Question

Suppose we have two galaxies that are sufficiently far apart so that the distance between them increases due to Hubble's expansion. If I were to connect these two galaxies with a rope, would there be tension in the rope? Would the tension increase with time? Is the origin of the tension some sort of drag between the expanding space and matter?

Answer 1

Yes, of course, there would be some tension in the rope. The rope would eventually break, and maybe it would be slowing the galaxies motion if it were a really tight rope (you can't get rope with the required rigidity to stop the motion of galaxies in Nature).

If one only considers a pair of galaxies only, the Hubble expansion doesn't really differ from the ordinary motion of two objects away from one another. They want to move along the natural trajectories - those we observe - so any rope trying to prevent them from doing so will be stretched by the force of inertia of these galaxies. If you prevent some objects to move in a natural way they like, you will always experience an inertial force. Whether you call this force (translated into a tension in the rope) as "inertial" or "gravitational" in the cosmological context is up to your taste: after all, the equivalence principle is what guarantees that the effects of gravity and acceleration are indistinguishable so both answers are "equivalent" from a GR viewpoint.

If the tension in the rope (well, I would say a spring) can be written as k times the excess proper length, then the problem of its tension as a function of time is equivalent to the problem of the proper distance between the two galaxies as a function of time. This is nothing else than the a(t) parameter used in cosmology. See some texts on the Friedmann equations

http://en.wikipedia.org/wiki/Friedmann_equations

that this a(t) satisfies. As a result, a(t) was given by various power laws as a function of time. As we're entering the era dominated by the cosmological constant, a(t) becomes exponentially increasing in t. So already today, the tension in the rope is increasing kind of exponentially.

Of course, one has to be careful about the literal interpretation of these things. The signals about the tension in any real "rope" are propagating by the speed of sound which is usually much slower than the speed of light. So it would take a lot of time for the most of the internal part of the "rope" to learn that it is attached to any galaxies at the endpoints. So most likely, the rope would get torn apart at the very endpoints very quickly while the internal bulk of the rope would stay at rest. You would have to specify more precisely what kind of a rope you want to consider if you want to solve the "engineering question" rather than the conceptual question about the changing proper distances in a cosmology.