Universal expansion

I understand that in a uniformly expanding universe of the Einstein-de Sitter type, the distances between galaxies is constantly increasing but that the red shift displayed by those galaxies is not due to the Doppler shift - rather it is due to the fact that, during the time it takes for their photons to reach us, the universe has expanded thus stretching the wavelengths of the photons.

This suggests that it ought to be possible in principle to measure the expansion of the universe in the laboratory by measuring how the apparent wavelength of a source of monochromatic light varies with its distance from the receiver. Assuming a value of the Hubble constant of H = 2.33 x 10-18 s-1, we should expect a red shift of H/c = 7.77 x 10-28 for every metre of extra separation.

Notwithstanding the fact that this shift is so small, it could not possibly be measured, I believe that it must exist. I have, however, heard it categorically stated that while the distances between the galaxies is increasing, the dimensions of the galaxies themselves, and everything in them (including planets and atoms) stays the same. As I see it, this statement is open to three interpretations:
1. It means exactly what it says - and therefore my suggested thought experiment would return a null result because, on a local scale, space is not expanding.
2. It means that, while the universe is expanding on a cosmological scale as evidenced by the galactic red-shift, on a local scale, any measurement that we make eg with a metre ruler, will stay the same because our measuring instruments expand as well. My thought experiment will give a positive result but the measured distance between source and receiver will not change.
3. The statement is not trying to say anything profound at all; it simply means that, while the galaxies are expanding in a cosmological sense, this expansion is completely overwhelmed by the force of gravity which holds them together.
Can anyone help me to resolve this issue?

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Ich
I understand that in a uniformly expanding universe of the Einstein-de Sitter type, the distances between galaxies is constantly increasing but that the red shift displayed by those galaxies is not due to the Doppler shift - rather it is due to the fact that, during the time it takes for their photons to reach us, the universe has expanded thus stretching the wavelengths of the photons.
That's what one gets told. It's misleading:
This "expansion but not motion" viewpoint refers to a specific coordinate system. This system is useful for doing cosmological calculations, for regarding the idealized universe as a whole, with all parts of it joining the "Hubble flow".
It's more than useless for the kind of calculations you're trying to do.
There, you'd adopt the usual static coordinates. There's no more expansion then, there's only relative motion with the according doppler shift. If things are not moving, like your laboratoy setup, there is no doppler shift, and that's it. The value of H is irrelevant.

Except that there is gravitation ,too. You would measure gravitational red/blueshift, depnding on the exact position and orientation of your lab. If there is Dark Matter or Dark Energy in your lab, this will generally also cause a very tiny gravitational red/blueshift.

bcrowell
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#1 is correct. The experiment will give a null result.

This may be helpful: http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.2 [Broken] (subsection 8.2.5)

the red shift displayed by those galaxies is not due to the Doppler shift
This is not really correct. You can say it's a Doppler shift, or you can say it's due to the expansion of space. There is no experiment that can distinguish between these two interpretations.

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Yes, I am sure you are right but I still can't quite get my head round it.

Surely if the light source and the detector were not physically connected together but were, say, 10 metres apart in empty space they would (in principle of course) detect a red shift - because in order for the cosmological coordinates to stay the same, they would have to be given a small recession velocity. Of course!

Now what about the size of atoms. If atoms stay the same size while space gets larger there is more room for them. Run the clock backwards however and we see that as we approach the Big Bang, there is less and less room for atoms to exist. If we ignore for the moment the fact that the very early universe didn't contain any atoms, only radiation, do we have any idea of the size and the epoch of the universe when its density equaled that of a neutron star?

One other question. What is the crucial difference between an atom and a photon that makes one independent of cosmological expansion and the other not?

bcrowell
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Surely if the light source and the detector were not physically connected together but were, say, 10 metres apart in empty space they would (in principle of course) detect a red shift - because in order for the cosmological coordinates to stay the same, they would have to be given a small recession velocity. Of course!
Take a look at the link I gave in #3. I gave some figures there. The effect is not predicted by GR to be exactly zero, but it's much, much smaller than you'd naively expect by applying cosmological expansion on the smaller scale -- many, many orders of magnitude too small to be detectable.

One other question. What is the crucial difference between an atom and a photon that makes one independent of cosmological expansion and the other not?
This is discussed in the same link.

[EDIT] Oops -- sorry, I realized that what I wrote above (at "Take a look...") was wrong. I was thiniking of the expansion of gravitationally bound systems, but that's not what you were asking about.

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Ich
Surely if the light source and the detector were not physically connected together but were, say, 10 metres apart in empty space they would (in principle of course) detect a red shift - because in order for the cosmological coordinates to stay the same, they would have to be given a small recession velocity. Of course!
Of course. But I understood that you would not give them a velocity, so there is no expansion and no redshift.
do we have any idea of the size and the epoch of the universe when its density equaled that of a neutron star?
Matter density (including DM) is now ~2.6*10^-27 kg/m³. A Neutron star has ~5*10^17 kg/m³. The Volume ratio is 2*10^44, so the universe was ~6*10^14 times smaller then (in one dimension). I don't know exactly how many seconds after the BB that would be the case.
What is the crucial difference between an atom and a photon that makes one independent of cosmological expansion and the other not?
Binding. The atom is in a bound state, which means that the electron is not moving away from the nucleus. No velocity = no expansion.
With light it's a bit more complicated, as you can't define a relative velocity of two parts of a light wave. The important thing is that light speed is the same for every comoving observer, which means that two parts of the wave stay at a constant comoving "distance" - you get the "real" distance by multiplying with the scale factor.

bcrowell
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Surely if the light source and the detector were not physically connected together but were, say, 10 metres apart in empty space they would (in principle of course) detect a red shift - because in order for the cosmological coordinates to stay the same, they would have to be given a small recession velocity. Of course!

Of course. But I understood that you would not give them a velocity, so there is no expansion and no redshift.

Sorry for my incorrect answer on this topic above. However, now that I realize that I misread the question, I think neither JollyOlly's answer to his own question nor Ich's answer to it is quite correct.

The answer to this question depends on where the source and detector are.

If they're out in deep space, in a region far from any galactic supercluster, then JollyOlly's answer is basically correct. This part of space is well described by an expanding cosmological solution to the Einstein field equations. Such a solution has a preferred rest frame, which can be determined, e.g., by finding a frame in which the Doppler shift of the CMB vanishes. If the detector and the source are each initially at rest by this criterion, then they are not initially at rest relative to each other. Furthermore, the expansion is currently accelerating, so the separation between the source and detector will not just increase, it will increase at an accelerating rate.

On the other hand, if the source and detector are inside the solar system, then they're in a region of space that is not well described by an expanding cosmological solution to the Einstein field equations. The cosmological effect on the separation between them is then nonzero, but many, many orders of magnitude too small to be measurable. It will be much smaller than the general cosmological expansion described by the cosmological solution. The gravitational binding of the solar system tends to prevent expansion from happening to the space within the solar system. The size of the effect is given numerically in link #3, and references therein.

bcrowell
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Binding. The atom is in a bound state, which means that the electron is not moving away from the nucleus. No velocity = no expansion.
With light it's a bit more complicated, as you can't define a relative velocity of two parts of a light wave. The important thing is that light speed is the same for every comoving observer, which means that two parts of the wave stay at a constant comoving "distance" - you get the "real" distance by multiplying with the scale factor.

I'm a little uncomfortable with this argument. I'm not saying it's wrong, but I think it may be more complex than one would think at first glance.

Re "With light it's a bit more complicated, as you can't define a relative velocity of two parts of a light wave," I don't see why not. You have two world-lines. What stops you from talking about the separation between the two world-lines, and the rate at which this separation is increasing or decreasing? In SR, the separation is constant. In GR, it isn't necessarily.

Naively, based on E=mc2, one would expect that the different parts of a (classical) electromagnetic wave-train would exert gravitational attractions on one another. But IIRC, light rays traveling in the same direction actually turn out to experience zero gravitational attraction. (I think this is the kind of thing that is studied by people who work on things like the theory of colliding E&M and gravitational waves.)

Therefore it's not totally obvious to me whether you can have bound systems in GR consisting of nothing but electromagnetic waves. Maybe such systems exist, but they're black holes? It's not obvious to me how one should even define "binding" in this context.

Quantum-mechanically, we expect a bound system to have a ground state with a certain well-defined size. For a hydrogen atom, this size is defined by a certain combination of universal constants. For an electromagnetic wave, the only thing I can imagine as playing the role of the "ground state" would be a state in which the wave's energy was compressed into a small enough region to form a black hole, but this is a purely classical idea, not a quantum-mechanical one.

So if the question is why photons behave differently from atoms in terms of cosmological expansion, I'm not convinced that just describing the difference in behavior in terms of binding versus no binding really works, or at least I'm not sure it works without a little more justification.

To me, the arguments I gave in the link at #3 seem more clearcut. There is one argument based on the existence of a uniquely defined ground state for atoms, with a definite size, which I think shows fairly clearly that expansion can't apply to atoms. The argument definitely can't be applied to show that expansion doesn't apply to electromagnetic wave-trains.

I also gave another argument in that link based on the fact that there is no such thing as intrinsic curvature in one dimension.

Thank you both for your thoughtful replies. I will chase up the link too.

In the meantime could I express my difficulties in the following way: I have heard it said by a respected author that the balloon analogy, while excellent in helping to visualise how space can be expanding but not expanding into anything and how space can be finite but unbounded etc, it is misleading in the following way. Galaxies are often represented in the analogy by spots drawn on the rubber with a felt tip pen. This gives the impression that the galaxies expand with the expanding universe. As you have both pointed out, this is not the case. The author (whose name I forget) goes on to suggest that the galaxies should be represented by coins superglued onto the surface of the balloon thus preventing any local expansion. This seems to represent the situation well but it does have some implications. It means, for example, that the universe cannot remain spherically symmetric and that the regions of space round the edges of the galaxies will undergo unusual distortion. All of which fits with the notion that the presence of a large mass of gravitating matter distorts the surrounding space.

But if the expansion of the universe is halted inside a galaxy by the presence of gravitating matter, it must surely be a matter of degree. The extent to which the expansion is reduced will depend (probably) on the local density and distribution of matter. Gravity does not make a galaxy rigid so in the balloon analogy, the coins must not be made of a rigid substance like metal but rather of thick rubber. This means that, even inside a galaxy, my thought experiment will not return a null result but rather an even smaller one. (is this what you were getting at in your post bcrowell 09:13?)

If this is correct then interpretation #3 is nearer the mark than #1.

bcrowell
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This means that, even inside a galaxy, my thought experiment will not return a null result but rather an even smaller one. (is this what you were getting at in your post bcrowell 09:13?.
Yes, the effect inside a galaxy is not zero, but it is extremely small -- too small to measure in any conceivable experiment.

Excellent. I am much happier with that than the idea that cosmological expansion is exactly zero inside a galaxy. That just didn't make any sense to me.

I am afraid that the mathematics of GR is completely beyond me (I have tried) but I would guess that the effect you are talking about (the reduction of cosmological expansion within a gravitational system) is dependent on either the gravitational field strength at the point in question or (more likely) the local gravitational potential. The reason I raise this is that I wish to be convinced that fundamental particles do not take part in the expansion of the universe. The GFS at the surface of a neutron is 9 x 10-8 ms-2 while the GP is a mere 1 x 10-22 m2s-2. Neither of these figures seem to me to be large enough to compare with the GFS and GP within a typical galaxy or near a star.

But I was not aware that other forces such as the nuclear force had any effect on the curvature of space - so why is it claimed that atoms and fundamental particles do not share in the cosmological expansion? Even a neutron is far from being a rigid body.

Ich
bcrowell #7 said:
I think neither JollyOlly's answer to his own question nor Ich's answer to it is quite correct.
Assuming that emitter and observer are at rest wrt each other, what's not correct with my answer?
The answer to this question depends on where the source and detector are.
Not really, if you choose the procedure you described. By making the CMB dipole vanish, you set up a relative velocity of H*d, no matter whether you're in a galaxy or in intergalactic space. You get the same two-way redshift everywhere, maybe with an additional one-way shift due to the potential difference.
So, if you choose this definition, you have to answer that expansion of space is unchanged in galaxies. Which shows that this sentence:
The gravitational binding of the solar system tends to prevent expansion from happening to the space within the solar system.
doesn't have physical meaning. Expansion is a state of relative motion of some canonical observers, not a local property of space.
The size of the effect is given numerically in link #3, and references therein.
Which numbers are you referring to?

I don't whether we have sime misunderstandings here:
bcrowell #8 said:
Re "With light it's a bit more complicated, as you can't define a relative velocity of two parts of a light wave," I don't see why not.
Because you get tared and feathered here in the relativity forum for suggesting relative velocities of photons, as "relative velocity" is meant to be measurable by one of the participants. Of course, you can track differrences in some coordinate speed of light.
Therefore it's not totally obvious to me whether you can have bound systems in GR consisting of nothing but electromagnetic waves.
Well, you could, but my point was rather that, contrary to the bound atom, two parts of a (unidirectional) wave are most obviously not bound, so that each follows it null geodesic.
It's not obvious to me how one should even define "binding" in this context.
Exactly. There isn't even a rest frame.

My argument: at any cosmological time t, two consecutive parts of a wave have the same velocity in comoving coordinates - that's a direct consequence of the cosmological principle.
Therefore, their separation in cosmological coordinates remains unchanged, which means that their cosmological proper distance scales with a.
That's (I think) a short and concise derivation of cosmological redshifth, the stretching of wave packets.
An atom's constituents do not all have the same velocity in comoving coordinates. They have the same velocity in static coordinates, which means that they stay unchanged (apart from Lorentz contraction in the case of high velocities, of course).

JollyOlly said:
The author (whose name I forget) goes on to suggest that the galaxies should be represented by coins superglued onto the surface of the balloon thus preventing any local expansion.
Think of each particle floating absolutely frictionless on the surface. That gives you a very accurate quantitative understanding of the cosmological dynamics of test particles. Of course galaxies are also bound, so you'd also have to imagine some attraction there.

JollyOlly said:
I am much happier with that than the idea that cosmological expansion is exactly zero inside a galaxy. That just didn't make any sense to me.
Again:
If you start with two object at rest wrt each other, you don't have to consider "expansion". Inside a galaxy, they will start accelerating towards each other due to the mass of the stars, clouds, dark matter and so on. Dark Energy would try to push them apart, but lose.
so why is it claimed that atoms and fundamental particles do not share in the cosmological expansion?
If their constituents have no relative velocity from the start, they do not move apart. It's as simple as that.
Internal binding easily counteracts the feeble additional gravitational forces within the atom (DE in that case, and maybe some DM particles zipping through).

bcrowell
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Gold Member
Not really, if you choose the procedure you described. By making the CMB dipole vanish, you set up a relative velocity of H*d, no matter whether you're in a galaxy or in intergalactic space.
No, that's wrong. The issue here is the level of approximation. The thought experiment, as proposed, requires an incredible level of precision in setting up the initial relative velocities of the source and receiver. At that level of precision, the metric inside the galaxy is not well approximated by a cosmological metric. Making the CMB dipole vanish is *not* the same as setting up a relative velocity of Hd, for the metric inside a galaxy.

Which shows that this sentence:

doesn't have physical meaning. Expansion is a state of relative motion of some canonical observers, not a local property of space.
There's some discussion of this in the link I gave at #3, with references to two papers that take opposite views. One is by Bunn and Hogg, one by Francis. This is an issue of philosophical preference.

Which numbers are you referring to?
At "For example, the predicted general-relativistic effect on the radius of the earth's orbit ..."

Well, you could, but my point was rather that, contrary to the bound atom, two parts of a (unidirectional) wave are most obviously not bound, so that each follows it null geodesic.
It's not at all obvious to me that this is true to all levels of approximation, for the reasons given in #8. If you look at exact electromagnetic plane wave solutions, http://en.wikipedia.org/wiki/Monochromatic_electromagnetic_plane_wave , they're very complicated, and the interpretation is complicated.

Further to my question as to why nucleons do not take part in the cosmological expansion of the universe I have been studying your (bcrowell) excellent book on General Relativity where you say:

It is more dicult to demonstrate by explicit calculation that atoms and nuclei do not expand, since we do not have a theory of quantum gravity at our disposal. It is, however, easy to see that such an expansion would violate either the equivalence principle or the basic properties of quantum mechanics.

I can't say that I quite follow the argument that follows but I appreciate that, as with Chiao's paradox discussed earlier in the book, there is a lot we do not know about the connection between our theories of gravity and quantum mechanics. I am therefore content to accept that fundamental particles behave in the way that they behave and we have no right really to apply theories designed to explain the structure of the cosmos to such small objects. Nor should we object to the idea that at some time in the past, the universe was smaller than the total volume of all the particles inside it because at that epoch, the only theory that has any chance of being correct is quantum gravity - a theory which is as elusive now as it was when Einstein first started looking for it.

So to summarise what I think I have learned: going back to my original posting, interpretation #1 is wrong because GR does predict a cosmological expansion even within the solar system - albeit an incredibly small one. Interpretation #2 is wrong because if the source and receiver are fixed to the same bench, their separation will not change and no red shift will be observed at all. #3 is wrong because the statement that galaxies and atoms do not partake in the expansion of the universe is indeed a profound one. Moreover, the reasons for this are not as simple as saying that the expansion is 'overwhelmed' by the force of gravity.

So can we construct a statement that really does state the case exactly? Will this do?

#4 GR predicts that, within gravitationally bound systems such as galaxies the distortion of spacetime produced by the gravitating masses also reduces the cosmological expansion to practically nothing. GR has nothing to say about the expansion or otherwise of atoms and fundamental particles but quantum mechanics seems to rely on them staying the same size.​

One more question then. Photons obviously are affected by cosmological expansion. All particles have a wavelength which is associated with their momentum so presumably the momentum, and hence energy of a neutral particle emitted by a distant galaxy is also reduced. Is this consistent with the laws of conservation of energy and momentum?

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Ich
Making the CMB dipole vanish is *not* the same as setting up a relative velocity of Hd, for the metric inside a galaxy.
How so?
Let's have 2 distant observers at rest wrt each other. If we neglect the galaxy's gravity, both will see a slightly different dipole. To make it vanish, they would have to go to different speeds, according to dv=H*dx.
Now let's superimpose the galaxy's gravity (we're talking about weak fields here). The notion of relative rest is unaffected. Both would see the CMB at slightly different temperatures (if they are on different potential), but would observe the exacly same relative dipole as before, as the photon energy depends on the potential only and not on the path of the photon.
Therefore, both would have to go to the same different (local) speeds as before to make the dipole vanish.
Local speed is exactly what defines redshift, so the result is the same.
This is an issue of philosophical preference.
Whether space is expanding or things are moving is philosophy.
That expansion is not a local property of space (read: spacetime) is fact. Or which component of the Riemann tensor exactly do you think codes for H? Expansion has to do with observers, and which observers are the appropriate ones can only be seen on large scales.
Ich said:
Which numbers are you referring to?
At "For example, the predicted general-relativistic effect on the radius of the earth's orbit ..."
This effect (as described in the paper you're referencing) has nothing to do with the specialties of GR.
They assume that the local matter density - not counting the sun - is exactly the average cosmological density. As this density decreases, so does the amount of matter inside the earth orbit. Therefore the orbit increases. You need Newtonian mechanics to calculate the effect, not necessarily GR.
(If you don't think it's that easy: follow their calculation and use the second Friedmann equation when appropriate to get the density. Carry on with Newtonian mechanics, you get exactly their numbers. This is not surprising, as we're in a very weak field with very low velocities anyway.)
As, in real life, the DM density in a galaxy does not scale with a^-3, but stays rather constant, there is no such effect.
In either case, it is consistent with my claim that the local density is all that counts, irrespective of H.
It's not at all obvious to me that this is true to all levels of approximation
Ok. It is obvious for light as we know it, as its gravity can definitely be neglected. It's less obvious for strong waves, but I don't think that this is relevant for cosmological redshift, where the single light wave is a test "particle" anyway.

There's some discussion of this in the link I gave at #3, with references to two papers that take opposite views. One is by Bunn and Hogg, one by Francis. This is an issue of philosophical preference.
Before different interpretations of some model can be reduced to philosophical preferences, mathematical
consistence of said interpretations must be assured. Interpretations that are not mathematically consistent
are invalid.

Interpreting the cosmic redshift as a Doppler effect in flat space-time means something specific
mathematically. That is why this interpretation is in general inconsistent with the geometry of the FRW
models. This is particularly easy to see for a FRW model with flat spatial sections. See a discussion I
had in the Cosmology forum around April 1. last year. Another, less straightforward way to see this is
given in arXiv:0911.1205 (however, the method used in this paper seems more dubious for FRW models
with hyperbolic space sections since these models do not have a simple product topology SxR where S is
the spatial sections at constant cosmic time and R is the real line).

So the B&H paper advocates an "interpretation" of the cosmic redshift that is mathematically inconsistent.
But, of course, for the true believers, hard mathematical facts in no way trump personal prejudices.

Ich
I'm getting tired of this argument.

Wikipedia said:
every point of an n-dimensional manifold has a neighborhood homeomorphic to the n-dimensional space Rn.
You know what this means? It means that topology arguments are irrelevant as long as we're concerned only with what happens in said neighbourhood.
Wikipedia said:
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection.
Now what do you read here?
"on a manifold with a symmetric affine connection" or "on a manifold with a symmetric affine connection unless it is meant to represent a spatially closed FRW model"?
And, surprise, in normal coordinates you find that comoving observers are, well, moving. Away from us. That's valid in, you guessed it, a local neighbourhood.

But, of course, for the true believers, hard mathematical facts in no way trump personal prejudices.
Obviously. But take it easy, maybe in a few years you'll laugh about your claims here. Until then, please, do not join discussions with pejorative or misrepresenting remarks about other members or past discussions.

I am extremely grateful to those of you who have helped me answer the question posed in the original post but many of the recent exchanges have gone way off the mark and have degenerated into a slanging match.
Please could I redirect your attention to my post #14. I would be grateful for confirmation that I have not made any glaring mistakes in my interpretation and, if possible, some further discussion of my question about the conservation of energy and momentum.
Thank you.

atyy
The "expanding universe" is one possible description of a solution of general relativity in which matter is evenly distributed throughout the universe. It is not the only possible description of the solution, but it is a conventional one.

This "expanding universe" solution obviously does not apply to our universe in all details, since matter in our universe is not evenly distributed. However, on very large scales, matter is approximately evenly distributed, and the "expanding universe" solution is very useful.

On very small scales, matter is just not evenly distributed, and the "expanding universe" solution does not apply.

The full solution at all scales is presumably the "expanding universe" solution plus some corrections due to the uneven distribution of matter on small scales. So it may be said that the uneven distribution of matter on small scales is what prevents "expansion" on small scales.

http://arxiv.org/abs/0707.0380
Expanding Space: the Root of all Evil?
Matthew J. Francis, Luke A. Barnes, J. Berian James, Geraint F. Lewis

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Ich
#4 GR predicts that, within gravitationally bound systems such as galaxies the distortion of spacetime produced by the gravitating masses also reduces the cosmological expansion to practically nothing. GR has nothing to say about the expansion or otherwise of atoms and fundamental particles but quantum mechanics seems to rely on them staying the same size.
I disagree completely, and that's why I tried to get into a discussion with bcrowell. I think the mentioning of quantum mechanics doesn't help to explain the non-expansion of particles. There is a common explanation for both the solar system and small particles.

Now I'll choose some illustrating words to make my point:

There is no such thing as expansion of space.

Which means:
Take some "small" (some lightyears, maybe) region of space around the particle in question. Take every measurement equipment you can conceive of, and try to tell me via local measurements wheter space is expanding or not: you can't.
That's not a matter of measurement resolution, it's a matter of principle (*). Expansion is not a local property of spacetime.
So how should the particle even know that it's supposed to expand?
What explanation should one give for the nonexistence of expansion when there is no reason to expand in the first place?

*
disclaimer: of course you could see light from distant stars, and the CMB and such, to make a model of the universe. That's not what I mean by "local" measurement.
If you're lucky, you're in a region where more Matter/Dark Matter particles leave than enter. With some big leap of faith, one could take this fact as evidence for expansion rather than, say, suggesting that they are diffusing for some other reason. My point is then that the effect on local dynamics is exactly that there is less matter than before, nothing more. Ah, and DE accelerates things outward. You could measure this, but it doesn't tell you whether the universe is expanding or contracting.

The "small effect" bcrowell is citing stems solely from a supposed reduction of matter density in the solar system. It has nothing to do with expansion, except in an oversimpified model where matter density is exactly homogeneous in the whole universe (with a sun that is not distributed over the universe, however).

I didn't intend to get into a slanging match. I may be sometimes harsh; in my native language, there would be some subtle humor to take the edge off my words, but my English isn't good enough to be subtle. Sorry.

atyy
There is no such thing as expansion of space.

Is your point that "expanding space" is meaningful language in one particular coordinate system, and since any coordinate system is purely conventional, there is no such thing as "expanding space"?

There is no such thing as expansion of space.

Which means:
Take some "small" (some lightyears, maybe) region of space around the particle in question. Take every measurement equipment you can conceive of, and try to tell me via local measurements wheter space is expanding or not: you can't.
That's not a matter of measurement resolution, it's a matter of principle (*). Expansion is not a local property of spacetime.
So how should the particle even know that it's supposed to expand?
What explanation should one give for the nonexistence of expansion when there is no reason to expand in the first place?

Gonna have to disagree with you on this one. Just because you can't measure cosmological expansion on the local level doesn't negate its existence. What if spacetime is a substance which exists on a scale that we can't yet examine because our current technological level is not advanced enough? In this instance, cosmological expansion can only be inferred from the movement of matter embedded in spacetime. If expansion is affecting all matter in the same way (as it appears to be doing), then local measurements could not possibly prove anything.

And we're right back to the question of why local matter isn't pulled apart by this expansion if the expansion is, as it appears to be, affecting all matter. In other words, why do planets, stars and even galaxies remain intact? I don't think this can be answered without fully understanding how cosmological expansion takes place.

Does distorted spacetime negate the effect? Can't answer this without knowing the mechanics of the expansion.

Is it because the scale of spacetime is so small that the expansion acts on only the most fundamental building blocks of matter...therefore affecting all local matter in the same way? Maybe, but this seems like it would be highly dependent on the scale at which spacetime is expanding. Again, can't answer this without knowing the mechanics of the expansion.

Combination of the two? At this point, it's all guesswork.

Ich
Is your point that "expanding space" is meaningful language in one particular coordinate system, and since any coordinate system is purely conventional, there is no such thing as "expanding space"?
It's a bit stronger:
I'm not only talking about words. For example, even when - in your words - the "expanding universe" solution applies, and even if I'd refuse to call it "expanding space" for some ideological reasons, one might still assume that something weird is happening, and that is has an effect on local dynamics. You might still search for reasons to prevent these effects:
atyy said:
So it may be said that the uneven distribution of matter on small scales is what prevents "expansion" on small scales.

My point is that, call it what you like, "expansion" is not a local property of spacetime. No matter how closely you examine it, there is nothing about local spacetime that hints at some expansion.(*) It's simply not there, and so you don't have to justify that it has no effect on local dynamics.

Expansion is a state of motion of a certain, generally preferred, class of observers. You don't need uneven distribution of matter to prevent expansion, it's enough when you simply don't start your test particles with a relative velocity. No motion, no expansion.

* disclaimer: as said before, in an idealized thought experiment, there might be a very slight tendency of the background fluid to flow away, which might be interpreted as expansion. Still, if you plug the known fluid in you Newtonian or Postnewtonian equations, you get the right dynamics, whether or not there is expansion of the universe. All you have to know is the local mass distribution.

Ich
What if spacetime is a substance which exists on a scale that we can't yet examine because our current technological level is not advanced enough?
I'm talking about our standard theory of gravity only, GR. I'm not speculating about other theories.
In other words, why do planets, stars and even galaxies remain intact? I don't think this can be answered without fully understanding how cosmological expansion takes place.
You're right. Understanding really helps.
I claim to have understood enough that I can reproduce the experts' calculations and see what they mean.
I am reasonably confident that my understanding is correct, because it's backed by textbooks.

So I know (or at least believe that I know) how expansion takes place in GR: Not at all (*), if you don't explicitly start it by giving the correct initial conditions.

* disclaimer etc. Dark Energy and such. I think I need a lawyer.

atyy
It's a bit stronger:
I'm not only talking about words. For example, even when - in your words - the "expanding universe" solution applies, and even if I'd refuse to call it "expanding space" for some ideological reasons, one might still assume that something weird is happening, and that is has an effect on local dynamics. You might still search for reasons to prevent these effects:

My point is that, call it what you like, "expansion" is not a local property of spacetime. No matter how closely you examine it, there is nothing about local spacetime that hints at some expansion.(*) It's simply not there, and so you don't have to justify that it has no effect on local dynamics.

Expansion is a state of motion of a certain, generally preferred, class of observers. You don't need uneven distribution of matter to prevent expansion, it's enough when you simply don't start your test particles with a relative velocity. No motion, no expansion.

* disclaimer: as said before, in an idealized thought experiment, there might be a very slight tendency of the background fluid to flow away, which might be interpreted as expansion. Still, if you plug the known fluid in you Newtonian or Postnewtonian equations, you get the right dynamics, whether or not there is expansion of the universe. All you have to know is the local mass distribution.

Let's see, I think I do understand that your objection is stronger than the inhomogeneity argument.

But I don't understand why your argument is stronger than the "particular reference frame" argument. You are saying that even if FRW applies exactly, the FRW universe is not expanding, because that is true only for a certain class of observers in the FRW universe. Is "preferred class of observers" different from "particular reference frame" (and aren't both of these nonlocal)?

Thank you guys for coming back to the point and giving us something to think about.

In the FWR solution to Einstein's equations (please don't start talking about them in detail because I won't have a clue what you are on about!) there is a (dimensionless) time dependent term a(t). Although it is dimensionless (in our universe) it seems to behave very much like a length. For example, the quantity a3rho (where rho is the density of matter in the universe) appears to stay constant. This suggests that it does have a dimension but it is not mass, length or time. Let us call the quantity a the pseudo-radius of the universe and measure it in a new unit - the 'rod'. The mean curvature of the universe is defined as 1/a2 and is measured in rods-2.

Now we can certainly measure the curvature of our local neck of the woods by simply measuring the angles of a triangle and I am pretty confident that if we were to do this experiment out in intergalactic space somewhere, we would find that the curvature of our universe was changing and that the pseudo-radius of the universe is increasing. If our universe is finite, it implies that the total volume of the universe is (measurably) increasing. Even if our universe is open (infinite) it still means that more 'space' is being added all the time. Can we all agree that this is what we mean by 'the expansion of the universe'?

If so my question can be rephrased in the following operational way. If you were to do the same measurement inside our galaxy and, having subtracted out all the light bending effects due to the local gravitational field, would you still detect a 'flattening' of the curvature and hence a change in the pseudo-radius of the universe as measured in rods. I think bcrowell would say (see post Mar 23 10:49) yes - but the effect would be incredibly small. My original view was that you would be left with the whole cosmological factor but I now see that, since the equations of state are non-linear, you can't just add a local curvature to a cosmological one to get the total.

Ich
But I don't understand why your argument is stronger than the "particular reference frame" argument. [...] Is "preferred class of observers" different from "particular reference frame" (and aren't both of these nonlocal)?
Maybe I simply misunderstood you. Yes, my argument is that expansion locally becomes motion by a simple coordinate transformation. And that "motion" is not exactly the kind of hyperdimensional weirdness that needs excessive knowledge of GR to understand.

You are saying that even if FRW applies exactly, the FRW universe is not expanding, because that is true only for a certain class of observers in the FRW universe.
No, that's not what I'm saying. The expanding universe is a perfecly legitimate description at large scales. I mean, an expanding universe is certainly different from a static one, so why should I say that it's not expanding?
It's not a problem that the "expanding universe" idea is coordinate dependent. Many things are. If I switch to different coordinates, the exact "expansion" idea migh look quite different, but it's still different from a static universe, and you could expect that the difference is something very strange that has an influence on local dynamics which is hard to calculate or understand. So it's not coordinate dependence that bothers me.
It's the fact that when you switch to static coordinates, there is nothing left of expansion than things moving apart. That's all.
For this reason, I said that my point is stronger than that expansion is a coordinate dependent concept.

Ich
Very good questions, JollyOlly!

I don't see how I could answer them without generating further confusion.

In the FWR solution to Einstein's equations (please don't start talking about them in detail because I won't have a clue what you are on about!) there is a (dimensionless) time dependent term a(t). Although it is dimensionless (in our universe) it seems to behave very much like a length.
The product a*r is a length.
That means that it's up to you whether treat a as a dimensionless scale factor and r as the "comoving distance" (with unit m), or take a as a "scale" (with unit m) and r as a dimensionless number - like a being the radius of a sphere and r an angle.
It means also that you're generally free to scale a and r as you like, as long as their product stays the same.
So, generally, a itself is a meaningless number at any time, and it's not the "pseudo-radius" of the universe. You can't measure a.
The closest thing to a (pseudo)-radius is cosmological spatial curvature (let's call it Kc).
For a closed universe, you would choose a such that Kc=1/a², so that's the special case where a has a meaning.
However, our universe for example is believed to be spatially flat, Kc=0. In such a case, Kc is useless for determining expansion.

Now we can certainly measure the curvature of our local neck of the woods by simply measuring the angles of a triangle and I am pretty confident that if we were to do this experiment out in intergalactic space somewhere, we would find that the curvature of our universe was changing and that the pseudo-radius of the universe is increasing.
Yes, you can measure local spatial curvature with triangles. But you won't get "cosmological spatial curvature". You get something I'd call static spatial curvature, Ks.
Ks is a function of local matter density only. It is totally insensitive (to leading order) to whether said matter is expanding or not. So, if you're measuring angles in a triangle, you could instead measure local density, you learn nothing new.
Now how does Ks evolve over time?
It will asymptotically approach a constant value, as the universe becomes vacuum dominated and hence exponentially expanding. You'll find expansion neither in Ks nor in Kc. You can't measure locally the universe's scale. You'll find expansion only in the motion of comoving particles.
If our universe is finite, it implies that the total volume of the universe is (measurably) increasing. Even if our universe is open (infinite) it still means that more 'space' is being added all the time. Can we all agree that this is what we mean by 'the expansion of the universe'?
Yes. It's an increase in the global volume of the universe if it's finite, otherwise it's increasing distances between comoving observers, which you could phrase as 'space' being added if you like.

If so my question can be rephrased in the following operational way. If you were to do the same measurement inside our galaxy and, having subtracted out all the light bending effects due to the local gravitational field, would you still detect a 'flattening' of the curvature and hence a change in the pseudo-radius of the universe as measured in rods.
Well, as I said, there is no local measurement that gives you the "pseudo radius" of the universe.
If we're talking about Ks (matter density), which is the actual result of such a measurement:
It's fairly constant in the voids, for some billioin years now. There, the universe is already vacuum dominated.
It's fairly constant in a galaxy too, but much greater (more density = smaller curvature radius). If it changes at all, it would be rather increasing as the Dark Matter density increases (the curvature radius getting smaller, contrary to the expansion idea).
I think bcrowell would say (see post Mar 23 10:49) yes - but the effect would be incredibly small.
I already commented that statement, but unfortunately bcrowell did not respond, so there are still two opinions.

My original view was that you would be left with the whole cosmological factor but I now see that, since the equations of state are non-linear, you can't just add a local curvature to a cosmological one to get the total.
You mean the field equations are nonlinear? True, but these are weak fields, you can add the curvatures.
But there is no "cosmological factor", there's just a matter distribution and its gravitational field. No "expansion" there.

I think we are at last coming to a mutual understanding, even though I prefer a slightly different viewpoint. I made the classic error in post Mar 27 10:44 of confusing an expanding universe with a closed one. If our universe is flat (which all the evidence appears to support) measuring the angles of a triangle will not prove anything.

Can I ask you something else? In the currently accepted LCDM model of the universe as described in the Wiki article http://en.wikipedia.org/wiki/Lambda_CDM_model" [Broken] the age of the universe is quoted as 13.72x109 y and the Hubble constant as 70.5 km/s /Mpc. This is consistent with the formula Tuniverse = 1 / H0 which in turn implies that the expansion rate is linear. (In a matter-dominated universe the expansion rate would vary as t2/3 and Tuniverse = 2/3 / H0)

Now the red shift at decoupling is 1090. In a linearly expanding universe, the red shift of an object is simply T2/T1 where T1 is the time when the photon was emitted and T2 is now. Dividing 13.72x109 by 1090 gives 12.6x106 which does not agree with the quoted age of decoupling of 3.77x105 years.

It is very odd that if we assume instead a matter dominated universe, the ratio T2/T1 is not Z but Z3/2 and if we divide 13.72x109 by 10903/2 we get exactly the right result.

What is going on?

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Ich
If our universe is flat (which all the evidence appears to support) measuring the angles of a triangle will not prove anything.
So I generated confusion.
If you build a large scale triangle in a "flat" universe, you'll find the sum of angles to be different from 180°. From the difference, you'd deduce a positive curvature of 1/H².
It's just that when you set up a "virtual" triangle with three comoving observers at the corners that you measure 180° (due to aberration). That's a different slice of spacetime that you call "space", and it has a different curvature.
... This is consistent with the formula Tuniverse = 1 / H0 which in turn implies that the expansion rate is linear.
No, that's a coincidence. The universe evolved matter-dominated in the early phases, until DE became important. DE acclerated the expansion just enough that the slope of a(t) matches that of a linear model. (see the attached diagram)
It is very odd that if we assume instead a matter dominated universe, the ratio T2/T1 is not Z but Z3/2 and if we divide 13.72x109 by 10903/2 we get exactly the right result.
In the early phases, the matter dominated model (with 1/H=13.7 GY) fits the LCDM model much better - it's the same behaviour up to a scaling factor, as the universe was indeed matter dominated.
The exact match for z=1090, however, is another odd coincidence.
The 380,000 years in the standard model are a result of an exact calculation, that takes the radiation-dominated era in the early universe into account.
Your 380,000 years result when you neglect radiation. If you take it into account, you'd get 317,000 years.

Check out the advanced version of http://www.astro.ucla.edu/~wright/ACC.html" [Broken]. By setting T0=0, you neglect radiation.

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There seem to be an extraordinary number of coincidences here!

Thank you for your reference to Ned Wrights site. I have found it very illuminating. In particular on http://www.astro.ucla.edu/~wright/cosmo_03.htm" [Broken] of his cosmology tutorial he plots various possibilities for the variation of a(t) with time. The black line is the matter dominated line which predicts an age for the universe of 9.2 Gyr. The green line is a linear one which I understand is not a solution to the Friedmann equations (see my post on is the cosmological constant constant) but which gives an age of 13.8 Gyr. The magenta line is the one currently favoured. It gives an age of coincidentally almost the same age and, to my mind, unacceptably puts us exactly at the epoch when the curve changes from slowing down to speeding up.

I can, however, see now that the coincidence of the age of the CMB is not a coincidence because we are all agreed that in that early universe, matter dominated so the magenta line approximates to the black one.

If you build a large scale triangle in a "flat" universe, you'll find the sum of angles to be different from 180°. From the difference, you'd deduce a positive curvature of 1/H².

I cannot agree with you here. In a 'flat' universe the angles of a triangle add up to 180° by definition. You must have some other definition in mind.

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Ich
I cannot agree with you here. In a 'flat' universe the angles of a triangle add up to 180° by definition. You must have some other definition in mind.
You have to understand that "space" depends on coordinates.
When cosmologists talk about space being flat, they mean the spatial slicings defined by constant cosmological time. These are dynamic coordinates.
Triangles don't know about cosmological expansion and cosmological coordinates. They are just there, defined by their structure, which is assumed to be as rigid as possible. They are static, and such is the relevant "space". It is curved, the sum of angles is >180°.
Operationally, if you set the corners in motion according to Hubble's law, the relevant angles are changed due to aberration. Thus, their sum becomes 180° again if the corners are moving.

Can't $$\Lambda$$ in GR be interpreted as an intrinsic expansion of the underlying space? In this case universal expansion is a local property of space.

Two objects at rest relative to each other seperated by a distance would tend seperate further if no other force or curvature is present. A volume element would tend to increase three times faster.

The existence of a cosmological constant or dark energy imply a fifth force of nature exists, we just don't know how it works yet. Or we don't know how one of the existing forces works completely and the perturbation from our understanding is what is causing the acceleratin of the universe.

The definition of a gravitationally bound mass is when the gravity is stronger than this undetermined force. At the solar system level this force is of the scale of the Pioneer anomoly, but this anomoly is a contraction not an expansion.

The reason we don't see it is because the effect is so small. If this effect is so much smaller than gravity at our scales it is even less significant at the scale of particles.

Based on a DeSitter universe with the observed hubble expansion the length of an object doubles every 2.8E17 seconds, or 8.8 billion years. An electron expanding at this rate would hardly blink at this effect.

When cosmologists talk about space being flat, they mean the spatial slicings defined by constant cosmological time. These are dynamic coordinates.
I am afraid you have lost me there Ich!
Are you saying that three comoving observers in a Friedmann universe in which k = 0 would still measure an angular defect?

Ich