Last month I was in a couple of conversations that explored question could something go faster than light. The particular idea explored was a moving beam of light. Imagine a beam of light from a flashlight aimed into the night sky. If you move that beam around, how fast is the distant part of light beam moving? To illustrate the issue I'll tell you about a dance routine I saw. My wife's parents are very accomplished square dancers and perform in a square dance company. One of their signature performances is dance where they form a line, side-by-side, link arms, and rotate their line. There is couple in the center, with linked arms rotating slowly. Each couple attached to them outwardly must move faster and faster to keep the straight line, with the last ladies at theends of this line are moving so fast that their feet are off the ground. Imagine this with a beacon of light, extending out from a flashlight. How fast is that beam moving as the beam of light stretches millions of miles out into space.
This leads me to propose a thought experiment. Thought experiments have a noble position in investigating questions. Of course they can be done wrong and lead to all sorts of fallacious notions. But if done right, a thought experiment can lead to critical insights about these kind of problems. In this thought experiment I have set up the parameters so that they can be diagrammed and illustrated in an animated gif.
I am using three different units of length: Imperial system miles (commonly used in the United States); the metric system meters (used by the rest of the world); and light seconds -- the unit of measurement being the distance spanned by light traveling in a vacuum for one second. Each of these can be used interchangeably as long as you use them consistently.
You are at the center of the sphere holding a marvelously engineered laser flashlight. It will send out a perfectly aligned, straight beam of light that won't disperse or spread out. The nice thing about thought experiments is that you can construct all sorts of expensive gadgets, such as this precision laser flashlight and giant sphere. In the diagrams the flashlight will not be drawn to scale.
We will rotate this beam of light at one revolution per minute. The light hitting the sphere will trace a circle in one minute. Tracing the circle is tracing the circumference of the circle, which is 113.1 light seconds (10,792,528,488 meters, or 6,706,166 miles). Tracing the circle at light speed would mean it would complete the revolution in 113.1 seconds -- but in this experiment it will trace it in 60 seconds -- faster than the speed of light!
The animation seen on the right runs for two minutes and then repeats itself (If you do not see the animation move, click on it so that the image shows up alone in your browser). All during this time the laser flashlight in the middle rotates around at one revolution per minute. During the first minute it starts emitting a stream of photons, one per second. As the flashlight turns you see the path of the photons as the move away from the flashlight. At 30 seconds a red arrow traces the path of one photon so you can see how an individual photon travels and so see that all the photons travel in a straight line out from the flashlight to the rim of the sphere. The photons appear to form a spiral that moves out from the flashlight, but each individual photon travels in a straight line away from the flashlight.
A few observations follow. First, it takes 18 seconds for each photon to travel from the flashlight to the rim of the sphere, all traveling at light speed of 299,792,458 meters per second. The beam starts sweeping the rim of the sphere at 18 seconds after the first photon is emitted from the flashlight. Secondly, the last photon hits the rim 18 seconds after the completion of the first revolution, which is 18 seconds into the second revolution when the flashlight is not emitting photons any more. Accounting for the delay at the start and at the end, the beam does a complete sweep of the rim in one minute. Finally, the thing I want to note is that the sweeping motion of the beam on the rim of the sphere is only an apparent motion. There are no photons that actually take a path around the rim of the sphere. Photons are hitting the rim in rapid succession in during one minute, but each photon that hits the rim, traveled at light speed traveling on a straight line from the flashlight to the rim. This demonstrates that a rotating beam of light does not violate the speed limit 299,792,458 meters per second, or 186,282 miles per second, or 1 light second per second.
Okay, some might say this is well and good for emitting one photon per second, but what about a laser flashlight emitting a continuous beam of light? Imagine our laser flashlight emits a photon every half second. The animation would look much the same, only the photon dots would be tighter together showing a more continuous spiral. Each photon still travels in a straight line from the flashlight to the rim. Now imagine a photon every tenth of a second, still the animation would be much the same. The dots would overlap, but it is still doing the same thing, each photon travels in a straight line from the flashlight to the rim. The sweeping motion on the rim is still only an illusion of a spot of light traveling around the rim, because it is still a succession of individual photons hitting the rim that traveled from the flashlight. No matter how small we make the interval of time between the photons, the overall principle holds true. All the photons are traveling from the flashlight to the rim in a straight line going only at light speed, no more, no less.