# On the expansion of space on small distances

After reading this post :Why does space expansion not expand matter? and after posting some comments to Peter Diehr, I was invited to make my argument to a question. The main things one needs from the post above are

The answer seems to be (from Marek in the previous question) that the gravitational force is so much weaker than the other forces that large (macro) objects move apart, but small (micro) objects stay together. However, this simple explanation seems to imply that expansion of space is a 'force' that can be overcome by a greater one. That doesn't sound right to me.

And

Hi. @PeterDiehr and Sushant23 . But why Brooklyn doesn't? If on the small scale we agree on the posted answer, that we cannot see the expansion since everything is expanding, then why see it on the big scale? Is it because it expands faster on the big scales but on the small the speed is the same for all objects? Thanks. – Constantine Black 1 hour ago

@ConstantineBlack: the expansion is equivalent to a very weak force - local binding forces always overwhelm it: atoms, molecules, people (eg, not a valid excuse for the waistline!), planets, solar systems, and galaxies. But you can see it over very great distances - hence the red shift due to cosmic expansion is a good proxy for distance, though other proxies are used to set the distance scale. See Hubble's Law – Peter Diehr 1 hour ago

@PeterDiehr Thanks for the fast response. I find it conceptually wrong to admit that expansion is a force such that you can use an equation like Newton's or any argument at least saying that: the total force on the object is expansion + other_forces so that the result in small scales is not-expansion. It' s more reasonable to either say that experiments say this or that or that in small scales, the expansion rate is the same for all objects( even inside the galaxy??) so that we don't observe it. Am I losing something here? Thanks. – Constantine Black 53 mins ago

from the comment discussion. I am posting them as they are, because in the end, I don't know if my question is something new besides the fact that the answers at the pre-mentioned post are not satisfactory( at least for me), and because indeed, my question is a function of the preceding discussion.

Thank you.

• If the other forces are strong enough to overcome gravitational expansion, then there is no expansion to observe in the first place, no matter what experiments you perform. Sorry if I misunderstood your point though – user108787 May 28 '16 at 19:12
• @count_to_10 Hi. You mean that if we consider a transposition dx due to expansion effect, then the local forces will provide a counter-transposition -dx so that in the the system will remain as it was? This is happening, for example, to the outer disc of the galaxy? Does this mean that the local forces do work all the time to counter effect the expansion of the universe, and this means that expansion is happening at small scales? Thanks. – Constantine Black May 28 '16 at 19:18
• @count_to_10 It would seem that the forces will do work, use energy, to counter-effect the expansion. What then happens to the conservation of energy of the system, if it use it's energy to counter the expansion? Doesn't it "lose" energy that way, since it is balancing not a force but a movement caused by the intrinsic expansion of the universe? – Constantine Black May 28 '16 at 19:21

In general relativity a free particle moves on a trajectory called a geodesic and to make it diverge from that geodesic you need to apply a force to it. To take an everyday example, an object momentarily at rest at the surface of the Earth would normally follow a geodesic that leads radially towards the centre of the Earth with an acceleration relative to the surface of $g$. To keep the object stationary at the surface we need to apply a force, and that force is of course just the gravitational force $F=mg$.

You'll hear people say that gravity isn't a force and while this is true it's also misleading. The curvature of spacetime does result in a force but that force is a bit different to the naive view. There isn't a force pulling objects down, but you do need a force to prevent them falling down i.e. to accelerate them away from the geodesic they would otherwise follow.

The point of all this is that I suspect your key concern is your statement:

I find it conceptually wrong to admit that expansion is a force such that you can use an equation like Newton's

You are correct that the expansion of space is not a force, but to make an object not follow that expansion you have to apply a force to it, and that is a real force that is in principle measurable (in practice it would be far too small to measure). So if you could tie two objects together with a string many light years long, to keep them at rest relative to each other, there really would be a tension in that string. That tension arises because you are forcing the objects to accelerate away from the geodesics they would otherwise follow, and it arises in the same way as the tension in the string if you suspend an object in Earth's gravity.

Having said this, I actually agree that it is meaningless to talk about the force on an atom due to the expansion, except just possibly in some idealised circumstances. By the expansion of spacetime we mean a spacetime geometry called the FLRW metric, but this is a large scale geometry due to a homogeneous distribution of matter and the distribution of matter isn't homogenous. If you looked at the spacetime geometry with some imaginary curvometer then you'd find at the small scale it didn't look anything like the FLRW metric. And if it locally doesn't look like the FLRW metric then that means locally there is no expansion and therefore no force needs to be applied to resist that expansion.

A footnote: when I started writing this I intended to go into more details about geodesics in an expanding spacetime, but this turns out to be somewhat complicated so it's a temptation I have resisted. If this is a subject that interests you it might be worth writing a new question.

• Good morning(on my behalf :) ) Thanks for the reply- your suspect on on my concerning hits bullseye, you are correct. Although new questions may arise from your answer I have only two to add, concerning the above: 1) what happens to the conversation on energy if the forces of the system have to outgo the intrinsic expansion of spacetime and 2) as for the geodesics at your footnote, maybe you could point me to some references and papers on arxiv or elsewhere, so I can study the details of geodesics in an expanding universe. Thank you again. – Constantine Black May 29 '16 at 7:36
• @ConstantineBlack: energy is not conserved in an expanding spacetime because the geometry violates time shift symmetry. Technically it has no timelike Killing vector. I don't know of a good source for geodesics in the FLRW metric. Perhaps I should write one :-) – John Rennie May 29 '16 at 7:40
• Well, this is outrageous :) (I am kidding of-course). Thanks anyway for your trouble. Is there a way, if I find something on the subject, to post it here or somewhere else on the site? Good day. – Constantine Black May 29 '16 at 7:45
• I would argue, that even at the solar system or in the galaxy, we cannot talk about expansion because the FRW solution is not valid. Would you agree with that? Thanks. – Constantine Black Jun 7 '16 at 14:03
• @ConstantineBlack: yes, and that's why I said in my answer you'd find at the small scale it didn't look anything like the FLRW metric :-) – John Rennie Jun 7 '16 at 14:05

Because there is no quantum theory of gravitation, we cannot express with confidence the microscopic view of what is happening. However, at the large scale the expansion of space is described by the Hubble parameter, and the Big Bang model, which is informed by the Cosmic Microwave Background (CMB), which once was 3,000 K,but today is about 2.7 K, which represents an expansion of space of slightly more than 1000-fold since the temperature fell below 3,000K, and the cosmos became electromagnetically transparent.

Since ordinary quantum theory considers electrons to be point particles, this expansion does not affect them. For the simplest atoms, such as hydrogen, the ground state orbitals have an average distance from the nucleus, and so does each increasing orbital - and as the electron is shifted from one orbital to another there is a fixed energy relationship based on the potentials calculated from these distances. Extended observation of the hydrogen line, 21 cm shows no variation with time of emission, even over cosmological time scales. So the sizes of atoms don't change.

We also observe galaxies, and the observed gravitational behavior is apparently unchanged over cosmological timescales.

The conclusion: even if space is slowly expanding around and through all matter, the binding forces keep things "put together" at their natural sizes. That is, the change in coordinates of a particular patch of space as it slowly grows with time, is different than the growth of a bar of metal as undergoes thermal expansion. The parts that are connected, stay connected in space.

This may have implications for our understanding of electromagnetic waves which require the comments of a specialist who has thought carefully about this.

• I very much like the mention of the observation of the 21 cm line. However, the statement "even if space is slowly expanding around and through all matter" implies that the local metric in the solar system, for example, has to be non-static and, in particular, expanding. But it really does not. The combination "net effect of local stuff plus rest of cosmos" can add up to a static local metric for the local spacetime. (c.f. ans from John Rennie). – Andrew Steane Nov 20 '18 at 20:00