Take a perfectly rigid metal rod of length $2\ell$ and some uniform linear density. Place one end (‘south’) at $(0,-\ell)$ and the other (‘north’) at $(0, \ell)$. Over some reasonably short time interval $t$, perhaps on the order of a fraction of a second, displace the center of the rod eastward from $(0,0)$ to $(1,0)$. In practice it's very easy to do this so that the entire rod moves one unit eastward; in particular the north end moves from $(0, \ell)$ to $(1, \ell)$.
But this is actually a classical view of the situation. To see this, make $\ell$ very long, say on the order of ten light-seconds, and large enough to be bigger than $t\cdot c$. Then at time $t$ the center of the rod is at $(1,0)$ but the north end is still at $(0,\ell)$, because the north end can't have noticed yet that the middle has moved.
But this has an implication for the material properties of the bar. I claimed in the first paragraph that it was perfectly rigid, but it now appears that it isn't as rigid as all that. Purely from speed-of-light considerations we can conclude that even a perfectly elastic bar must temporarily deform in the process of being translated from $x=0$ to $x=1$.
It seems to me that if one assumed that the rod had length $2\ell$ and uniform linear density $\rho$, then one could calculate the amount of force required to translate it from $x=0$ to $x=1$ by pushing on the midpoint. Then supposing that the rest of the rod followed as quickly as speed-of-light propagation allows, one could calculate the stiffness of the rod, and this would be a theoretical upper bound on the maximum stiffness of any material whatever.
But I don't have enough expertise or understanding of materials calculations to do actually perform this one. Also I suspect I must have left out something important, for the same reason.
My questions are:
- Can this calculation be done, or is there some reason the whole idea is unsound?
- If it does make sense, what upper limit on material stiffness does this method produce?
I suppose that if it does work, the upper bound is vastly greater than the stiffness of any real material, but I don't mind that.
(I found the question Extended Rigid Bodies in Special Relativity, which is clearly related to this, but doesn't get at what I want. My earlier question Behavior of shock waves at relativistic speeds started out as an attempt to ask this one, and somehow went in a completely different direction by the time I posted it.)