Is Space Expanding?

I think I’ve just got time for a quick post this lunchtime, so I’ll pick up on a topic that rose from a series of interchanges on Twitter this morning. As is the case with any interesting exchange of views, this conversation ended up quite some distance from its starting point, and I won’t have time to go all the way back to the beginning, but it was all to do with the “expansion of space“, a phrase one finds all over the place in books articles and web pages about cosmology at both popular and advanced levels.

What kicked the discussion off was an off-the-cuff humorous remark about the rate at which the Moon is receding from the Earth according to Hubble’s Law; the answer to which is “very slowly indeed”. Hubble’s law is v=H_0 d where v is the apparent recession velocity and d the distance, so for very small distance the speed of expansion is tiny. Strictly speaking, however, the velocity isn’t really observable – what we measure is the redshift, which we then interpret as being due to a velocity.

I chipped in with a comment to the effect that Hubble’s law didn’t apply to the Earth-Moon system (or to the whole Solar System, or for that matter to the Milky Way Galaxy or to the Local Group either) as these are held together by local gravitational effects and do not participate in the cosmic expansion.

To that came the rejoinder that surely these structures are expanding, just very slowly because they are small and that effect is counteracted by motions associated with local structures which “fight against” the “underlying expansion” of space.

But this also makes me uncomfortable, hence this post. It’s not that I think this is necessarily a misconception. The “expansion of space” can be a useful thing to discuss in a pedagogical context. However, as someone once said, teaching physics involves ever-decreasing circles of deception, and the more you think about the language of expanding space the less comfortable you should feel about it, and the more careful you should be in using it as anything other than a metaphor. I’d say it probably belongs to the category of things that Wolfgang Pauli would have described as “not even wrong”, in the sense that it’s more meaningless than incorrect.

Let me briefly try to explain why. In cosmology we assume that the Universe is homogeneous and isotropic and consequently that the space-time is described by the Friedmann-Lemaître-Robertson-Walker metric, which can be written

ds^{2} = c^{2} dt^{2}-a^{2}(t) d\sigma^{2}

in which d\sigma^2 describes the (fixed) geometry of a three-dimensional homogeneous space; this spatial part does not depend on time. The imposition of spatial homogeneity selects a preferred time coordinate t, defined such that observers can synchronize watches according to the local density of matter – points in space-time at which the matter density is the same are defined to be at the same time.

The presence of the scale factor a(t) in front of the spatial 3-metric allows the overall 4-metric to change with time, but only in such a way that preserves the spatial geometry, in other words the spatial sections can have different scales at different times, but always have the same shape. It’s a consequence of Einstein’s equations of General Relativity that a Universe described by the FLRW metric must evolve with time (at least in the absence of a cosmological constant). In an expanding universe a(t) increases with t and this increase naturally accounts for Hubble’s law, with  H(t)=\dot{a}/a but only if you define velocities and distances in the particular way suggested by the coordinates used.

So how do we interpret this?

Well, there are (at least) two different interpretations depending on your choice of coordinates.  One way to do it is to pick spatial coordinates such that the positions of galaxies change with time; in this choice the redshift of galaxy observed from another is due to their relative motion. Another way to do it is to use coordinates in which the galaxy positions are  fixed; these are called comoving coordinates.  In general relativity we can switch between one view and the other and the observable effect (i.e. the redshift) is the same in either.

Most cosmologists use comoving coordinates (because it’s generally a lot easier that way), and it’s this second interpretation that encourages one to think not about things moving but about space itself expanding. The danger with that is that it sometimes leads one to endow “space” (whatever that means) with physical attributes that it doesn’t really possess. This is most often seen in the analogy of galaxies being the raisins in a pudding, with “space” being the dough that expands as the pudding cooks taking the raisins away from each other. This analogy conveys some idea of the effect of homogeneous expansion, but isn’t really right. Raisins and dough are both made of, you know, stuff. Space isn’t.

In support of my criticism I quote:

 Many semi-popular accounts of cosmology contain statements to the effect that “space itself is swelling up” in causing the galaxies to separate. This seems to imply that all objects are being stretched by some mysterious force: are we to infer that humans who survived for a Hubble time [the age of the universe] would find themselves to be roughly four metres tall? Certainly not….In the common elementary demonstration of the expansion by means of inflating a balloon, galaxies should be represented by glued-on coins, not ink drawings (which will spuriously expand with the universe).

(John Peacock, Cosmological Physics, p. 87-8). A lengthier discussion of this point, which echoes some of the points I make below, can be found here.

To get back to the original point of the question let me add another quote:

A real galaxy is held together by its own gravity and is not free to expand with the universe. Similarly, if [we talk about] the Solar System, Earth, [an] atom, or almost anything, the result would be misleading because most systems are held together by various forces in some sort of equilibrium and cannot partake in cosmic expansion. If we [talk about] clusters of galaxies…most clusters are bound together and cannot expand. Superclusters are vast sprawling systems of numerous clusters that are weakly bound and can expand almost freely with the universe.

(Edward Harrison, Cosmology, p. 278).

I’d put this a different way. The “Hubble expansion” describes the motion of test particles in a the coordinate system I described above, i.e one  which applies to a perfectly homogeneous and isotropic universe. This metric simply doesn’t apply on the scale of the solar system, our own galaxy and even up to the scale of groups or clusters of galaxies. The Andromeda Galaxy (M31),  for example, is not receding from the Milky Way at all – it has a blueshift.  I’d argue that the space-time geometry in such systems is simply nothing like the FLRW form, so one can’t expect to make physical sense trying to to interpret particle motions within them in terms of the usual cosmological coordinate system. Losing the symmetry of the FLRW case  makes the choice of appropriate coordinates much more challenging.

There is cosmic inhomogeneity on even larger scales, of course, but in such cases the “peculiar velocities” generated by the lumpiness can be treated as a (linear) correction to the pure Hubble flow associated with the background cosmology.  In my view, however, in highly concentrated objects that decomposition into an “underlying expansion” and a “local effect” isn’t useful. I’d prefer simply to say that there is no Hubble flow in such objects. To take this to an extreme, what about a black hole? Do you think there’s a Hubble flow inside one of those, struggling to blow it up?

In fact the mathematical task of embedding inhomogeneous structures in an asymptotically FLRW background is not at all straightforward to do exactly, but it is worth mentioning that, by virtue of Birkhoff’s theorem,  the interior of an exactly spherical cavity (i.e. void)  must be described by the (flat) Minkowski metric. In this case the external cosmic expansion has absolutely no effect on the motion of particles in the interior.

I’ll end with this quote from the Fount of All Wisdom, Ned Wright,in response to the question Why doesn’t the Solar System expand if the whole Universe is expanding?

This question is best answered in the coordinate system where the galaxies change their positions. The galaxies are receding from us because they started out receding from us, and the force of gravity just causes an acceleration that causes them to slow down, or speed up in the case of an accelerating expansion. Planets are going around the Sun in fixed size orbits because they are bound to the Sun. Everything is just moving under the influence of Newton’s laws (with very slight modifications due to relativity). [Illustration] For the technically minded, Cooperstock et al. computes that the influence of the cosmological expansion on the Earth’s orbit around the Sun amounts to a growth by only one part in a septillion over the age of the Solar System.

The paper cited in this passage is well worth reading because it demonstrates the importance of the point I was trying to make above about using an appropriate coordinate system:

In the non–spherical case, it is generally recognized that the expansion of the universe does not have observable effects on local physics, but few discussions of this problem in the literature have gone beyond qualitative statements. A serious problem is that these studies were carried out in coordinate systems that are not easily comparable with the frames used for astronomical observations and thus obscure the physical meaning of the computations.

Now I’ve waffled on far too long so  I’ll just finally  recommend this paper entitled Expanding Space: The Root of All Evil and get back to work…

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21 Responses to “Is Space Expanding?”

  1. A bit of self-promotion: David Hogg and I wrote an American Journal of Physics article on this subject a couple of years ago: http://arxiv.org/abs/0808.1081 . If I understand your point of view correctly, then Hogg and I agree with you.

    One other comment, about the belief that the Hubble expansion is “trying” to push the Moon away from the Earth, but the effect is very weak and is counteracted by gravity. I think that the “tethered-galaxy” thought experiment is a salutary thing to think about there. Suppose that, a long time ago, someone stretched a long rope between us and a distant galaxy, holding the galaxy still with respect to us. After the rope is cut, does the galaxy start to recede due to the expansion or not?

    (Hogg and I discuss this in our paper, and, unless we were grossly negligent, we included a reference to a discussion of it by Davis and Lineweaver.)

  2. telescoper Says:

    That reminds me I should have mentioned the paper by Tamara Davis and Charlie Lineweaver, but forgot. Here ’tis

    http://arxiv.org/abs/astro-ph/0310808

  3. First, a nitpick: Einstein’s original static universe was certainly described by GR, yet the metric does not evolve. (There is also the trivial static case in which everything (density, curvature, cosmological constant) is zero, which can be seen as the far-future limiting case of the Einstein-de Sitter model.)

    Second, it is interesting that the Robertson-Walker metric contains essentially no physics; it follows from the assumptions of homogeneity and isotropy. (Thus, I prefer to leave out Friedmann and Lemaitre in the name for the metric, thinking more of Friedmann-Lemaitre equations which involve the density, cosmological constant etc; there is more physics here, such as the gravitational constant, equations of state etc.)

    Third, I believe that Davis and Lineweaver show quantitatively somewhere that, while bound systems don’t partake of the Hubble expansion, their equilibrium size is slightly larger than it would otherwise be.

    Fourth, I urge readers to check out the discussion of this topic on Ted Bunn’s blog: http://blog.richmond.edu/physicsbunn/2009/12/02/interpreting-the-redshift/

    Fifth, on a related note, check out this discussion on Ted Bunn’s blog:
    http://blog.richmond.edu/physicsbunn/2010/04/08/more-on-the-cosmological-redshift/

    Like Lineweaver, I was heavily influenced by Harrison in my youth. I think he is less well known than he should be. (Probably most people don’t realise that the author of the wonderful cosmology-for-poets textbook Cosmology: The Science of the Universe is the same Harrison as the one in the Harrison-Zeldovich fluctuation spectrum.) As a result, I spent a lot of time trying to convince people that the cosmological redshift is not a Doppler shift. I was almost always right, but, as Hogg and Bunn point out, there is a way in which one can correctly think of it as a Doppler shift. (What is confusing is that a Doppler shift involves a velocity, and velocity is distance changing with time, and in cosmology one learns about many different distances, but none of these changes with time in such a way that the corresponding velocity is that given by the Doppler formula. In particular, the Robertson-Walker metric implies a velocity-distance relation where the distance is the proper distance and the velocity the change of this with time, measured now. It can be arbitrarily large, much larger than the speed of light. In this case, using the relativistic Doppler formula is not even wrong. However, as Bunn and Hogg point out, there is a way of seeing things such that the relativistic Doppler formula is valid, but it doesn’t involve velocities or distances which appear in other contexts in cosmology.) Ted gave an interesting example: no-one would question interpreting the changing redshifts of the components of a binary star as a Doppler shift, but the corresponding velocity does not correspond to the change of any distance with time.

  4. The way I’ve always looked at it starts with Birkhoff’s theorem and models inhomogeneities as perfectly spherical overdensities with uniform distribution of matter. By Birkhoff’s theorem the metric within that overdensity is FRW but it obeys Friedmann equations for a closed universe (if the background is flat) with the density internal to the sphere; the composition of the background (density, curvature, etc.) doesn’t factor in at all, so clearly this overdensity doesn’t “feel” the outside expansion (whatever that would even mean). Inhomogeneities in real life are much more complicated but once you get down to the scale of bound structures the general conclusion should still hold true.

  5. I can thoroughly recommend the “Root of all Paper”. A fantastic read 🙂

  6. John Peacock Says:

    I’ve already had my say in print on this, but perhaps it’s worth stressing just a couple of points. I don’t like the frequent reference to “bound objects” not expanding: this gives the picture that the expansion of the universe is trying (in some mysterious way) to tear the earth away from the sun, and only the sun’s gravity prevents this.
    This is completely wrong, and the existence of gravitational force in the solar system is a distraction from the real issues. it’s better to think about a pair of massless test particles: if their relative velocity is ever cancelled, they will subsequently tend to begin to separate if the universe is vacuum dominated. But this is entirely due to the antigravity properties of the vacuum, and nothing to do with the expansion of the universe per se: the tendency to separate would be the same if the universe was in a collapsing phase. If vacuum energy is negligible, then our test particles actually tend to approach each other – drawn together by the gravitational force due to the mass between them. Again, this just depends on how much mass there is, and is independent of whether it is expanding or contracting.

    We should remember that the notion of “expanding space” is utterly in conflict with a fundamental of general relativity, namely the equivalence principle. Freely-falling observers inhabit the Minkowski spacetime of special relativity. This doesn’t expand, and appealing to any such concept is simply wrong when discussing scales where spacetime curvature can be neglected (up to 100 Mpc, say).

    In the end, you can’t do better than quote Woody Allen from Annie Hall: “What has the universe got to do with it? You’re here in Brooklyn! Brooklyn is not expanding! “. Woody Allen gets it, but it’s amazing how many smart scientists continue to get completely confused over the issue.

    • Massless test particles drawn together by gravity? Maybe you mean in the limit of negligible mass. (If they are completely massless, then they must move at the speed of light, so can’t have their relative velocity cancelled (unless moving in parallel counts).)

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  8. Space expanding…

    INTO WHAT is space “expanding”?

    Into more space obviously!

    All this talk about space “expanding” is completely incomprehensible to me. It just doesn’t make any SENSE.

    • telescoper Says:

      Since the Universe is, by definition, everything that exists then it can’t be expanding into anything else…but that doesn’t mean it can’t be expanding.

  9. interesting article, but i had to copy/paste it to notepad because my eyes bleed from reading white text on black background.. yes its fitting for this blog, bu it is unreadable

  10. […] accounts for the fact that lasts more than two minutes. More importantly, though, is the content. Here’s an old  discussion of mine on this question. Let me know what you think via the comments […]

  11. Somewhat tangentially … you say ” as someone once said, teaching physics involves ever-decreasing circles of deception”. Just for the record, I think that someone was Hugh Montgomery, lecturing me on Thermodynamics in 1974. At least that’s the story I seem to recall telling you in the pub after some AGP meeting recently… and maybe he got it from Nick Kemmer who got it from Peter Guthrie Tait etc etc….

    • telescoper Says:

      I think I’ve had that phrase in my head for much longer than I’ve been on the AGP…

      …but you may well be right, as I’m always in a vegetative state after those meetings.

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