Tanelorn said:
I still do not get what you mean by this: "Only in a coordinate sense. Once again, thinking of these "superluminal speeds" as causing other things to happen will only lead to confusion."
I mean that this "superluminal speed" you are talking about is not directly measured by anybody. It's just a mathematical result you get when you use a particular system of coordinates, and divide the increase in "distance" in those coordinates over a given time interval in those coordinates, by the time interval. Coordinates don't cause anything.
Tanelorn said:
the sliding telescopic stick thought experiment helps and allows all frames of reference to not exceed relative super luminal speeds.
What does "all frames of reference to not exceed superluminal speeds" mean? Jorrie is only talking about local comparisons--relative velocities between a given piece of the stick and "comoving" objects (objects moving along with the Hubble flow) at the same spatial location. He is not talking about the "relative speed" between opposite ends of the stick (which is the "coordinate speed" I described above). That can still be "superluminal", but that doesn't matter because it's only a coordinate speed. No piece of the stick is moving faster than light rays at the same spatial location, which is what the rule "nothing can go faster than light" actually means, physically.
Tanelorn said:
I agree that as we start to stiffen the stick something has to give, but I am still not sure what.
What "has to give" as you stiffen the stick is that the maximum possible length of the stick decreases.
In the limiting case of zero stiffness (i.e., zero force between neighboring pieces of the stick), the stick's length can be anything; there is no maximum. That's because each piece of the stick just follows a "comoving" worldline--i.e., it moves along with the Hubble flow in its vicinity. To observers looking at a particular local section of the stick, the stick appears to stretch as neighboring sections move further away (as the universe expands). This doesn't cause any force or stress anywhere because the stick has zero stiffness; one part can't restrict the motion of any other part. Also, in this scenario, the piece of the stick that is exactly one Hubble radius away from the Earth is moving "at the speed of light" relative to the Earth end of the stick, in "comoving" coordinates. Pieces further away are moving "faster than light" in those coordinates. (But, as I said, this is just a coordinate speed and nobody actually measures anything moving faster than light beams at the same spatial location.)
If we make the stiffness of the stick nonzero, each part of the stick now exerts some force on neighboring parts. The boundary condition you appear to be assuming is that one end of the stick, the end anchored to the Earth, is moving along with the Hubble flow, i.e., it is "comoving". (The Earth actually isn't exactly "comoving", but we can ignore that here.) So at that end of the stick, there is zero stress. But that end of the stick is now pulling on the neighboring piece, exerting some force, so that neighboring piece is
not exactly comoving; it is moving away from the Earth end of the stick, but not as fast as a "comoving" object would. That means the neighboring piece is under some stress, because it is feeling a force pulling it off of a "comoving" trajectory.
As we move further and further along the stick, away from the Earth, we can apply the same argument: the first neighboring piece next to the Earth end of the stick pulls on the second, the second pulls on the third, etc. At each stage, the stress in the stick increases; the motion of each piece of the stick is a little more different from that of a "comoving" object at the same spatial location, and the force the piece feels is larger. So now, when we get to the piece of the stick exactly one Hubble radius away from the Earth, that piece will have a coordinate velocity of
less than ##c##, relative to the Earth end of the stick, in "comoving" coordinates, because it is moving away from the Earth end of the stick more slowly than a "comoving" object at the same spatial location. And the stress in the stick at this point will be, I believe (but I haven't done the calculation) small enough that the stick can withstand it.
However, if we continue along the stick, we will reach some point, at some finite "comoving" distance from the Earth, where the stress in the stick exceeds the maximum possible structural strength imposed by relativity (i.e., that the speed of sound in the stick would have to be greater than the speed of light for it to withstand the stress). That point determines the maximum possible length of the stick.
As we increase the stiffness of the stick, the force each part exerts on neighboring parts increases, so the speed of a given piece of the stick relative to a "comoving" object at the same spatial location increases faster. In the (unphysical) limiting case of infinite stiffness, no piece of the stick could move at all relative to any other piece; and in this case, the stress in the stick would go to infinity at the Hubble radius (which would be the maximum possible length of the stick). But, as I said, this case is unphysical, because relativity imposes a finite limit on stiffness--an infinitely stiff material would have an infinite sound speed, and the sound speed in any actual material cannot be greater than the speed of light. So any real stick would stretch some as the universe expanded--each piece would move to some extent relative to neighboring pieces, because of the finite limit on stiffness.