# Nothing can travel faster than the speed of the light, but could this be an exception? [duplicate]

I often show off by asking people how fast electrons flow through wires. Then I tell them it's actually only a few millimeters per second. To clear the disbelief off their faces and boost my ego, I make the analogy as follows:

If I have a 10 meter long stick, and I poke them with it, the effect is nearly instantaneous, I can poke them well under one second, yet the stick is moving far less than 10 meters per second.

Now what I want to know, is if I had an extremely long stick, made of something lighter and stiffer than say carbon nanotubes, whacked one end with a hammer, would it be possible for the effect to be detected at the other end faster than light could get from the hammer side to the other?

I'm guessing it wouldn't, and I'm guessing I won't be able to understand the answer, but I'll ask anyway! Perhaps someone can give a simple and also a suitably comprehensive explanation?

Nope, not an exception.

A stick, like any solid object, is essentially a bunch of atoms held together at a particular equilibrium distance. Atoms of a solid are pretty well modeled by a lattice of spheres connected by springs (with damping).

What actually happens when you hit one end of a stick, is that you compress the surface atoms inwards towards their neighbors. They in turn exert a force on the next layer in, and so forth down the entire length of the stick. The effect of your hammer strike appears to the human eye to have an instantaneous effect on the other end of the stick, but in reality it is transmitted as a wave through the length of the stick, and it takes a minimum time of $t = \frac{L}{v_{sound}}$ to make it through the stick.

So even if you have a super long, super light, super strong stick, it will take a finite, speed-of-light obeying amount of time for the other side of the stick to react.

As I wrote this, I realized I don't have a good analytical argument why the speed of a compression wave $c = \sqrt{\frac{K}{\rho}}$ (where $K$ is the bulk modulus and $\rho$ is the density) has to be limited to the speed of light. I'm curious if anyone can shed light on that.

• Very nice illustrative gif. +1 – Steeven Jun 11 '15 at 9:40
• Thanks - It's the one in the Wikipedia article I linked to. – Brionius Jun 11 '15 at 9:49
• Changes in the electromagnetic field propogate at the speed of light, and that field is what primarily governs interactions between atoms, so light speed limits force propogation in the material. – Alan Jun 11 '15 at 9:51
• See the addition to my answer for the simple explanation of why the compression wave you describe is still limited by the speed of light. – Steeven Jun 11 '15 at 9:59

If you punch one end, the atoms at that end are displaced. They push the neighbouring atoms, who push their neighbours etc.

This "push" comes in the form of an increase electric field, because the atoms get closer. Such changes in electric field propatages with the speed of light. So if you could move an atom close to another atom instantaneously, then the neightbour would feel the new atom pushing after the field has propagated at the speed of light $c$.

Your punch will propagate as a mechanical pressure wave from one end to the other. Depending on the material rigidity this will be fast or slow - but is not in any way comparable with the speed of light.

• But would it be theoretically possible, with an ultra advanced material? – CL22 Jun 11 '15 at 9:37
• You mean for perfect rigidity? In a model, yes. But in reality, relativistic effects will show up. The speed of light is the max-speed of matter - you cannot cross that limit. – Steeven Jun 11 '15 at 9:40