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Universal expansion

  1. Mar 23, 2010 #1
    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?
    Last edited: Mar 23, 2010
  2. jcsd
  3. Mar 23, 2010 #2


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    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.
  4. Mar 23, 2010 #3


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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)

    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.
    Last edited by a moderator: May 4, 2017
  5. Mar 23, 2010 #4
    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?
  6. Mar 23, 2010 #5


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    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.

    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.
    Last edited: Mar 23, 2010
  7. Mar 23, 2010 #6


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    Of course. But I understood that you would not give them a velocity, so there is no expansion and no redshift.
    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.
    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.
  8. Mar 23, 2010 #7


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    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.
  9. Mar 23, 2010 #8


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    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.
  10. Mar 23, 2010 #9
    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.
  11. Mar 23, 2010 #10


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    Yes, the effect inside a galaxy is not zero, but it is extremely small -- too small to measure in any conceivable experiment.
  12. Mar 24, 2010 #11
    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.
  13. Mar 24, 2010 #12


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    Assuming that emitter and observer are at rest wrt each other, what's not correct with my answer?
    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:
    doesn't have physical meaning. Expansion is a state of relative motion of some canonical observers, not a local property of space.
    Which numbers are you referring to?

    I don't whether we have sime misunderstandings here:
    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.
    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.
    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).

    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.

    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.
    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).
  14. Mar 24, 2010 #13


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    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.

    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.

    At "For example, the predicted general-relativistic effect on the radius of the earth's orbit ..."

    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.
  15. Mar 24, 2010 #14
    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:

    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?
    Last edited: Mar 24, 2010
  16. Mar 24, 2010 #15


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    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.
    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.
    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.
    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.
  17. Mar 24, 2010 #16
    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.
  18. Mar 25, 2010 #17


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    I'm getting tired of this argument.

    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.
    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.

    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.
  19. Mar 25, 2010 #18
    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.
  20. Mar 25, 2010 #19


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    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.

    Expanding Space: the Root of all Evil?
    Matthew J. Francis, Luke A. Barnes, J. Berian James, Geraint F. Lewis
    Last edited: Mar 25, 2010
  21. Mar 25, 2010 #20


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    Sorry, I missed your #14.
    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.
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