The Light-Year Long Plank

A while back I stumbled across the following video, and the conundrum in it resurfaced in my thoughts later and I presented it to a few friends. It provided an interesting hour or two at the college cafeteria for discussion, and tons of disagreements.



One person in particular suggested the following:

The speed of the information traveling through the plank would approach the speed of light (c) as the bonds joining the atoms in the plank become 'stiffer', and if you had a completely rigid atomic structure, the plank's other end would move nearly instantaneously as the information traveled faster than light. The whole structure would move in unison if we reached complete stiffness, as there is no room for contraction waves and every particle moved at the same time.

The first part of this is correct as described in the following formula for the speed of sound:


 
(from Wikipedia)
Where:

E is a coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gases),

is the density

The speed of sound decreases with density p and increases with bulk modulus (stiffness), which in solids is Y, Young's modulus.. The question then is, what is the highest that E can be?

The transfer of this movement through the plank is a pressure wave that moves the atoms within. When an atom is pushed, its electrons create a force against the electrons of the neighbouring atom, moving it and causing eventually the whole system to be displaced when the movement reaches the end of the plank. Diamond, the hardest bulk material known to man, has a speed of propagation of 1200 m/s (as opposed to air's speed of about 340. This is because diamond is not very elastic at all and it has the density of carbon. However, dealing with a theoretical material with total stiffness, we run into a problem.

 Faster than light or near instant?

Can infinitely high bulk modulus allow sound to surpass c, the speed of light? Is this property possible? Firstly, atoms have a limit on their bonds' rigidity; and there are reasons why infinite bulk modulus does not make sense.

A completely rigid system would mean that the initial atom would not be able to move in relation to the one next to it, with no relative change in distance between them, and no push exerted. This means all the energy would be absorbed and none would be transferred onto the next atom. Clearly this is a problem. It might seem counter-intuitive but the idea of completely stiff atomic bonds does not make sense since atoms need at least some degree of movement in order to act on nearby atoms.

The idea is a lazy one because it doesn't explain how atoms could transfer energy to each other. Infinite stiffness does not make sense and could allow one to decide its outcome as anything because it is undefined. Dividing infinity by a finite number and taking its square root will give you an undefined answer.

This model ignores the restrictions of our current model of the universe, and therefore the current model of the universe cannot tell us anything useful about it. Ok, so if instant movement does not make sense by definition, we need to consider the next issue issue: the possibility of information traveling faster than c, the speed of light. Tachyons are the only exception to this rule and as far as science has seen, they do not exist. Furthermore, other areas of astronomy can shed some light on this, as this passage I quote from here, about the largest theoretical size of neutron stars before they collapse into black holes:

Since the conditions in a neutron star are very difficult to duplicate on Earth, nobody is exactly sure just how big a neutron star can be. One indirect argument is based on the fact that as a neutron star becomes more massive, it must become stiffer to maintain itself, and the speed of sound through the star increases accordingly. Above six solar masses, the speed of sound exceeds that of light, which is ruled out by Einstein's theory of relativity. 

Six solar masses is only an upper bound on the size of a neutron star. More practical calculations estimate the upper limit as three solar masses. No objects confirmed as neutron stars are known that are larger than two solar masses; the mass is typically about 1.35 solar masses.

At the speed of light, or below?

Once we discard the possibility of faster-than-light sound-waves, one begins to wonder; what is the speed of a force? This might sound strange, but firstly, the energy is transferred from one end of the plank to the other through electromagnetic force that, as mentioned before, keeps atomic and subatomic particles apart. According to the standard model and the theory of relativity, forces travel at the speed of light, so the weak and strong nuclear forces, gravity and electromagnetism are not instantaneous as may be perceived in the small scales in which we might perceive them.
 

What is the maximum speed of sound? Theoretical and observed?

 In the link above, the following possibilities are explored by some people:

• Plasma: matter in this state goes by slightly different mechanics, and bulk modulus would be the energy associated with electron degeneracy, and the guesstimated speed would reach about
  500,000 m/s. (100 times slower than light).
• Neutron stars: electron degeneracy is followed by neutron degeneracy which makes matter collapse onto itself. This causes the bulk modulus to rise exponentially (as predicted by my friend). To quote the result:

"At that point, information from the center of the star can barely reach the edge, whether it's light or sound, so the local speed of sound at a neutron star just about to become a black hole would probably be just about light speed.

References: I used no references and I also don't know what I am talking about. Please do not substitute this answer for medical advice."

The distance the electromagnetic forces communicate movement through is not the space between the atoms (atoms are mostly empty space); it's the space between the electromagnetic fields of the electrons that interact with each other. That space is VERY CLOSE to a light year long in our experiment. The whole process of movement includes this space plus the distance particles (which can't be completely rigid) must travel to influence one another. In theory, as long as the formula used to calculate it does not provide undefined answers, none of the parameters stray off the current model of the universe we understand and the answer reaches c but isn't equal to it, there would be little reason to dismiss claims of the possibility that some materials may be able to transmit at almost such speeds. The best example I could find was in this following paper that concludes that evaluating equations at different densities for neutron stars all yield values for the speed of sound inferior to c.

Wait a minute! What about black holes?

Most of the theory before this deals with matter before it collapses into black holes. Most mathematical models explaining spacetime and force break down when they deal with black holes since they consist of matter that has collapsed further than a neutron star would have and has become so compressed it ceases to have actual volume.

More info:
Why is the elastic modulus relatively insensitive to changes in chemistry/heat treatment(...)
The Nature of Sound - The Physics Hyperbook
The Superluminal Scissors (similar Gedankenexperiment)
Compressibility of a Black Hole
More on the speed of sound in neutron stars: http://qr.ae/RPawRn