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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:

  1. Can this calculation be done, or is there some reason the whole idea is unsound?
  2. 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.)

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Mark Dominus
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  • I would doubt there is such a limit. Rigidity is a statement about the material, which could be formulated in some local rest frame, whereas special relativity is essentially about kinematics. – Isidore Seville Dec 09 '13 at 03:57
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    @IsidoreSeville Er...rigidity is related to the speed of sound $v_s$ in the body and there is a hard limit $v_s < c$, so relativity does put a limit on rigidity. This is one of the arguments that without a modification to relativity all fundamental particles must by dimensionless. – dmckee --- ex-moderator kitten Dec 09 '13 at 03:59
  • @dmckee are you saying that there is a hard limit or there isn't? Your first sentence says there is, but your second sentence seems to be saying that the first sentence needs to be modified. – Brian Moths Dec 09 '13 at 04:04
  • @dmckee By rigidity, I mean the stiffness, i.e. the bulk modulus K in the linear elasticity. I doubt SR can put any limit on K. I think we can imagine the relation between K and speed of sound gets modified in the regime where SR effects become significant. – Isidore Seville Dec 09 '13 at 04:04
  • @IsidoreSeville I agree with this. If we imagine a rod that is already compressed and there is some force required to keep it compressed, I see no way SR puts an upper bound on this force. You can think of potentials that are arbitrarily sharp and these potentials are consistent with the framework of SR. – Brian Moths Dec 09 '13 at 04:06
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    @NowIGetToLearnWhatAHeadIs: There are so called energy conditions, they do pose limitations on strains and tensions. (Although they are usually employed in general, not special relativity). – user23660 Dec 09 '13 at 04:22
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    It comes down to: there can't be a perfectly rigid body, exactly because of special relativity. No matter what the macro force source, it comes down to fields imparting force on particles, and they can't be accelerated beyond said limits. – Carl Witthoft Dec 09 '13 at 12:39

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I believe that one could rephrase the question as "if the limit of the speed of sound in a medium must be the speed of light in vacuum, what does that mean for the limit on rigidity of an object?"

Speed of sound is given by $$c=\sqrt{\frac{E}{\rho}}$$ - it depends on both density and Young's modulus. I would consider "rigidity" to be just the modulus, and if there is no theoretical limit on density then there is no theoretical limit on rigidity (following your logic).

Of course from a materials science and quantum mechanics perspective there is always going to be a finite force-distance relationship for atoms - this sets realistic limits on elastic modulus that are well below the theoretical one calculated above. At 12,000 m/s, diamond (a very rigid material) is still far away from the limit.

Floris
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