Shape of Universe - What would a very long stick do

[CMaso]

if it were extended out from Earth, was perfectly straight, and could be any length desired? If I understand the prevailing theories it would either A) Just keep going forever (assuming infinite mass were possible), or B) Seem to travel in a straight line as far as we could tell, but eventually return to Earth from the opposite direction. Going with the popular "balloon surface" analogy, B seems the more likely of the two. 2-dimensional beings on the surface of a balloon would perceive its surface as a flat plane, and perceive their very long stick to be extending away in a straight line, but it would eventually go around the balloon and return to its point of origin.

 

[HiggsBoson1]

That is fascinating!

 

[PeterDonis]

Going with the popular "balloon surface" analogy, B seems the more likely of the two.

That analogy is misleading in this respect (and also in a number of other respects, which have been discussed in plenty of other threads in this forum). The current best-fit model of our universe says that it is spatially flat, which means the stick would just go on forever.

 

[Chalnoth]

And if it got large enough, it'd get ripped apart by the cosmological constant.

 

[timmdeeg]

The current best-fit model of our universe says that it is spatially flat, which means the stick would just go on forever.

Is the 3-torus which is spatially flat too already ruled out? But apart from that it is my impression that cosmologist indeed prefer the 3-plane.

 

[Chalnoth]

No, a 3-torus isn't ruled out. I don't think it can be.

 

[CMaso]

The shape of the universe is widely discussed on this forum and elsewhere -- my apologies for creating a new thread, I just wanted to approach the question from a different angle. It gets a little confusing when scientists describe the big bang, saying the universe expanded to be x wide in the first y seconds, as though the universe has some central point of origin, which it does not. Another way to ask the question might involve lowering the bridge rather than raising the river, so to speak -- if one were here on Earth observing this very long stick extending straight outward into space during a big crunch event, what would the stick be doing...would it still appear to be stretching on forever, even though all objects in the universe were in much, much closer proximity to each other? (assuming it were physically invulnerable to the immense heat and gravity...)

 

[marcus]

if it were extended out from Earth, was perfectly straight, and could be any length desired? If I understand the prevailing theories it would either A) Just keep going forever (assuming infinite mass were possible), or B) Seem to travel in a straight line as far as we could tell, but eventually return to Earth from the opposite direction. Going with the popular "balloon surface" analogy, B seems the more likely of the two. 2-dimensional beings on the surface of a balloon would perceive its surface as a flat plane, and perceive their very long stick to be extending away in a straight line, but it would eventually go around the balloon and return to its point of origin.

Hi Maso, welcome to PF!
I think that's a good basic "thought experiment" type question. to make it work you should imagine that you temporarily PAUSE expansion of distances (or contraction if distances happened to be shrinking).
Then what you find is there are two popular ideas of large-scale spatial geometry A) flat infinite and B) slight overall positive curvature, analogous to a balloon surface but 3d instead of 2d
So you freeze the geometry of space at a particular instant and A) you find there is no limit on how long and straight the stick can be, it goes "forever". OR you find that it is analogous to the balloon picture and B) the stick comes around and rejoins from the opposite direction.

There are other more complicated possibilities but those models of spatial geometry are probably the most commonly considered and cosmologists keep MEASURING the large scale spatial curvature in the hopes of getting a decisive answer. You might ask how they do it, sometime, there are clever ways to judge overall large scale curvature. but so far what they get is that SPATIAL curvature is either exactly zero or very small===so small that it is "as good as zero". I think lot of people would be willing to admit that it might be case B) with a very small curvature but they don't BOTHER to include it in calculations because all the calculations wouldn't change very much.

The other thing to mention is that if you don't pause the expansion process then it would wreck the stick.
If the stick were a fixed 14.4 billion LY long, then its tip end would find itself in local space that was getting farther away from us at speed c. So for the stick to remain intact that tip end would have to be moving towards us (thru its local surroundings) at speed c. but material things don't do that. so the tip end couldn't make it, and the stick would come apart.
14.4 billion LY is called the "Hubble radius". It is the size of distances which are growing at exactly the speed of light. Other cosmic scale distances are growing in proportion. One 7.2 billion LY long would be growing at speed c/2.

Anyway, welcome and keep asking questions :D

 

[timmdeeg]

-- if one were here on Earth observing this very long stick extending straight outward into space during a big crunch event, what would the stick be doing...would it still appear to be stretching on forever, even though all objects in the universe were in much, much closer proximity to each other? (assuming it were physically invulnerable to the immense heat and gravity...)

We talk about accelerated motion of those objects towards each other. Which means that there are tidal forces tending to stretch (during expansion) or to squash (during contraction) them. So, what happens to the stick seems to depend merely on its physical properties. In my opinion any thought-experiment should obey the physical laws und thus ideal rigidity of the stick isn't possible then. Note that tidal force is proportional to distance, so the longer the stick ... .

 

[CMaso]

Hi Maso, welcome to PF!
The other thing to mention is that if you don't pause the expansion process then it would wreck the stick.
If the stick were a fixed 14.4 billion LY long, then its tip end would find itself in local space that was getting farther away from us at speed c. So for the stick to remain intact that tip end would have to be moving towards us (thru its local surroundings) at speed c. but material things don't do that. so the tip end couldn't make it, and the stick would come apart.
14.4 billion LY is called the "Hubble radius". It is the size of distances which are growing at exactly the speed of light. Other cosmic scale distances are growing in proportion. One 7.2 billion LY long would be growing at speed c/2.
Anyway, welcome and keep asking questions :D

Thank you very much Marcus, and everyone, for your comments - great forum. :) I really mean the stick to be a virtual one; just a convenient gauge of what's happening to 3-d space. But as a follow-up question - I get that the tip end of a 14.4 billion LY-long stick would be stretching away from us at a rate of c, and eventually come apart, but then, what if someone were at the other end of that stick, pointing an identical stick back at us? Because of relativity, they would perceive *us*, and the tip of their stick, to be stretching away from them at c. So would the stick(s) start coming apart at their end, or ours?

 

[Jorrie]

But as a follow-up question - I get that the tip end of a 14.4 billion LY-long stick would be stretching away from us at a rate of c, and eventually come apart, but then, what if someone were at the other end of that stick, pointing an identical stick back at us? Because of relativity, they would perceive *us*, and the tip of their stick, to be stretching away from them at c. So would the stick(s) start coming apart at their end, or ours?

I think a hypothetical "ideal stiffness stick" of less than the Hubble length will experience a stretching force (in our present universe) between its two ends that depends on both its proper length and the deceleration parameter (q = Ω_m/(2a³) - Ω_Λ). It does not matter to which end ("ours" or "theirs") it is anchored. One cannot say where it will break; I suppose an 'ideal stick' will just be breaking up into many smaller pieces over its length.

 

[TEFLing]

I think you would need to calculate the deviation from geodesic motion for the two ends relative to the middle

Then the geodesic equation would tell you the required force

 

[Jorrie]

I think you would need to calculate the deviation from geodesic motion for the two ends relative to the middle

Then the geodesic equation would tell you the required force

That would be one method, yes. Davis et. al give another approach in their "Tethered Galaxy problem", which is what I based my comment on.

 

[CMaso]

This brings up another question which I haven't found an answer for anywhere online yet -- relative to one's point of reference, how does one distinguish if objects are moving through space vs. moving *with* space as it expands?

 

[Jorrie]

This brings up another question which I haven't found an answer for anywhere online yet -- relative to one's point of reference, how does one distinguish if objects are moving through space vs. moving *with* space as it expands?

It is rather difficult to determine the peculiar velocities of distant galaxies. A good description of the problem and methods is given by Jeffrey Willick of Stanford in MEASUREMENT OF GALAXY DISTANCES: http://ned.ipac.caltech.edu/level5/Willick/Willick_contents.html, specifically
"1.1. Peculiar Velocities versus H0"

 

[TEFLing]

A light bridge between galaxies, composed of photons ( say in a standing wave ), would redshift with expansion so as to keep in connection, yes?

 

[timmdeeg]

This brings up another question which I haven't found an answer for anywhere online yet -- relative to one's point of reference, how does one distinguish if objects are moving through space vs. moving *with* space as it expands?

One can tell that the velocities of galaxies belonging to the local group are obeying Special Relativity, because those galaxies being gravitationally bound can be treated as moving in flat space-time. Thus they don't 'feel' any expansion.
Farther away both, peculiar velocities and influence of expansion will contribute to the observed redshift. However it seems very difficult to distinguish one from the other, if that is possible possible at all.
Regarding galaxies in cosmological distances peculiar velocities are negligible.

 

[Tanelorn]

If the stick were fixed at our end then space at other end of the stick would be hurtling away from that end at superluminal velocities and such local velocities are not allowed.
Perhaps we are not allowed to something like this except perhaps as a thought experiment to demonstrate the fact.

 

[timmdeeg]

If the stick were fixed at our end then space at other end of the stick would be hurtling away from that end at superluminal velocities and such local velocities are not allowed.

In that case the other end of the stick would move with superluminal velocity relative to the CMB. That isn't forbidden, provided the stick survives its length.
You have a similar situation, if you imagine a stick dipped into a black hole.

 

[PeterDonis]

In that case the other end of the stick would move with superluminal velocity relative to the CMB

No, it wouldn't. Each point of the stick would be moving slower than light, relative to the CMB in its local vicinity.

You have a similar situation, if you imagine a stick dipped into a black hole.

No, you wouldn't. You would find that the stick would have to either fall into the hole or break; but in either case, each point of the stick would be moving slower than light, relative to light in its local vicinity.

 

[Nugatory]

In that case the other end of the stick would move with superluminal velocity relative to the CMB. That isn't forbidden, provided the stick survives its length.
You have a similar situation, if you imagine a stick dipped into a black hole.

In both situations you describe, there's no "provided the stick survives". It will break, as the stress in the stick increases without bound as the velocity of the far end approaches $c$ relative to its local surroundings.

 

[timmdeeg]

No, it wouldn't. Each point of the stick would be moving slower than light, relative to the CMB in its local vicinity.

Yes, agreed and thanks. Perhaps one should distinguish between real and imagined, regarding the stick in this discussion. The length of a real stick grows with a velocity $< c$ and therefore remains shorter than Hubble length. Whereas an imagined stick has arbitrary length and besides talking about proper distance speculations regarding the velocity of his end are of no use. Would this make sense?

 

[timmdeeg]

In both situations you describe, there's no "provided the stick survives". It will break, as the stress in the stick increases without bound as the velocity of the far end approaches $c$ relative to its local surroundings.

Well, I think that the far end of a "physically real stick" can't approach $c$ and whether it gets broken or not depends on material properties and tidal forces. In my opinion (today) it makes no sense to discuss an imagined stick in a physical context like this one.

 

[Tanelorn]

I meant the stick being long enough to reach space moving away from us at superluminal velocities..

 

[timmdeeg]

I meant the stick being long enough to reach space moving away from us at superluminal velocities..

Yes, understood. Therefore I started reasoning about the 'nature' of that stick.

 

[PeterDonis]

The length of a real stick grows with a velocity $< c$ and therefore remains shorter than Hubble length.

It depends on how long the stick is, and how strong the inter-atomic forces in the stick are. It's quite possible for one end of the stick to be moving "faster than $c$" relative to the other end of the stick, if the stick is long enough. But this "relative speed" is not a relative velocity in the sense of Special Relativity, so there's nothing preventing it from being faster than $c$. A light beam emitted at one end of the stick, in the direction away from the other end, would recede faster than the end of the stick itself does--so the light beam would also be moving "faster than $c$" relative to the other end of the stick. (See further comments below.)

The important "relative speed" in determining how the stick behaves is the relative speed between neighboring parts of the stick, parts close enough together that a single local inertial frame can cover them both. This relative speed will always be less than $c$--how much less depends, again, on how strong the inter-atomic forces in the stick are. Those forces have to resist some amount of tidal gravity, due to the universe's expansion, that is "trying" to pull the neighboring pieces of the stick apart. If the forces are strong enough, the neighboring pieces of the stick will not move apart at all--they will just experience some internal stress. In this limiting case, then the opposite ends of the stick will not be moving apart either; the proper distance between the ends will be constant--because each small piece of the stick is keeping a constant proper distance from neighboring pieces, and the proper length of the stick as a whole is just the sum of all those small proper distances between neighboring pieces.

Note that this means that "comoving" objects at either end of the stick, objects which are moving with the expansion of the universe, will be moving away from each end of the stick. It also means that the stress on a given piece of the stick will get larger as you move towards either end (it will be zero at the center of the stick). So there is a limit in this case on how long the stick can be--basically it can't be equal to the Hubble length (or twice the Hubble length, i.e., the stick's diameter cannot span the Hubble sphere), because if it were, the stress at the ends of the stick would be infinite.

The other limiting case is that in which the forces between neighboring pieces of the stick go to zero--the inter-atomic bonds are so weak that they have a negligible effect on the worldlines of each individual piece of the stick. In this case, each individual piece of the stick will follow a "comoving" worldline--i.e., it will move along with the "flow" of the universe's expansion in its vicinity. In this case, the opposite ends of the stick can indeed be moving apart "faster than $c$", as described in my first paragraph above, if the stick is long enough. There is no limit in this case on the length of the stick (but of course calling it a "stick" in this case is kind of a misnomer, since it does not behave like a single object, it's just a collection of particles).

The case of interest is an intermediate case between these two--neighboring pieces of the stick are moving apart, but not as fast as "comoving" worldlines would, i.e., the inter-atomic forces do affect the motion of the pieces of the stick, but not enough to keep neighboring pieces the same proper distance apart. In this case, the stick does behave more or less like a coherent object, but an elastic one--it's more like a rubber band than a stick, getting stretched as the universe expands. In this case, the opposite ends of the "band" can be more than a Hubble diameter apart, and can be moving faster than $c$ relative to each other (but note that, at the Hubble diameter, they won't be, because the pieces of the stick there are not following "comoving" worldlines). There will still be (I think) a limit on how long the stick can be, the limit will just be larger than the Hubble diameter. (I do not, however, think this limit will be the same as the point where the ends of the stick are "moving at $c$" relative to each other. I haven't had time to do a calculation to confirm this, though.)

 

[Tanelorn]

I meant an extremely long, infinitely stiff, stick. I guess it is very hypothetical :)

What force would be acting on the end in which space is moving near c or superluminal away from us?

 

[PeterDonis]

an extremely long infinitely stiff stick.

There's no such thing; relativity places a finite limit, even in principle, on the structural strength of materials. One way of stating it is that the speed of sound in the material can never be greater than the speed of light. (An "infinitely stiff" material would have an infinite sound speed.)

 

[Tanelorn]

There's no such thing; relativity places a finite limit, even in principle, on the structural strength of materials. One way of stating it is that the speed of sound in the material can never be greater than the speed of light. (An "infinitely stiff" material would have an infinite sound speed.)

What is the force which would be acting on the end of the stick in which space is moving near c or superluminal away from us?
Assuming empty space.

 

[PeterDonis]

What is the force which would be acting on the end of the stick in which space is moving near c or superluminal away from us?

Whatever force is produced by the interactions between the end of the stick and the neighboring piece of the stick. "Space moving" doesn't exert a force on anything.

 

[Tanelorn]

So why does the stick need to be stiff? Consider an extendable stick like a telescopic antenna being extended from earth, how would space eventually tear it apart?

 

[PeterDonis]

Consider an extendable stick like a telescopic antenna being extended from earth, how would space eventually tear it apart?

It wouldn't. As I said, "space moving" doesn't exert a force on anything. If you could somehow make a telescopic antenna that could extend for fifty billion light years, the expansion of space wouldn't stop it from working.

 

[Jorrie]

What is the force which would be acting on the end of the stick in which space is moving near c or superluminal away from us?
Assuming empty space.

Google "Davis Tethered Galaxy problem", where eqs. 12 to 15 give the tidal accelerations between any two free particles over a distance D in empty space. From this you may be able to calculate the stresses on your hypothetical "stick".

My 2cent is that the ends of the stick will always 'join the Hubble flow', minus some local speed differential (< c), depending on the stiffness of the stick against stretching. This means that the two ends can in principle be farther away from each other than the Hubble radius, as Peter has already stated.

If you had the "infinite telescopic stick", it does not have to break at any distance, but if there is some resistance to extending, it would not quite follow the Hubble flow. Does this make sense?

 

[Tanelorn]

Jorrie, Thanks for the tethered galaxy problem paper. Unfortunately I am still not getting this. Is this because the thought experiment is just not valid?

All I am trying to understand is how do we deal with the relative superluminal speeds between the stick and the space and matter at the far end of the stick, with our end of the stick firmly tethered here on earth? The stick is stiff so something has to give right?

 

[PeterDonis]

how do we deal with the relative superluminal speeds between the stick and the space and matter at the far end of the stick

By recognizing that this "relative speed" does not work the way you are thinking it does. As I said in a previous post, it is not a "relative velocity" in the sense of SR. And it is not a "relative speed" that is exerting a force on anything. It's just a coordinate velocity in comoving coordinates. Thinking of it as though it were driving any of the physics will only cause confusion.

The stick is stiff so something has to give right?

Not necessarily. See above and my previous posts.