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?
The most interesting version of this question (I think) is this: Suppose that the rope has been attached long enough to bring the two galaxies to a stop relative to each other. (d[physical distance]/d[cosmic time] = 0). For subsequent times, does the rope need to maintain any tension to keep the galaxies from starting to accelerate away from each other? If we cut the rope, does the distance to the galaxies start to increase, decrease, or stay the same?
The answer, it turns out, is that if the expansion of the Universe is decelerating (e.g., if there's no cosmological constant), then there need not be any tension in the rope after the galaxies have come to a stop, and if the rope is cut, the galaxies do not recede from each other. On the contrary, they fall toward each other. With a sufficiently large cosmological constant, on the other hand, they do recede after the rope is cut.
Charley Lineweaver and Tamara Davis wrote some quasi-pedagogical articles on this, http://arxiv.org/abs/astro-ph/0104349 being the main one. Personally, I think that this question is helpful in clarifying intuition about the "meaning" of expanding space. The tethered galaxy problem is part (but by no means all) of the argument we tried to make in http://arxiv.org/abs/0808.1081 . (Sorry for the self-promotion, but I honestly do think it's a relevant citation, albeit not quite as relevant as the Davis-Lineweaver work.)
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
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.
A related question seems to be whether one can get useful work from the expansion of the universe. The rope could be a long tether with inductive coils and magnets in a chain. As the tether is pulled the magnets run through the solenoids and induce a current. The tether could then transmit the power back to the “home galaxy.” There would be loss of this energy due to $z~\simeq~Hd/c$ from more distant inductors along the tether, but you could in principle generate energy this way.
This is curiously the dream of the ZPE energy guys who think they can hook a cable to the vacuum and get “free energy.” However the cosmological constant $\Lambda$ is very small so the amount of energy in any volume, or in this case length, of space is miniscule. Tethering two galaxies together is a bit tough to accomplish I should think.
In my view, the whole theoretical experiment of the tethered galaxy is intrinsically flawed, because the presence of a tether, rod, or any other implement attached to the two galaxies introduces such an inhomogeneity in the environment that it invalidates the basic assumptions on which the FLRW metric is based.
Lest this statement seems unwarranted, I will provide some further explanation. To note, the inhomogeneity problem lies in the very existence of the rope or rod fastening the galaxies, and not in the masses of the galaxies or the tether. And to see why this implement flaws the whole experiment, we have first to realize that keeping galaxy B at a fixed proper distance to galaxy A (by any means) results in a symmetrical situation, in which it can also be rightly said that galaxy A is kept at a fixed proper distance to galaxy B. And since neither of the galaxies is "nailed" to the comoving background, the result is that both galaxies are no longer comoving.
Now, since the galaxies are no longer comoving, the times ticked by their clocks are no longer comoving time. Let's assume that both galaxies are of the same mass, so that they are both moving against their surrounding comoving background at the same relative velocity. Therefore clocks at each galaxy are ticking proper time t' where the relationship between t' and comoving time t is given by simple SR time dilation based on the relative velocity. On the other hand the middle point of the rope/rod is comoving, and therefore a clock attached to that point is ticking comoving time t. So we have a rigid object (the rope/rod with the two galaxies attached at its ends) where clocks at different parts thereof tick different proper time! This logical inconsistency is the necessary consequence of the sheer existence of the rope/rod itself, which violates the basic assumptions on which the FLRW metric is built.
Therefore, instead of dealing with a "tethered galaxy" it is much more insightful to study the case of a "stationary rocket" flying towards the origin of coordinates, with the crew continually fine-tuning the impulse in order to compensate exactly for the expansion of space (which they have calculated) so as to keep constant the proper distance of the rocket to the origin, as observed from the origin. At t = tr the crew stop applying impulse so that the rocket starts moving freely along a radial timelike geodesic. This scenario not only does not invalidate the assumptions of the FLRW metric but also is perfectly physical.
For t < tr, the proper velocity of the rocket as observed from the local comoving frame of reference (denoted as vsr) is equal to the recession velocity of said FR as observed from the origin, which is the constant proper distance to the origin Dr times the Hubble parameter:
vsr(t) = - Dr H(t)
The analogue to the tension in the rope is the acceleration that the rocket needs to have with respect to its local comoving FR in order to keep Dr constant, which is:
dvsr(t)/dt = - Dr dH(t)/dt ("needed" acceleration)
If as usual dH(t)/dt < 0, the needed acceleration is positive (away from the origin).
But in order to know whether the crew must apply impulse, and in what direction, that "needed" acceleration must be compared to the "free" acceleration that the rocket would have if left free in a timelike geodesic:
dvs(t)/dt = - H(t) vs(t) [1 - vs(t)^2/c^2] ("free" acceleration)
If vs(t) < 0 as in this case, the free acceleration is also positive.
Comparing both accelerations for different models:
in the empty linearly expanding model the needed acceleration is greater than the free acceleration, so the rocket must apply impulse away from the origin (i.e. throwing their exhaust towards the origin). (BTW, the proverbial "rope" would be useless, and what it would be needed would be a rod to keep the galaxy away.)
in the lambda-only exponentially expanding model the free acceleration is greater than the needed (which is 0 as H is constant), so the rocket must apply impulse towards the origin.
For t > tr (engines off), the proper velocity of the rocket starts decreasing according to the equation of the timelike geodesic of the FLRW model in use. The evolution of the distance of the rocket to the origin depends on the FLRW model:
In the empty linearly expanding "Milne" case, the recession velocity of the rocket's initial comoving background Dr / t decreases faster, so that the rocket starts getting closer to the origin. This in turn starts decreasing Dr(t) and therefore causes the recession velocity Dr(t) / t of the rocket's comoving background to decrease even faster, so that the rocket will keep getting closer to the origin until it reaches and passes it.
In the exponentially expanding "de Sitter" case, the recession velocity of the rocket's initial comoving background Dr H is constant, so that the rocket starts getting away from the origin. This in turn starts increasing Dr(t) and therefore causes the recession velocity Dr(t) H of the rocket's comoving background to increase, so that the rocket will keep getting away from the origin.
I think it might be clearest to first consider the case of a rope NOT connected to anything; a free, straightened rope in empty space. Each point along the rope feels the same effective force due to Hubble expansion; each point along the rope is slightly repelled by each of its neighboring points. Assuming the expansion is isotropic in space, there is no need to consider the speed of any forces propagating along the rope; the rope experiences a uniform tension that increases linearly with the rope's length. If the rope is long enough the stress on the rope will overcome its tensile strength, and it will snap.
Against all the above opinions: The rope will shrink
and maintain the normal tension between atomic structure
A space expansion scenario is dual (almost) to a shrinking matter scenario. Unless we have a way to decide I will support my point.
Already explained here: A relativistic time variation of matter/space fits both local and cosmic data and here: Cosmological Principle and Relativity - Part I (arxiv astro-ph 0208365)