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.
Answer
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.
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