In a universe that is expanding at a constant rate, do objects that are attracted to each other feel a force opposite to their attraction?

Question

In this article, the authors make the claim (pg 44) that "Expansion by itself—that is, a coasting expansion neither accelerating nor decelerating—produces no force."

I'm having a hard time convincing myself of this. I think about two bonded atoms and the space between them expanding. While I don't doubt that the structure remains intact, it seems to me that the attracting force will always have to play "catch-up" against the expansion which is pushing the particles apart. Where am I going wrong in thinking of expansion as a force which acts counter to the attraction? Does anyone have a metaphor handy?

Another way of stating my question: with large enough (but not accelerating) rate of expansion, would we get to a point where molecules broke apart?

EDIT: In this related question, the agreed-upon answer seems to be that the expansion of space manifests itself as a force.

Answer 1

You need to be careful what you mean by a force in general relativity. The usual definition of a force is that you get a non-zero reading on an accelerometer you are holding, but this can lead to some surprising conclusions.

To illustrate this suppose you are falling towards some massive body (with no atmosphere to complicate the issue). Does the gravity of the massive body create a force? The answer has to be no, because you're in free fall so you're weightless and any accelerometer you were carrying would read zero. Suppose know we give you a harness and tie you to some support fixed wrt the massive body. Now you feel a force, and your accelerometer reads non-zero. But this force is due to the fact you have been restricted (by the harness) from following the geodesic you would otherwise follow. It's the harness that exerts the force on you (and you on it) because it's pulling you away from geodesic motion and therefore imparting a non-zero four acceleration. The force isn't due to gravity, it's due to the harness and without the harness there will be no force.

Incidentally, although it's peripheral to this issue there is a nice calculation of the four acceleration and force in twistor59's aswer to What is the weight equation through general relativity?.

Let's go back to expanding spacetime. Hopefully you'll now see why we say that the expansion of spacetime doesn't create a force. Suppose we place you and me at some distance apart in an FRW universe and constant comoving position and we wait to see what happens. We'll each have an accelerometer so we can tell if we're accelerating.

If we now wait the expansion of spacetime will increase the proper distance between us - that is, we will move apart. However because both of us are at constant comoving position we are moving along a geodesic and experience no acceleration. Our accelerometers will read zero, which means we feel no force. In this sense the expansion of spacetime does not produce a force. This is exactly analogous to the claim I started out with, that the gravity of a massive body doesn't create a force either.

Now suppose we tie ourselves together with a rope. Once we have done this we cannot remain at constant comoving position and this means we must be accelerating. Our accelerometers would now register a non-zero acceleration towards each other and we'd feel a force. Any objects we drop will fall away from us. This is exactly analogous to using a harness to support yourself against the gravity of a massive body. It's the rope between us that generates a force not the expansion of spacetime.

By now you're probably thinking that this is all a bit of a swindle and I've just redefined what is meant by force to make it zero. Well, yes, but this is key to understanding general relativity. We don't often talk about force, but four acceleration is precisely defined in GR and can be calculated as described in the question I linked. When a general relativist talks about force they implicitly mean four acceleration. This does mean they are using the term in a different way to the general public - hence the confusion.