The thread thread: Strangeness of the expanding space paradigm

[Hurkyl]

Also, I get the feeling the basis of your argument is based on something like:

"I can't figure out how to answer this qualitative question with pure logic, therefore there must be a problem."

But physical theories are strongly quantitative. If you put numbers to everything in the problem, you could (in principle) turn the numerical crank to get the answer.

 

[Zanket]

To address this issue without the math, it's already been argued that a very long string must break in an expanding universe, when it passes the Hubble horizon.

The Hubble horizon is everywhere. The spot you are at is on the Hubble horizon for some hypothetical observer. Saying it must break there is saying it must break everywhere.

 

[Zanket]

"outside the scope of this topic", you say. "start a new thread", you say. As I understand it, the entire question is on how a thread (and all it inter-molecular parts) would respond to expanding space. But if those molecular parts are extended objects, then this is exactly equivalent to asking how those extended objects, strings, respond to expanding space.

The questions I pose here are as to whether the expanding space paradigm is consistent, i.e. free of self-contradiction. That should be determinable without including string theory in the discussion.

 

[Zanket]

The Hubble constant is an observed quantity. It's value is determined using General Relativity... I can't really make sense of just what you're trying to suggest in this paragraph.

Chronon cleared it up some. What still doesn’t make sense is, why is the expanding space paradigm attributed to general relativity (as in the quote from the book, above), which pre-dated the paradigm’s creation by 15 years? My understanding is that the paradigm implies that a floating intergalactic thread will break due to expanding space. Was that really predicted by GR, and if so, why was there a need to create a paradigm 15 years later?

Maybe it would do you better to consider individual particles. Consider a two-particle system, and then a three-particle system, and so on, to get an idea what's going on.

I had done that. Take particles A and B next to each other as AB. The space between them expands to try to make them become A__B, but the binding force between them makes the particles slip past the expanding space so that the system becomes _AB_. Likewise, expanding space tries to make ABC become A__B__C, but the binding force keeps it __ABC__. And so on. But for a very long system, that obviously doesn’t work. It seems that an intergalactic thread must break. The problem with the paradigm seems to be related to the fact that the slippage must occur toward some direction, but the direction is arbitrary, hence nonsensical, allowing a paradox to be created. I think I gave a better example of that above.

 

[Zanket]

If you put numbers to everything in the problem, you could (in principle) turn the numerical crank to get the answer.

A theory can be self-contradictory and still give an answer. Then to determine self-contradiction, you cannot rely on the math. I'm trying to determine whether the paradigm is self-contradictory. So far, it seems to be. So far as has been determined (I haven't got a reasonable answer as to why not), the floating intergalactic thread breaks at an arbitrary spot and the new ends fly apart, and the length of the thread is irrelevant to that conclusion. Then there is no explanation as to why a galaxy, held together only by the weaker force of gravity, does not break apart and the pieces fly apart.

 

[Hurkyl]

Take particles A and B next to each other as AB.

But they aren't next to each other -- remember that intermolecular forces are also repulsive when they're too close.

Their natural equilibrium state might be A...B. When placed in a region of expanding space, they would settle into A...B. (Actually, the difference wouldn't even be that large, but the separation would settle to something slightly larger than "normal")

But for a very long system, that obviously doesn’t work.

Why is it obvious?

It seems that an intergalactic thread must break.

If you mean a thread whose ends are anchored to galaxies, you would be correct. (I'm assuming that the string won't be strong enough to actually keep the galaxies from being carried along with expansion -- I have no idea just how much tension would be generated)

The problem with the paradigm seems to be related to the fact that the slippage must occur toward some direction, but the direction is arbitrary, hence nonsensical,

This is what I meant by my comment in post #31. If you precisely set down the scenario, then you could simply work through the kinematics and determine what happens.

The fact is, the problem you've specified is very vague, and there are any number of things that could happen depending on the precise details of the problem. That is not nonsensical, nor paradoxical.

A theory can be self-contradictory and still give an answer. Then to determine self-contradiction, you cannot rely on the math.

That is 100% wrong. Being self-contradictory means precisely that if you work through the math of one problem in two different ways, you get answers that are not compatable. You can't talk about self-contradiction of a mathematical theory without doing math.

 

[pervect]

The Hubble horizon is everywhere. The spot you are at is on the Hubble horizon for some hypothetical observer. Saying it must break there is saying it must break everywhere.

You are missing the point. If you can't get a light beam from point A to point B, because there is a horizon in between A and B, you can't have a string connecting A and B. The exact form or location of the horizon doesn't matter to the argument - whether it is the Hubble horizon, a black hole horizon, or a Rindler horizon is irrelevant.

To assume that there could be a string connecting A to B yields a contradiction. A string has the following characteristic - when you pull on either end of the string, the other end of the string moves (not instantly, but delayed by the speed of sound in the string).

Now, if light cannot go from A to B because there is a horizon in the way, they are causally disconected.

Therefore there cannot be an intact string connecting them either - when you pull on the string at A, the signal cannot reach B. What happens phyiscally is that the string breaks. (It didn't break from the pull - it broke when you tried to first stretch it from A to B).

Another way of thinking about this is that a light beam is the strongest possible string - it's a string that's so strong that the speed of sound in the string is the maximum velocity possible in the universe, the speed of light. If you can break a light beam, any weaker string must necessarily fail.

 

[pervect]

There is something I should probably point out.

To summarize, it's true that there are no tidal forces when an expanding universe is totally empty (i.e. it contains no energy or matter). But there isn't really any universe in this case. It's only when the universe actually contains matter that one finds that there are tidal forces due to it's expansion.

The details:

The metric for the flat-FRW space-time is

ds^2 = a(t)^2 (dx^2+dy^2+dz^2) - dt^2

The tidal force in terms of the above parameters as I mentioned in another thread is

-\frac{\ddot{a}}{a}

This is computed directly from the Riemann from the metric above.

This tidal force isotropic, the same in all directions (it's actually an acceleration per unit length, so it has units of 1/sec^2).

Now let's look at the the solution for a(t).

First, let's look at a totally empty universe.

When there is no matter in the universe, we have the boring solution a(t) = k*t, and \ddot a is zero, therefore there are no tidal forces. But there is no universe, either, really - it's just empty space-time.

Things get a lot more interesting when our universe has matter in it. To make it easy, let's assume there is basically no pressure, i.e. the expansion is matter dominated, and there is no radiation pressure.

The pressure term must be zero, which leads to the equation

\dot{a}^2+2a\ddot{a} = 0

(take my word on this, or if you really want to, look up or compute the Einstein tensor for the metric above, and remember that G_ab = 8*pi*T_ab, and we are assuming that T_ab has no pressure terms)

This has the solution

a(t) = t^(2/3)

The matter density is now non-zero, and equal to

3 (\frac{\dot{a}}{a})^2

And, we now have tidal forces, because -\frac{\ddot{a}}{a} is nonzero as well.


So the tidal forces are not appearing out nowhere or in any way "mysterious" - from one point of view, they are due to the gravitational interaction of the "string" with the rest of the universe.

 

[Garth]

Can you elaborate? What is SCC?

Self Creation Cosmology

One error in a previous post of mine was that in the static Jordan conformal frame of SCC, in which particle masses increase exponentially (exp(Ht)) and atomic diameters shrink exponentially (exp(-Ht)), the cosmic thread does not increase in length, but as its atoms shrink it will still break. The cosmic string breaks in both the Einstein and the Jordan conformal frames of measurement.

Garth

 

[Mike2]

There is something I should probably point out.

To summarize, it's true that there are no tidal forces when an expanding universe is totally empty (i.e. it contains no energy or matter). But there isn't really any universe in this case. It's only when the universe actually contains matter that one finds that there are tidal forces due to it's expansion.

Thanks for the calculations. I'm wondering what happens as galaxies leave the cosmological event horizon? If the galaxies left can no longer feel the gravitation force of those galaxies that have disappeared behind the cosmological event horizon, then is the FRW model affected by this loss of matter? Does this loss of matter further accelerated the expansion? Thanks.

 

[pervect]

Thanks for the calculations. I'm wondering what happens as galaxies leave the cosmological event horizon? If the galaxies left can no longer feel the gravitation force of those galaxies that have disappeared behind the cosmological event horizon, then is the FRW model affected by this loss of matter? Does this loss of matter further accelerated the expansion? Thanks.

I'm not sure what sort of experiment would exactly answer this question.

However, if you accept the case of a black hole forming as an example (an example of matter going out of sight behind a horizon), you can see that the gravity before the object collapsed into the black hole is the same as the gravity after the collapse.

The sci.physics.faq "how does the gravity get out of a black hole" goes into this a little more - people who are overly attached to the "graviton" point of view get confused by this question a lot, people with either a field-oriented view or a geometrical view don't have any problem with the gravity existing after the object has passed beyond the event horizon. The philosophical explanations vary somewhat, but everyone agrees that information can't get out of a black hole, while gravity (and the electrostatic field) can.

 

[chronon]

You are missing the point. If you can't get a light beam from point A to point B, because there is a horizon in between A and B, you can't have a string connecting A and B. The exact form or location of the horizon doesn't matter to the argument - whether it is the Hubble horizon, a black hole horizon, or a Rindler horizon is irrelevant.

No, no, no. The Hubble sphere is not a horizon. See http://www.chronon.org/Articles/cosmichorzns.html, or if you don't believe me then read the Lineweaver & Davis article http://xxx.arxiv.cornell.edu/abs/astro-ph/0310808 .

 

[chronon]

That again implies to me that a floating intergalactic thread will not physically stretch. But if not, then a paradox arises in determining in which direction relative to one or both galaxies (on either end of the thread) will move, given that no force pushes the thread in a particular direction.

Where do you find a dog with no legs?
My assumption is that the thread is initially put into place to be stationary with respect to some galaxy, and the question is then 'Does it stretch because of the expansion of the universe'. My assertion is that it does not (Rather it begins to contract).

I recommend that you read the relevant part of Ned Wright's FAQ
http://www.astro.ucla.edu/~wright/cosmology_faq.html#MX , in particular the second paragraph

In the absence of the cosmological constant, an object released at rest with respect to us does not then fly away from us to join the Hubble flow. Instead, it falls toward us

Of course if you try to pull the ends of the thread apart, or attach it to two objects which are moving apart then it is likely to break, but this is nothing to do with cosmology.

 

[Zanket]

But they aren't next to each other -- remember that intermolecular forces are also repulsive when they're too close.

I think the question as to whether the paradigm is consistent can be answered without going into that level of detail.

Why is it obvious?

Because, for example, if the galaxies near either end of a floating thread are receding from each other due to expanding space between them, the ends of the thread presumably stay at rest with respect to either galaxy (no force exists to move the ends of the thread in any particular direction), and presumably the thread cannot expand to any length without breaking.

If you mean a thread whose ends are anchored to galaxies, you would be correct. (I'm assuming that the string won't be strong enough to actually keep the galaxies from being carried along with expansion -- I have no idea just how much tension would be generated)

Or not anchored, as above.

This is what I meant by my comment in post #31. If you precisely set down the scenario, then you could simply work through the kinematics and determine what happens.

The fact is, the problem you've specified is very vague, and there are any number of things that could happen depending on the precise details of the problem. That is not nonsensical, nor paradoxical.

I may not be putting the query in the best way (discussion help me improve that), but it’s not vague to me. It should not take any math to figure out the basic result for the intergalactic thread (like “it breaks” or “it moves relative to one or both galaxies” or “it stretches forever”). Presumably others cranked out such answers long ago (if that was even necessary, since the paradigm was created to match observation) and wrote about it. Lots of what I’ve read suggests that the thread will break, which seemingly leads to a paradox, as I noted. And the other two possibilities have problems.

That is 100% wrong. Being self-contradictory means precisely that if you work through the math of one problem in two different ways, you get answers that are not compatable. You can't talk about self-contradiction of a mathematical theory without doing math.

That’s a good point. I was talking about one answer. I don’t need to do the math myself, though, if someone else already wrote about it (like “it breaks” or “it moves relative to one or both galaxies” or “it stretches forever”).

 

[Zanket]

You are missing the point. If you can't get a light beam from point A to point B, because there is a horizon in between A and B, you can't have a string connecting A and B. The exact form or location of the horizon doesn't matter to the argument - whether it is the Hubble horizon, a black hole horizon, or a Rindler horizon is irrelevant.

I don’t see how that can be true in the case of a Hubble horizon. I’m on some hypothetical observer’s Hubble horizon as I write this. Am I not?

Now, if light cannot go from A to B because there is a horizon in the way, they are causally disconected.

But light can cross a Hubble horizon, by which I’m assuming you mean the surface of a Hubble sphere. My Hubble sphere is my observational limit. You have your own Hubble sphere. Whereas a black hole’s event horizon, say, is the same observational limit for everyone.

Another way of thinking about this is that a light beam is the strongest possible string - it's a string that's so strong that the speed of sound in the string is the maximum velocity possible in the universe, the speed of light. If you can break a light beam, any weaker string must necessarily fail.

That’s a good way of thinking in the case of a black hole’s event horizon, say (so long as you're careful; a thread can cross an event horizon intact so long as it's falling). But I don’t see it applying to a Hubble horizon. And why involve a Hubble horizon in this case at all, when galaxies fast-receding from each other due to expanding space exist within our Hubble sphere? We can put the thread in between those galaxies to answer our questions.

 

[Zanket]

My assumption is that the thread is initially put into place to be stationary with respect to some galaxy, and the question is then 'Does it stretch because of the expansion of the universe'. My assertion is that it does not (Rather it begins to contract).

I recommend that you read the relevant part of Ned Wright's FAQ
http://www.astro.ucla.edu/~wright/cosmology_faq.html#MX , in particular the second paragraph

Ned Wright’s FAQ, which I’ve spent a lot of time on, including this question, seems to dance around issues rather than get to the point, leading to confusion. For example, you took from the FAQ that the thread does not stretch. But consider… What happens to a thread released at rest with respect to both of two galaxies fast-receding due to space expanding between them? That is, either end of the thread is at rest with respect to a respective galaxy. That’s the intergalactic thread example I’ve used above. According to the FAQ, I presume that either end falls toward its respective galaxy, which means that the thread physically stretches, eventually to a breaking point. Then the paradox aforementioned arises.

 

[Zanket]

I think I can now better put the paradox that I gave above. I think no reasonable resolution has been given so far:

According to the expanding space paradigm of cosmology, the galaxies do not expand along with the intergalactic expanding space because gravity holds them together. Let a floating thread span the distance between two galaxies receding from each other due to space expanding between them. The ends of the thread are not anchored to their respective galaxies. According to Ned Wright's Cosmology FAQ here, either end falls toward its respective galaxy. Then the thread physically stretches. Presumably the thread cannot stretch forever, so eventually it must break. The thread breaks at an arbitrary spot (if you disagree, then tell me, at what spot does an infinitely long thread break?) and the pieces fly apart. Even the strongest binding force of the thread is not strong enough to keep the thread intact. The galaxies are not in expansion-free zones, and they exist in arbitrary spots. Then how can gravity, a binding force far weaker than the strongest binding force, keep the galaxies from breaking and the pieces flying apart?

 

[chronon]

OK, so let's ignore cosmology for a moment. The initial state of the thread is one of uniform expansion, and so the tension will be increasing, creating a force towards the centre of the thread, counteracting the expansion. Whether the thread breaks depends on whether the tension manages to stop the expansion before it reaches breaking point.

So what happens when we include cosmological effects. There will now be three forces.

1) Tension
2) Gravitational effects due to the matter in the universe. (Which is assumed to be evenly distributed)
3) A stretching effect due to dark energy/non-zero cosmological constant.

I'm ignoring (3). To start with (2) will be zero. However due to tension, each part the thread will begin to lag behind the surrounding matter (except for the centre). The important point to note is that each point will now be stationary with repect to some part of the universe nearer to the centre of the thread, and gravity will pull it towards that point, rather than causing it to catch up with the surrounding matter. Hence the thread will stretch less than in the non-cosmological case, and so is less likely to break.

 

[chronon]

Note that q is a negative number for the numbers quoted, as far as sign issues go.

q is a negative number (-0.6) according to recent measurements, but this needs a positive cosmological constant to explain it. If the cosmological constant is zero then q>=0. For instance, for the critical density model, where a=t^(2/3) you have q=1/2.

 

[pervect]

OK, so let's ignore cosmology for a moment. The initial state of the thread is one of uniform expansion, and so the tension will be increasing, creating a force towards the centre of the thread, counteracting the expansion. Whether the thread breaks depends on whether the tension manages to stop the expansion before it reaches breaking point.

So what happens when we include cosmological effects. There will now be three forces.

1) Tension
2) Gravitational effects due to the matter in the universe. (Which is assumed to be evenly distributed)
3) A stretching effect due to dark energy/non-zero cosmological constant.

I'm ignoring (3). To start with (2) will be zero. However due to tension, each part the thread will begin to lag behind the surrounding matter (except for the centre). The important point to note is that each point will now be stationary with repect to some part of the universe nearer to the centre of the thread, and gravity will pull it towards that point, rather than causing it to catch up with the surrounding matter. Hence the thread will stretch less than in the non-cosmological case, and so is less likely to break.

I'm not sure I follow your logic here, but I think I agree with the conclusion, which I take to be:

If (contrafactually) we had no cosmological constant, the deceleration parameter q would be positive, and our string would be in compression rather than tension.

[add]
In fact I get q=+.5 with no cosmological constant, though I haven't double checked my calculations.
[end]

As was pointed out in another thread by SpaceTiger

https://www.physicsforums.com/showthread.php?t=76405

in the same contrafactual case (no cosmological constant), there would be no event horizon - anyone would eventually be able to see the whole universe, if they waited long enough - so we don't have the impossibility of a string going through an event horizon.

 

[Hurkyl]

I think the question as to whether the paradigm is consistent can be answered without going into that level of detail.

I think these details are important. It explains at the microscopic level exactly what is happening with the thread. Especially so, because of the confusion you're having with the macroscopic analysis.

At the microscopic level, it seems clear -- in free space, each particle can be in equilibrium by remaining a fixed distance from each other. (As long as the rate of expansion remains constant)


If the ends are in the gravitational well of galaxies, the system cannot remain in equilibrium while the ends remain fixed with respect to the galaxies.

There are lots of ways things could be. The ends could be accelerating into the galaxies, because gravity is overcoming tension (Though, any dust next to the end of the thread would fall faster into the galaxy), and the string will eventually break someplace, or even multiple places. Where and how many depends on the actual conditions.

The ends could be accelerating away from the galaxies as tension overwhelms gravitational force, yet the string could still be expanding everywhere, and break.

Either of the above could occur when the ends start at rest with the galaxies.

Or, the string could start off by expanding everywhere, but the tension forces overwhelm gravity fast enough and the string pulls itself into an equilibrium in free space.

Or, the string could start off contracting everywhere, even though the ends begin at rest with respect to the galaxies.

Or...


In particular:

(no force exists to move the ends of the thread in any particular direction)

This statement is patently incorrect.

(1) If there were no forces, the ends of the thread would fall into the galaxy, as they traveled along a geodesic. (In particular, if the thread were made of dust instead of interacting particles, that is precisely what would happen)

(2) There are forces. The end particles are attracted to the next-to-end particles. (Assuming the string doesn't start in a compressed state)

 

[Zanket]

OK, so let's ignore cosmology for a moment. The initial state of the thread is one of uniform expansion, and so the tension will be increasing, creating a force towards the centre of the thread, counteracting the expansion. Whether the thread breaks depends on whether the tension manages to stop the expansion before it reaches breaking point.

I can buy that logic for a scenario involving tugging on the ends of the thread, but it doesn’t seem relatable to the expanding space paradigm, in which the expansion force is equal in all directions presumably even at a subatomic level. Space expands between the teeniest of adjacent particles, pushing them apart with equal force in all directions. There’s no excess (non-cancelled) force to transmit along the thread. This applies to your application of the logic:

I'm ignoring (3). To start with (2) will be zero. However due to tension, each part the thread will begin to lag behind the surrounding matter (except for the centre). The important point to note is that each point will now be stationary with repect to some part of the universe nearer to the centre of the thread, and gravity will pull it towards that point, rather than causing it to catch up with the surrounding matter. Hence the thread will stretch less than in the non-cosmological case, and so is less likely to break.

I addressed this thought process above, when I ask, “at what spot does an infinitely long thread break?” According to the paradigm, about which I am much clearer after browsing some books today, an infinitely long thread will break eventually, and of course it has no center. That the thread breaks at an arbitrary point means that all points on the thread are equivalent.

 

[Zanket]

Self Creation Cosmology

Keeping in mind I’m a layman... It seems a good paper on its face. At least it addresses experimental tests of GR, the first thing I looked for. Assuming you wrote it, why not include a section where you plug in the values into the equations to show that they do indeed pop out 42.98 arc seconds for Mercury, 1.75 arc seconds for light deflection by the Sun, etc.? That would save serious readers significant time in reproducing that. If you did that and also as well to reproduce the value of the anomaly of the Pioneer craft to within some tiny margin, that would be a more obvious coup.

 

[Garth]

Keeping in mind I’m a layman... It seems a good paper on its face. At least it addresses experimental tests of GR, the first thing I looked for. Assuming you wrote it, why not include a section where you plug in the values into the equations to show that they do indeed pop out 42.98 arc seconds for Mercury, 1.75 arc seconds for light deflection by the Sun, etc.? That would save serious readers significant time in reproducing that. If you did that and also as well to reproduce the value of the anomaly of the Pioneer craft to within some tiny margin, that would be a more obvious coup.

You will find all those calculations in the following paper:
http://www.kluweronline.com/oasis.htm/5092775

or free eprints:
gr-qc/0212111 ] The Principles of Self Creation Cosmology and its Comparison with General Relativity[/URL]

gr-qc/0302026 ]Experimental tests of the New Self Creation Cosmology and a heterodox prediction for Gravity Probe B[/URL]

gr-qc/0302088 ]The derivation of the coupling constant in the new Self Creation Cosmology[/URL]

astro-ph/0401136] The Self Creation challenge to the cosmological concordance model[/URL]

and preprint:
gr-qc/0405094] Self Creation Cosmology - An Alternative Gravitational Theory [/URL]

The Pioneer anomaly is not a 'clean' measurement of whatever it is detecting. The various possible causes, including out gassing and an anisotropic radiation field, although they cannot explain the entire anomaly probably each do contribute something to it. gr-qc/0310088] Can conventional forces explain the anomalous acceleration of Pioneer 10/11?[/URL]

The clock-drift explanation offered by SCC predicts a value of exactly cH or 7.0 +/- 0.1 x 10^-8 cm.sec^-2 as compared with the observed value of 8.74 +/- 1.3 x 10^-8 cm.sec^-2.

Garth

 

[Zanket]

The clock-drift explanation offered by SCC would predict a value of exactly cH or 7.0 +/- 0.1 x 10^-8 cm.sec^-2 as compared with the observed value of 8.74 +/- 1.3 x 10^-8 cm.sec^-2.

Impressive! Given that I'm a layman, is there a layman's synopsis by an independent party yet?

 

[chronon]

I can buy that logic for a scenario involving tugging on the ends of the thread, but it doesn’t seem relatable to the expanding space paradigm, in which the expansion force is equal in all directions presumably even at a subatomic level. Space expands between the teeniest of adjacent particles, pushing them apart with equal force in all directions. There’s no excess (non-cancelled) force to transmit along the thread.

I think you need to be clear about the difference between the expansion of the universe and the acceleration of that expansion.

A history of the cosmological constant
1) 1916ish: Einstein introduces the cosmological constant to give a static universe
2) Early 1930's: Hubble finds the universe is expanding, and the cosmological constant is unnecessary.
3) 1950ish: The age of the universe predicted from the expansion is less than that of the earth. Reintroducing the cosmological constant increases the predicted age of the universe
4) 1970ish: More accurate values for the Hubble constant mean that the cosmological constant is no longer needed to explain the age of the universe
5) 1980ish: The most distant objects detected are traveling away faster than light in the normal cosmological coordinate system. The "expanding space" paradigm is introduced to deal with this. I think that this was a mistake.
6) late 1990's: The expansion of the universe is found to be accelerating, implying a non-zero cosmological constant.
(Of course there might be an era (7) in which the cosmological constant is thought to be zero again)

As you talk about the expanding space paradigm, I have based what I say on era (5). If you are talking about what we know now - era (6) - then you should talk about acceleration of expansion, or the effect of dark energy or a positive cosmological constant - not about the effect of expanding space.

 

[chronon]

I addressed this thought process above, when I ask, “at what spot does an infinitely long thread break?” According to the paradigm, about which I am much clearer after browsing some books today, an infinitely long thread will break eventually, and of course it has no center. That the thread breaks at an arbitrary point means that all points on the thread are equivalent.

But you don't need expanding space, or cosmology to tell you that. Newtonian mechanics will tell you that an infinitely long thread which is set in motion so as to be expanding uniformly is bound to break.

 

[Zanket]

A history of the cosmological constant

Good history lesson providing fodder for more research. Thanks! My books sometimes mislead me. One book says about the expanding space paradigm, “Slowly [beginning early 1930s] emerged the idea that the universe consists of expanding space!” With further reading in other books I come to find out this really means only (2), which doesn’t seem like much of a theoretical discovery to me (observational, yes). None of my books have this complete history you gave.

As you talk about the expanding space paradigm, I have based what I say on era (5). If you are talking about what we know now - era (6) - then you should talk about acceleration of expansion, or the effect of dark energy or a positive cosmological constant - not about the effect of expanding space.

Definitely not (6). I’ve been talking about (5), and didn’t realize it differed from the 1930s outlook (2).

But you don't need expanding space, or cosmology to tell you that. Newtonian mechanics will tell you that an infinitely long thread which is set in motion so as to be expanding uniformly is bound to break.

Makes sense to me. But I can’t focus on Newtonian mechanics when I’m trying show strangeness of the current expanding space paradigm. So do you agree that the current paradigm suggests, where the cosmological constant is zero, that an infinitely long thread breaks? (I assume that if the constant is positive then the thread breaks more easily.) Ned Wright’s FAQ seems to say yes. And if yes, do you agree that there’s an inconsistency with the paradigm (threads break but galaxies don’t)? If not, why?

 

[Zanket]

I think these details are important. It explains at the microscopic level exactly what is happening with the thread. Especially so, because of the confusion you're having with the macroscopic analysis.

By my comment I mean that the paradigm was created to match observations, so if the creators of the paradigm thought that a long thread must break to match observations, then presumably the math predicts that. Rather than do the math, I can just examine the goal of the paradigm. I’ve done further reading since yesterday. It now seems clear that the paradigm calls for a long thread to break. That’s all I need to know to create a seeming paradox.

This statement is patently incorrect.

Right you are. It’s hard to describe the movement of the thread that I meant by that comment, but now I consider the point unnecessary to create the paradox.

 

[Hurkyl]

If the thread is sufficiently long, it must break, unless I've made a horrible math error, or am modelling the problem incorrectly.

Assuming:
(1) There's a coordinate chart where the expansion of the universe can be modeled as a pseudoforce whose strength is proporitional to the distance from the origin, and in which the string remains stationary.

(2) Only neighboring molecules interact, and the binding force is proportional to the displacement from natural equilibrium. (i.e. like an ideal spring)

Then, the distance between the end particle and its neighbor is:

L + Kx

where L is the natural equilibrium distance, and x is the distance to the midpoint of the string.

Then, as you move from the end particle towards the center, the separation between particles increases. (the rate of growth is proportional to the distance to the center)

Specifically, the change in the distance between neighboring particles is Kx.

IOW, it looks something like:

*..*...*...*...*...*..*


So, if the thread is sufficiently long, the condition for being in equilibrium would require particles to be too far apart. Thus, in a given region of free expanding space, there is a threshold such that strings of length greater than that threshold cannot be in equilibrium.


Others have told you exactly this before, incidentally. (Just not in this manner)