Saturday, August 5, 2017

Sending A Signal Faster Than Light - Solution

My patent on a yarn-and-laser system for faster-than-light signalling isn't going to get approved, because it has a fatal flaw. The thing that burns the yarn is photons, and they stream out of the laser at precisely the speed of light. So you won't be able to burn yarn arbitrarily far away in an arbitrarily short time.

Each photon emerging from the laser follows a straight-line path, maintaining the direction in which it was first launched. It isn't as if the beam keeps the photons rotating…so the analogy to the rotating line fails altogether. The laser situation is more like this:

It's a "pew-pew" thing, like the old Asteroids video game.

I derived a relationship between the horizontal progress a of the signal, and the time t since the laser began rotating:

\[ t = \sqrt{\tau_1^2 + \frac{a^2}{c^2}} + \tau_2 \tan^{-1}\frac{a}{L}\]

Here L is the vertical distance from the laser to the horizontal line, c is the speed of light, and τ1, 2 are the two timescales in the problem, τ1L/c  and τ2 = 1/ω, where ω is the angular rotation rate of the laser. The expression has the limiting behavior one expects—specifically, when a >> L, we have


This is what it ought to look like, since the photons striking the yarn very far away are those that have traveled approximately distance a after leaving the laser at approximately the end of its rotation period.

Once when I was in college—I think I was a freshman—I asked my physics professor, William Wootters, a question about faster-than-light signaling. I said, "If you and I hold onto either end of a broomstick, and I jerk my end of the broomstick, then haven't I signaled to you just as fast as we please?" At that time, I didn't have the understanding of solid objects that I would eventually gain during the physics major—that a broomstick isn't an undifferentiated chunk of stuff, but is well thought of for these purposes as an atomic slinky made of balls and springs. If you push one end of the broomstick, you set up a compression wave in the slinky (a sound wave), which travels through the broomstick with a speed determined by its material properties. Key to those material properties are the electromagnetic forces that play the role of the "springs" in the slinky analogy. And so there's no obvious mechanism to achieve faster-than-light signaling with a broomstick because disturbances in electromagnetic fields propagate with—you guessed it—the speed of light.

I believe Bill answered my broomstick question by saying that relativity would seem to rule out the existence of a perfectly rigid body. That makes sense, otherwise the rotating line analogy could be made real by swinging a broomstick. Given things as they are, if you try to swing a rod so fast that its tip moves faster than light, you will find yourself limited by the finite propagation speed within the material. Parts of the rod near the distant end will take some time to "find out" that they ought to be moving, because of the finite speed of the "signal" that your wrists are transmitting. In simple terms, the rod will bend. Or break. And if you could somehow make the rod more and more rigid by improving its material properties, then at some point maybe you will have created such a stress-energy field that the rod would turn into a singularity.

Well maybe that's not right, and anyway I certainly haven't proved that faster-than-light signalling is impossible. Nor have I studied professionally the arguments of those who believe they have done that. Probably everything is settled and airtight…however, like John Bell, whose work I once read carefully, I tend in general to "suspect that what is proved by impossibility proofs is lack of imagination."

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