My friend Pete (he’s the one who looks like John Tesh) is trained as a physicist, so naturally I thought of him when I couldn’t get this thought experiment out of my head:

Imagine a spaceship, one light-year in length.  Assume it’s built in such a way that it magically has zero ability to compress or flex.  Humans are in the front, and engines are in the back.

If the engines turn on, I know that it takes one year from the perspective of the crew until they could see the light from the engines.  Obviously, the vibrations from the engines would take much much longer.

What I don’t know is when the humans would notice motion.  If the motion was bound by the speed of light, then it would take them one year to notice that they were moving.  But based on my rudimentary physics knowledge, this doesn’t seem to make sense.  I never heard of any physics that made the head of an object move long after the tail was pushed.

Luckily, Pete saved me from a series of sleepless nights in which I half-wondered if I’d derived some form of really clunky FTL communication technology.  His response was so thorough and informative that I wanted to share it with the world.  Well, the subset of the world who reads my blog.

The speed of light is not just a limit for light transmission, but for all interactions. From a particle viewpoint, interactions are mediated by particles. In the case of electromagnetism, this particle is the photon, which moves of course at the speed of light. The mediator particles for other types of interactions are also limited by the speed of light. Generally, massless particles travel at the speed of light, and all others are limited by it (but can never reach it.)

The macroscopic effects of what you described, aside from any resulting motion, might be called material stress or a material shock wave.

With regard to the humans in the front, not only is there no instant notification of what’s happening in the back, there can be no effect at all for 1 year. That is, no stresses, no motion, no nothing.

So what you’re thinking right now is, well then how can I conserve momentum? If the engines are spewing things out backwards, the center of mass of the spaceship taken as a whole must be moving the other way to balance that. In fact, there is momentum produced in the other direction (as there must be), and if you found a way to calculate precisely the center of mass motion of the spaceship as a whole you’d find it to be moving, but it is in the form of either material deformation in the rear, or microscopic dynamics that on a macroscopic level could be described by a type of wave in the material which can be associated with an overall macroscopic momentum.

In this context, an analogy between the spaceship material and fluid medium might be useful. If you’re in the middle of the lake and row your boat, the boat goes one way, and the momentum is balance by the motion within the water. The water at the edge of the lake doesn’t know anything about it until the waves you’re making reach there. If you were to calculate the momentum of all the water molecules at once and add them all together, you’d find that it cancels out the momentum of your boat. You can think of the momentum of your engine exhaust being balanced by the momentum associated with the shock wave traveling forward through the material of your spaceship.

You might not believe that the effects of the engines could be absorbed entirely by material waves with no macroscopic motion. This is because the scale of your spaceship is rather extreme. If you made the spaceship too rigid and not large enough width-wise, and the engines were too powerful, then the rear of your spaceship would just bust apart. If you imaged that the spaceship was a borglike cube of solid iron 1 light year per side, then you can start to believe that the engine thrust could be “absorbed” by wave disturbances within the material.

The apparent paradox in your thought experiment is due to the false assumption that you can make something that’s completely rigid. This is not physically possible. On a logical level, your thought experiment itself, combined with the speed of light restriction, proves this. On a microscopic level, you can make arguments regarding microscopic interactions as I did above for why 100% rigidity is not possible.

This reminds me of a special relativity problem I used to give my students that is related in one respect, but otherwise different:

Imagine a garage that’s 20 ft long, and a pole vaulter with a 30 ft pole that can run really fast into the garage.  As you may know, things that are moving get contracted in length. This is not because they are being compressed or anything like that – it is a result of the nature of space time.

Suppose the vaulter runs so fast that in the rest frame of the garage the pole is only 20 ft long. (He would have to run really really fast to get that kind of effect.) Then he would be able to fit the pole inside the garage, and you could close the door quickly and the entire pole would be contained in the garage undamaged for a split second until everything hits the back wall with massive destruction ensuing.

So that’s all well and good, but what about the rest frame of the vaulter? In that frame, the pole is still 30 ft. To make matters worse, the garage, which is in this frame speeding towards the vaulter, is contracted to only 10 ft. So in that frame when the front of the pole hits the back wall, there’s still 20 ft of pole outside the garage.

I was actually aware of the pole-vaulter example, but I hadn’t considered how it applied to my own thought experiment.  Physics is filled with such awesome stuff like this.  Thanks, Pete!