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Imagine a measuring stick (a rod or ruler) made of ordinary matter which has a length of a billion lightyears. The space surrounding the ruler expands. Will the ruler keep the same length by the accumulating electromagnetic forces opposing expansion? Or will expansion tear the ruler apart?

I have read somewhere here, in an accepted answer (I have no intention exposing the answer so you have to trust me on this) that the ruler expands along but I have my doubts. Will it not oppose expansion like the gravity of galaxies in a cluster opposes expansion?

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    $\begingroup$ @JamesK I'm not sure either. If locally expansion is opposed through the global ruler, it seems globally expansion is opposed too. Which would mean you would get accelerated holding the ruler at one end. $\endgroup$
    – Gerald
    Sep 8, 2022 at 22:34
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    $\begingroup$ @Gerald I've never been able to get answers to local expansion questions that I found truly satisfying. They usually end up in one of three categories; 1) "No, because of (insert short phrase to advanced concept without explanation)" 2) "No, because of (insert ballon with galaxies drawn on it or a piece of raisin cake)", 3) "No, because of (insert long, detailed GR answer indicating that there are unanswered questions or multiple formulations)". It seems that any experiment (thought, or real) that tries to see local expansion, can't. I still don't know why. :-) $\endgroup$
    – uhoh
    Sep 8, 2022 at 22:56
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    $\begingroup$ Would order 1AU metric expansion of space begin if the solar system were not inside a galaxy? and Is the metric expansion of space relatively uniform on different length scales? (you may find the answer here of interest) and Local expansion measured, near zero via Lunar Ranging - what about deep space probes? There are many, many more in Physics SE. $\endgroup$
    – uhoh
    Sep 8, 2022 at 22:58
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    $\begingroup$ @uhoh It confuses me uhoh.... Gravity between local cluster galaxies keeps them from expanding (for which a ruler obviously isn't enough, if it was "tied" between the galaxies and attractive gravity shut off). But what about the ruler itself? How big will the pulling force be without huge masses attached? Will it be like pulling a short ruler and will the long ruler just elongate along with space? Uhoh... :-) $\endgroup$
    – Gerald
    Sep 8, 2022 at 23:06
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    $\begingroup$ The Friedmann equations describing expansion of the universe were derived under the assumption that the mass density is the same everywhere. If you add a measuring stick (or any object with mass) you are violating that assumption. General Relativity is nonlinear, so there is no easy way to determine the effects of that addition precisely. Any combination of effects that you propose will be an approximation at best. $\endgroup$
    – D. Halsey
    Sep 9, 2022 at 20:50

3 Answers 3

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There is no Aristotelian/frictional force that drags things along with expanding space. The motion of objects is governed by

  • inertia
  • ordinary gravitation, which is an inward acceleration (note: acceleration, not velocity)
  • the cosmological constant, which is an outward acceleration (if $Λ>0$)
  • nongravitational forces

and nothing else. The reason matter has expanded for most of the universe's history is that it began with outward inertia (for unknown reasons – perhaps related to inflation), and the attractive force of gravity was too small to slow it to a stop. In the present era, attractive gravity has become weak enough that the repulsive $Λ$ force dominates.

If you plop a very long rod (of negligible mass) into this system, in the present era, there will be a net outward force on the ends, so it will be under tension. The result will be the same as if you were tugging on both ends: if the rod is too weak to survive the tension, then it will break; otherwise, it will be slightly longer than it would have been if it weren't under tension, but it won't grow over time (except see below).

If it's an earlier era, before $Λ$ dominance, then there will be a net inward force, and the rod will be under pressure. This is basically the same but with everything negated: either it will break or it will be slightly shorter than if it weren't under pressure.

If you leave the rod for a very long time, the matter responsible for the attractive force will continue to expand and thin, so the attractive force will decrease over time, and the rod will get slightly longer. Eventually, the matter density goes to zero, while the repulsive force remains constant, and that's the longest the rod will ever get (unless it breaks or deforms plastically). The repulsive force over a distance of 1 billion light years, given the measured $Λ$, is an acceleration of under $10^{-10}\text{ m/s}^2$.

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    $\begingroup$ I think we should also take into account the fact that masses in space tend to take up a certain shape for a reason. For example, the planets ended up as spheres, and galaxies ended up as spirals etc. I'm pretty sure there would be more forces at play on the ruler. $\endgroup$
    – Julia
    Sep 9, 2022 at 10:53
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    $\begingroup$ Good thing you added "of negligeable mass" there, since, if I read your answer correctly, a sufficiently massive ruler would be in the regime of "ordinary gravity applies," in which case we might have to invoke the "mole of moles" rule (what-if.xkcd) and watch the ruler collapse into a spherical object, probably quickly into a black hole $\endgroup$
    – Carl Witthoft
    Sep 9, 2022 at 11:19
  • $\begingroup$ @CarlWitthoft (and Julia) The whole thing is ridiculous, I'll grant you, but OP asked for a billion-light-year long stick and I was trying to honor that in some way. $\endgroup$
    – benrg
    Sep 10, 2022 at 0:00
  • $\begingroup$ This is what I always thought, but there are a lot of people who write narratives that are "in a zillion years, galaxies don't hold together, then stars don't hold together, then planets don't hold together, then atoms hold together, and eventually everything is lonely subatomic particles". I take it the real version is clumps of matter with ever larger and emptier expanses of nothing between them. $\endgroup$
    – hobbs
    Sep 10, 2022 at 23:04
  • $\begingroup$ @hobbs There are two ways they might be correct. First, if they're talking about a Big Rip, which is not standard cosmology. Second, even in standard cosmology, everything will eventually fall apart because of thermal and quantum fluctuations, but it takes an extremely long time. At time scales of tens of billions of years from now, it's clumps of matter separated by huge voids as you said, and some clumps are very large (superclusters). $\endgroup$
    – benrg
    Sep 11, 2022 at 0:58
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I'll give it a try, and I'm perfectly willing to take my answer back if it's wrong beyond repair.

The first thing I take issue with is your wording "the space surrounding the ruler expands". I would rather say "the space the ruler is embedded in expands". The ruler doesn't "know" anything about it.

Does the expanding space change the material properties of the ruler, e.g. by weakening or stressing the electromagnetic forces holding its atoms together? I don't think so. I have not heard that star physics or chemistry have changed during the past phase of "normal" (past the inflationary period) expansion. (Of course it's possible that I just missed it, I'm open for additions and corrections.)

That the space in and around clusters of galaxies doesn't expand is owed to the fact that gravity works on space proper, "holding it together", if you want. Electromagnetism doesn't (apart from the comparably tiny amount of energy which electromagnetic fields contain).

Lastly you establish a nice paradox: How do we measure the expansion of the space a ruler is embedded in? It's a paradox because in the typical gedankenexperiment the ruler is used to measure distances, but now we want to gauge our gauge.

The only way to measure the expansion of space, I would think, is to measure the time needed to traverse it with a given fraction of the speed of light. If it takes longer than before, space has expanded; one could say "new space" has been mixed in.

In this sense the ruler would lengthen, yes. But it is not an elastic stretching ("against" the space it is embedded in); instead, the space itself has lengthened.

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  • $\begingroup$ +1 just for "That the space in and around clusters of galaxies doesn't expand is owed to the fact that gravity works on space proper, "holding it together", if you want. Electromagnetism doesn't", a key concept I hadn't understood before (I thought space expanded but the galaxies clung together in it) simply explained. But I'm still a bit unclear on your conclusion, "the ruler would lengthen ... [because] the space itself has lengthened." If I've understood, you mean, the ruler (like everything) contains space "between" its fundamental particles, and since the ruler's gravity over that ... $\endgroup$
    – lessthanideal
    Sep 9, 2022 at 17:11
  • $\begingroup$ ... length is too small to hold its interior space together, that space will expand and so the ruler's constituent particles will move further apart? [I'm assuming fundamental particles don't get bigger in diameter as space expands since the other forces prevail, and so we can say the ruler got longer measured against, e.g., width of one of its atoms. Using the model of a particle can be said to have a defined diameter for simplicity] $\endgroup$
    – lessthanideal
    Sep 9, 2022 at 17:14
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Let's see.. suppose the ruler masses 0.001 g per meter. 1E9 ly * 0.001 g $\approxeq$ 9.5E18 kg , roughly equal to 324 Bamberga.
Leaving aside the obvious questions about rigidity in the presence of galactic gravity fields, if we take Benrig's value $\Lambda_{accel} = 10^{-10} \frac{m}{s^2}$ , and then compare with, say, the attractive force between two 1-kg masses separated by 1e9 ly due to regular gravity, that's around 7.052632e-36 N . For our ruler, take a 1-kg chunk at one end and perform a crude integral of the remainder, using $F = \frac{Gm_1m_2}{R^2}$ , I get roughly 0.11 N total force exerted on that chunk (again, assuming that $\Lambda$ force hasn't ripped the ruler apart, and that we somehow managed to wait long enough for the gravitons to propogate a billion ly).

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