# Does an upper limit on speed (the speed of light, $c$) also imply another universal constant: an upper limit ($R_{max}$ ) on the stiffness of matter?

I was imagining the following thought experiment:

This question is linked to the question:

Is it possible for information to be transmitted faster than light by using a rigid pole?

but it is not identical.

The question referred above asks if one could theoretically do an instantaneous action at a distance, using a perfectly rigid—i.e., infinitely stiff—rod (and if, thereby, special relativity could be challenged with that thought experiment).

This question assumes on the contrary that special relativity is correct, and asks if this unambiguously implies that there ought to be a universal constant $$R_{max}$$ corresponding to the upper limit of material stiffness materials (regardless of any underlying microscopic model of the object (the rod) implied) or if another explanation is possible (perhaps involving GR).

• If there existed a material with infinite stiffness, one could make a rod of any given length (say 1 meter or one kilometre,...) which we could use it to activate a light switch (for example) remotely and instantly. Why instantly? Simply because if the rod is really perfectly rigid, then if we move one end of it, the other end moves simultaneously without delay (by definition of the material being rigid).

• If, on the contrary, the chosen material has a finite stiffness, then a pressure exerted at one end will take a certain time (very short, but non null) to reach the other end, the pressure propagating from end to end, like a compression wave propagating through a spring.

• the stiffness of the rod must be such that the speed at which an action exerted at one end cannot exceed that of light.

• As a consequence, the stiffness of a bar of material cannot be infinite.

So there ought to be an absolute theoretical limit to the stiffness of a material.

Am I correct as to the conclusions drawn from that thought experiment?

• I meant rigidity. The confusion comes from the fact that the coefficient of stiffness k of a spring is often called (coefficient d'élasticité in French (my native tongue)) Commented Oct 15, 2022 at 9:15
• Does this answer your question? Is it possible for information to be transmitted faster than light by using a rigid pole? Commented Oct 15, 2022 at 9:20
• The speed of sound can not exceed the speed of light, but in a relativistic universe the speed of light is "infinite". It can not be exceeded by any phenomenon. What is not infinite is the ability of a physical action to return to the local past that it came from. That is a restriction on causality, not a restriction on "speed". I am not sure these finer points are adequately captured in non-relativistic material properties like "stiffness". Commented Oct 15, 2022 at 9:24
• What is rigidity? I know what the modulus of elasticity is, but I don’t even know what units your $R_{max}$ would have
– Dale
Commented Oct 15, 2022 at 11:51
• I edited the question to change "rigidity," "elasticity," "incompressibility," and "hardness" all to "stiffness" to avoid ambiguity and confusion. I believe these edits retain the intended meaning. The dichotomy of elastic deformation can be articulated as stiffness (e.g., the bulk modulus, shear modulus, and Young's modulus, which mediate the speed of various types of acoustic waves) vs. compliance. Commented Oct 15, 2022 at 22:39