## Very Long String vs. Speed of Light

Here’s a physics riddle that I just thought up. The only problem is that I’m not sure what the answer is. But I like the question, so here we go:

Let’s say you have a very long piece of string, and it has absolutely no elasticity, it’s perfectly straight, and it’s totally frictionless. The string is floating out in the void of space and it takes light 1 year to travel from one end of the string to the other.

What if you and I are at each end of this string, and I pull on it (or I have an incredibly powerful machine do it for me). How fast would you feel the pull?

Intuitively, it seems like it would be instantaneous, but that can’t be right because it would be traveling faster than light. The answer is probably that it would depend on the mass of the string and the amount of energy in the pull, but that whatever those variables are, it would take at least 1 year, and that while that isn’t intuitive, our brains haven’t evolved to intuitively think about this kind of situation. But what if the string is massless? But then, if I add this premise, is it even a real question for physics anymore? Anyway, I thought it was a fun thought experiment.

### 8 Responses to “Very Long String vs. Speed of Light”

1. Jesse Says:

Imagine you have a lightyear-long metal rod of negligible but non-zero mass, of minimal but non-zero thickness. In short, you have enough to have a crystal lattice structure, but nothing more.

When you pull on one end, you deform the crystal lattice — lengthening it ever so slightly. The lattice adjusts itself back to equilibrium down the length of the rod, until the deformation reaches the end of the rod, which moves in the direction you originally pulled. The rate at which a plastic deformation of a lattice propagates is, by definition, the speed of sound (sound is carried on a series of compressive and tensive deformations) in the material. In this case, the mass of each differential slice of the rod is what opposes the instantaneous stretch of the lattice and slows down the propagation rate.

Imagine the rod is somehow massless — some form of contrived matter where the electron-analogue and proton-analogue have charge but no rest mass. Each differential slice of the rod can be accelerated instantaneously (no mass, no momentum change needed). Then the rate of deformation of the lattice is the rate at which electron-analogues can be forced out of the way. That is the speed of light in the material (indeed, that is how electricity flows (look up drift current if you want to see how slow the individual charge carriers move)). Incidentally, all real-world conductors have parasitic electromagnetic properties, so if you were to use a massless copper analogue, for example, you’d be limited to a propagation speed of 2.3×10^8 m/s.

My credentials are a BSc in EE. I concede to a physicist, EE, or matsci type any mistakes I’ve made.

2. Michael Graham Richard Says:

Thanks Jesse, that makes a lot of sense. I wasn’t expecting the speed of sound to come into it, though. I’ll have to look that up.

3. Selit Says:

I think the string would move at the speed of sound in the material that it was made out of. Your saying it’s not elastic but I think it is still supposed to work this way. I don’t have personal knowledge but I remember a similar question being asked on the discovery channel years ago about bending a ‘super rod’…

4. Mike Says:

So here’s an important question… how do you pull on a frictionless string?

5. Michael Graham Richard Says:

Ha! Good question, Mike.

I guess when you start to make these theoretical scenarios, you have to decide where to stop making unrealistic assumptions… I decide to stop a bit past that point where you can actually pull on the string

6. Ali Says:

Physics get funner when you start spinning the rod around you.

7. David Says:

Mathematically infinities are usually not handed directly, but in the case of limits: instead of asking what happens with a massless infinitely stiff string, we can ask what happens to a string of mass M stiffness K as M and K become larger and larger.

I think you have probably already realised the signal will travel at c (the speed of light), and are trying to build a mental model that helps you understand why your intuition that it should be instant is wrong.

Here’s how I would look at the problem: given that the string has mass per unit length rho and modulus K, we can work the wave speed v. When solving with purely Newtonian mechanics, we get v=sqrt(K / rho). We are interested in the limiting case of K->infinity, which suggests v->invinity. This Newtonian solution is not valid.

Instead we should work out the Special Relativity equivalent, but it’d be a lot of maths. however we can recognise that when solved using special relativity, the solution will give a speed that ->c as K-> infinity, perhaps something like v=Sqrt(K/rho) *Sqrt(1-K^2/(c^2*rho^2)).

In simpler terms, our intuition that the wave speed goes to infinity is based on Newtonian mechanics, which is invalid in such cases. Special relativity will give a correct speed.

8. Kevin Says:

You’ve just shown that the theory of relativity does not allow perfect rigidity or perfect inelasticity. These are possible only if you have instantaneous transmission of force, which cannot occur.

It makes sense that perfect rigidity should be impossible when you think about the fact that your string is composed of smaller particles (atoms, molecules) with forces acting to hold them together while maintaining a particular pattern of distances between them. Even under Newtonian mechanics you couldn’t have perfect rigidity for any force that varies with the distance between particles — one particle has to move relative to the others before any forces can change, and it takes time for the other particles to accelerate in response to the change in the forces acting on them.