Suppose there is a strong stiff rope tied to a distant galaxy. The rope is so long that it passes by us here on Earth. If the galaxy is so far away that it moves away from us faster than the speed of light, does the rope move faster than light relative to us, here on Earth?
Just to make it clear: The string is not tied to anything in our galaxy, it is tied only to -an object in- the distant galaxy.
No, the rope would either tether the attached planet and pull it out of its galaxy, or it would snap.
Imagine the rope just stretched out across space. Would the expanding universe stretch the rope? No. The molecular forces holding the rope together would easily overcome the space time forces.
PS Just realized that you meant a loosely trailing rope, not fixed to the Earth or anywhere else at this end. A rope that long would be very heavy. The force needed to accelerate it would be enormous. Those forces would snap the rope.
To start to address this question the following set up is needed. The FLRW (Friedmann-Lemaitre-Robertson-Walker) energy constraint equation $$ \left(\frac{\dot a}{a}\right)^2~=~\frac{8\pi G\rho}{3c^2}~-~\frac{k}{a^2} $$ indicates the derivative of a scale factor $a$ that gives the distance $d~=~ax$, for $x$ a ruler distance scale, to another galaxy. The left hand side of this equation is the square of the Hubble factor $H$. The last term on the right is a factor involving the geometry or topology of the universe, where $k~=~1,~0,~-1$ corresponds to a spherical or closed universe, a flat universe and a hyperbolic or saddle shaped universe. The de Sitter cosmology with a constant vacuum vacuum energy $\rho~=~{\rm const.}$ results in the exponential expansion $$ a(t)~=~a_0~\exp\left(t\sqrt{\frac{8\pi G\rho}{3c^2}}\right), $$ where the assumption $k~=~0$ is employed. Observations support the idea of a universe with a flat space.
To consider the problem of a tether tying two galaxies we use the Newtonian energy $$ E~=~\frac{1}{2}mv^2~+~V(x). $$ It is interesting that we can use Newtonian physics, for the above FLRW energy constraint is very Newtonian with a zero total energy. That this happens has some interesting and deep reasons based on holography. This energy equation is employed with the idealization that the tether is anchored to a large mass in each galaxy, and the motion $v~=~\dot a x$ is due to the mass in a galaxy relative to the first the observer is in. The total energy, as with the FLRW energy equation, is zero. The kinetic energy $K~=~\int\vec F\cdot d\vec r$ has a clear relationship with the potential consistent with $\vec F~=~-\nabla V(x)$.
We then clearly have the kinetic energy of the mass associated with the tether mass $$ K~=~\frac{mx^2}{2}\left(\frac{\dot a}{a}\right)^2~=~\frac{m}{2}H^2d^2 $$ where the distance grows exponentially. This means if the observer wants to place some energy extraction system on the tether that energy can be generated. This seems rather odd, for it means energy is being generated from "nothing." However, the FLRW is an energy constraint $E_{total}~=~0$. In addition cosmologies are spacetimes with no timelike Killing vector, which means energy conservation is not defined in the usual way.
Is anything like this really possible? In principle there is nothing preventing this, though it is implausible. The tether might be a cosmic string, an ordinary type of rope will obviously break, which is a defect in the vacuum with a larger energy. The expansion of distance between the two galaxies would increase a linear distance for this vacuum defect, which means energy is being stored in this cosmic string that linearly depends on distance. This would then be a spring-like system, and our hyper-advanced being might extract energy this way.
