simply exist."
Didn't Fred Hoyle say something like that?

• Bran
"The Universe,would never come into existence. It would simply exist."
Didn't Fred Hoyle say something like that?
You can't really conclude from this kind of theorem whether or not the universe is eternal into the past, or how it started. The correct conclusion is that General Relativity can't be entirely correct for very high densities.

• Bran
A phase change seems to be the only reasonable conclusion.
(Except for the fact that we have no clue about the properties of stuff before the change occurred)

• Bran
There's still a past singularity in inflation models:
http://arxiv.org/abs/gr-qc/0110012

Interesting paper! A key observation to me is their reason for why de Sitter spacetime, which is geodesically complete, cannot serve as a model for our universe with no initial singularity: because extending timelike or null geodesics indefinitely into the past extends them into regions where the "universe" is contracting, not expanding. Heuristically, any such model of the universe would be a "bounce" type cosmology, not an inflationary cosmology.

I also notice, however, that their description of the geodesic incompleteness shown by their result does not use the word "singularity". What they say, at the start of the discussion section (section IV, bottom of pg. 3) is: "we have shown under reasonable assumptions that almost all causal geodesics, when extended to the past of an arbitrary point, reach the boundary of the inflating region of spacetime in a finite proper time (finite affine length, in the null case)".

In other words, geodesic incompleteness here does not necessarily mean a singularity in the sense of spacetime curvature increasing without bound. It means the geodesics must reach a past boundary of the inflating region of spacetime--but that boundary does not have to be a curvature singularity. They consider several possibilities for the boundary, and none of them, as far as I can tell, are a curvature singularity.

So I'm not sure this paper shows that there must be a past singularity in inflation models; I think it shows that there must be a past boundary in inflation models, and further investigation is needed to see what that past boundary could be.

• Bran
• Bran
I want to thank everyone who has responded to my question in this thread, and helped me to understand where some of my thoughts concerning this subject were heading astray! I can say that while this discussion has shown me very clearly that I need to learn a lot more about this (the subject really does fascinate me), it has also whetted my appetite to do just that. Because I wasn't sure whether or not asking for book recommendations would technically remain on-topic, I thought I would give that request a thread of its own. I will still follow the conversation in here; but I can tell I've got a ways to go before I can hope to contribute anything meaningful to it! :-)