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Author Topic: Is the Universe infinitely large?  (Read 7667 times)

Offline JP

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Re: Is the Universe infinitely large?
« Reply #25 on: 28/07/2013 19:33:44 »
Bill, I can answer you on the Dirac delta function in quantum mechanics, but I'll leave relativity to Pete.  The delta function is basically a mathematical tool to extend the idea of distributions over space to the case where everything is localized at one point.  Imagine a quantum particle within a box in one dimension.  If the particle has fixed charge of 1 unit and can be found anywhere within the box with equal probability, we can say that unit of charge is spread equally over the box.  If the box is width L, the height of the function describing how the charge is distributed over space is 1/L.  As we squeeze the box smaller and smaller, the particle takes up less space and the height of this function describing the charge goes up.  As L decreases in width without bound (which corresponds to a particle at a particular point), the height of this function diverges to infinity, but it does so in such a way that the total area under the function is still 1.

It's not a true function and one reason for this is that, like the concept of infinity, defining a distribution this way is done in terms of limiting processes.  The phrase "the height as L goes to zero" is not a number.  Despite this, the rules of calculus allow this to be used in various expressions and for the most part physicists get away with treating it like a function with a few special (usually memorized) rules.  Mathematicians are generally horrified by this.  :p
« Last Edit: 28/07/2013 21:40:45 by JP »
 

Offline JP

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Re: Is the Universe infinitely large?
« Reply #26 on: 28/07/2013 19:38:16 »
It does suggest that time and space, as well as galaxies, existed before the Big Bang. 

I sort of understood it has saying… If every point in an infinite universe is the centre of its own observable universe, then running the infinite model backwards results in a singularity at every point, in other words, an infinite expanse in a state of singularity. Correct me someone…singularity is a state not a point??
That is to say, we can’t understand the spacetime and physics when it’s in a state of singularity.
Winging it now… Our present understanding of spacetime and physics pops into existence with the big bang.

The context in which you use singularity matters.  Technically, it's a point where equations break down such that your mathematical quantity becomes undefined.  I usually see this used to describe cases in general relativity where certain quantities diverge to infinity, meaning that at that point, the equations break down and things become undefined (since infinity isn't a number that you can plug into equations.)  There is some debate on this, but I think most physicists (myself included) generally see singularities as a sign that our current models are incomplete rather than viewing them as a real physical object.
 

Offline Bill S

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Re: Is the Universe infinitely large?
« Reply #27 on: 28/07/2013 21:46:19 »
Thanks JP, your explanation gives me heart to believe that, even at my age, it's not too late to gain some basic mathematical understanding.  The problem is generally twofold; finding a simple enough explanation and then retaining it. :)
 

Offline yor_on

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Re: Is the Universe infinitely large?
« Reply #28 on: 28/07/2013 21:58:27 »
To me unbounded speaks about the impossibility of me placing myself 'outside', looking in. In that way similar to the idea of a infinity to me. With a boundary on the other hand, speaking about a possibility of being 'outside', looking in. So from that point of view maybe infinity is the one I would lean to. How else would one define a boundary? If not assuming something existing, defining that boundary?

On the other hand you have gravity, becoming a intrinsic property, stating our observations of SpaceTime to 'bend' comes from inside, not relative us being 'outside' of SpaceTime, looking in to find it to bend. On the third tentacle, can there be observations from a 'outside'? If there can't be any such observations, How would one prove a boundary?
 

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Re: Is the Universe infinitely large?
« Reply #29 on: 29/07/2013 14:06:51 »
The context in which you use singularity matters.  Technically, it's a point where equations break down such that your mathematical quantity becomes undefined.

Would that be equations breaking down before or at the big bang?

we can’t understand the spacetime and physics when it’s in a state of singularity. ...
 what happens to spacetime in a state of singularity is beyond current physics...  I think.

JP, looking at the link I gave about an infinite universe…
Here http://www.astro.ucla.edu/~wright/infpoint.html
In the context my post was about an infinite universe…
If all points can be considered to be at the centre of their own observable universe,
then what happens in the context of an infinite universe when it is run backwards so that each observable universe ‘closes down’ to its respected point? How many points in an infinite universe? The equations break down ‘everywhere’? singularity everywhere? My sentence ‘an infinite expanse in a state of singularity’ could read as ‘an infinite expanse in an indefinable state.’ :)
« Last Edit: 29/07/2013 14:14:35 by lean bean »
 

Offline JP

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Re: Is the Universe infinitely large?
« Reply #30 on: 29/07/2013 16:44:19 »
Good questions, lean bean. I'm not a general relativity expert but I'll answer to the best of my ability.

The big bang is generally called a singularity because our equations treat it as an infinitely dense state--that's an infinity and since infinity isn't a number we can't really get numbers out of our equations, so we say they're singular or that this is a singularity.  I believe this happens at the big bang, as plugging in any finite numbers will likely get you a number out.  The bigger question is what happens when you get "close enough" to the instant of the big bang.  Is there some extremely large but finite density at which general relativity is not longer an accurate theory?  I think most physicists believe so and that's why they're hunting for a theory of quantum gravity, which would cover the very tiny length scales that matter at those densities.

2) Yes, as I understand it the observable universe would be smaller in the past.  This is because only so much time has passed since the big bang, so light can only have traveled so far to reach us.  The less time between us and the big bang, the smaller region of the universe we can see.  (Expansion plays a role as well, so we actually see stars that have since moved further away due to expansion).  The closer you get to the big bang, the smaller the observable universe would be.  At 1 picosecond after the big bang, assuming we could be there to "see" objects, we'd see things 1 light-picosecond plus whatever expansion had happened distance from us.  (Obviously we can't necessarily imagine "seeing" things in this early universe since it was so hot and dense, so what we're really calculating is the maximum distance at which things can interact with each other since all interaction travels at the speed of light or less). 

One of the toughest things for me to grasp about this was that if the universe is currently infinite in size, but has finite average density, the big bang could have been infinite in size with infinite density.  It would still be a singularity because infinite density makes the equations break down.  Maybe Pete can correct me on this, but I was under the impression that if the universe was finite in size (even at the big bang) it would still have to be infinite in size, and if it was infinite in size at the moment of the big bang, it would still be infinite in size.
 

Offline Pmb

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Re: Is the Universe infinitely large?
« Reply #31 on: 29/07/2013 17:47:32 »
Quote from: JP
It's not a true function and one reason for this is that, like the concept of infinity, defining a distribution this way is done in terms of limiting processes.
One of the formal names for the Dirac delta function is distribution. The other formal name for it is generalized function since it can be defined as a map from functions to functions.

We use them in electrodynamics, quantum mechanics and classical mechanics. E.g. Poison's equation for the gravitational potential for a point mass is given as the Laplacian being proportional to the Dirac delta function (in 3 dimensions).

Fun stuff!
 

Offline Bill S

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Re: Is the Universe infinitely large?
« Reply #32 on: 29/07/2013 23:01:03 »
Quote from: JP
I was under the impression that if the universe was finite in size (even at the big bang) it would still have to be infinite in size, and if it was infinite in size at the moment of the big bang, it would still be infinite in size.

Assuming your first "infinite" should have been "finite"; this is a major part of the line of reasoning I have been trying to follow.  Namely that (physical) things that are finite, remain finite, and those that are infinite, remain infinite.  For a few ecstatic moments I thought Pete agreed with me, but then his “Although I said no at first I'd like to change my mind” dashed my euphoria.  Hopefully he will address your “impression”.

 

Offline Pmb

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Re: Is the Universe infinitely large?
« Reply #33 on: 30/07/2013 00:56:04 »
Quote from: JP
The big bang is generally called a singularity ...
Were you aware that there is no event in the big bang which is actually called "the big bang"?

From Principles of Physical Cosmology by P.J.E. Peebles, Princeton University Press, 1993, page 6
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If there were an instant, at a "big bang," when our universe started expanding, it is not in our cosmology as now accepted, because no one has thought of a way to adduce objective physical evidence that such an event really happened.
 

Offline JP

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Re: Is the Universe infinitely large?
« Reply #34 on: 30/07/2013 01:55:53 »
Quote from: JP
I was under the impression that if the universe was finite in size (even at the big bang) it would still have to be infinite in size, and if it was infinite in size at the moment of the big bang, it would still be infinite in size.

Assuming your first "infinite" should have been "finite"; this is a major part of the line of reasoning I have been trying to follow.  Namely that (physical) things that are finite, remain finite, and those that are infinite, remain infinite.  For a few ecstatic moments I thought Pete agreed with me, but then his “Although I said no at first I'd like to change my mind” dashed my euphoria.  Hopefully he will address your “impression”.



Ah yes, that should have been "infinite."  I'm always wary of answering questions based on pop-sci books/websites and I don't know enough general relativity to answer the question definitively.  But based on the pop-sci I have read, I haven't heard of a universe that's bounded (finite) becoming unbounded (infinite).  In my limited mathematics, I don't know how it could happen, but the topology involved in GR is a bit beyond me. 

As I've said before, I agree in spirit with your idea that "nothing finite can become infinite" but I disagree strongly with the statement itself.  If you want such a statement to hold, you have to be precise.  Pete gave the example of 1/(distance remaining) as you move between two points becoming infinite.  Ligharrow gave the example of a series with infinite terms that sums to a finite number.  These are certainly things.  Perhaps they're abstract things, but you'd better state what you're precluding from your rule before you can claim it holds in all cases.  I suspect that finding the exact dividing line of what it precludes will be very tricky.
 

Offline JP

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Re: Is the Universe infinitely large?
« Reply #35 on: 30/07/2013 01:58:15 »
Quote from: JP
The big bang is generally called a singularity ...
Were you aware that there is no event in the big bang which is actually called "the big bang"?

From Principles of Physical Cosmology by P.J.E. Peebles, Princeton University Press, 1993, page 6
Quote
If there were an instant, at a "big bang," when our universe started expanding, it is not in our cosmology as now accepted, because no one has thought of a way to adduce objective physical evidence that such an event really happened.

I was using it in the hand-wavy pop-sci way that it's the limiting case as time since the universe started -> 0.  My understanding (from pop-sci, as the equations are beyond me) is that this is a well defined limit, even though it's probably a non-physical state and it's unlikely that general relativity holds without quantum corrections.  If there's a good reason why taking that limit doesn't work, I'd love to learn more.
 

Offline Pmb

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Re: Is the Universe infinitely large?
« Reply #36 on: 30/07/2013 02:45:01 »
Quote from: JP
If there's a good reason why taking that limit doesn't work, I'd love to learn more.
Well my friend, I'll do my best to find out. I'm meeting a couple GR gurus this week and will bring it up. Anything in particular I should ask them or phrase the question?
 

Offline JP

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Re: Is the Universe infinitely large?
« Reply #37 on: 30/07/2013 03:24:03 »
I'd be interested to know what cosmology says as you take the limit T->0, where T is the age of the universe.  Does it predict an infinitely dense state we could tall "the big bang"?  (This is almost certainly non-physical, but does GR predict it?)  I've been led by pop-sci to believe that it does predict this "infinite density." 

But my biggest question, stated above, is whether a bounded universe could become unbounded or if bounded must remain bounded for all time and unbounded must remain unbounded for all time.  This ties into Bill's question on whether a finite universe can become infinite. 
 

Offline Pmb

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Re: Is the Universe infinitely large?
« Reply #38 on: 30/07/2013 13:30:28 »
I'd be interested to know what cosmology says as you take the limit T->0, where T is the age of the universe.  Does it predict an infinitely dense state we could tall "the big bang"?  (This is almost certainly non-physical, but does GR predict it?)  I've been led by pop-sci to believe that it does predict this "infinite density." 

But my biggest question, stated above, is whether a bounded universe could become unbounded or if bounded must remain bounded for all time and unbounded must remain unbounded for all time.  This ties into Bill's question on whether a finite universe can become infinite. 
I was talking to Alan Guth about this very thing. In cosmology as it is an unbounded universe cannot become bounded etc. If a universe is closed or open then they remain as such.
 

Offline Bill S

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Re: Is the Universe infinitely large?
« Reply #39 on: 30/07/2013 21:43:23 »
Quote
If there were an instant, at a "big bang," when our universe started expanding, it is not in our cosmology as now accepted, because no one has thought of a way to adduce objective physical evidence that such an event really happened.

Quote
Pete gave the example of 1/(distance remaining) as you move between two points becoming infinite.  Ligharrow gave the example of a series with infinite terms that sums to a finite number.

Absolutely right, undoubtedly mathematically sound, but has anyone adduced “objective physical evidence that such an event really happened”?
 

Offline Bill S

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Re: Is the Universe infinitely large?
« Reply #40 on: 30/07/2013 21:48:20 »
Quote from: JP
Ah yes, that should have been "infinite."

Tricky stuff, this infinity.  :)
 

Offline JP

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Re: Is the Universe infinitely large?
« Reply #41 on: 30/07/2013 23:42:47 »
Quote
Pete gave the example of 1/(distance remaining) as you move between two points becoming infinite.  Ligharrow gave the example of a series with infinite terms that sums to a finite number.

Absolutely right, undoubtedly mathematically sound, but has anyone adduced “objective physical evidence that such an event really happened”?


Now we're getting into the definition of "real," which is slippery.  But I can move from point A to point B, so the quantity 1/(remaining distance) diverges to infinity.  Similarly, the series is real--I can write it in notational form (though for obvious reasons I can't write it out explicitly). 

The problem is that there's a huge class of finite things which can't become infinite, so I agree with you for the most part.  But there's a smaller set of things that can, so I don't agree completely. 
 

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Re: Is the Universe infinitely large?
« Reply #41 on: 30/07/2013 23:42:47 »

 

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