Does it contradict special relativity that an electron beam in a television picture tube can move across the screen faster than the speed of light?

While looking at some exercises in my physics textbook, I came across the following problem which I thought was quite interesting:

It is possible for the electron beam in a television picture tube to move across the screen at a speed faster than the speed of light.

Why does this not contradict special relativity?

I suspect that it's because the television is in air, and light in air travels slower than light in a vacuum. So I suppose they're saying the the electron could travel faster in air than the speed of light in air, like what causes Cherenkov radiation?

Answer

This is an example of what is sometimes called the "Marquee Effect." Think of the light bulbs surrounding an old-fashioned movie theater marquee, where the light bulbs turn on in sequence to produce the illusion, from a distance, of a light source which is moving around the the marquee.

There is no limit on how short the time interval is between one light turning on and the next turning on, so the perceived light source position can move arbitrarily fast, but in fact nothing is actually moving at all.

In the case of the television screen, the phosphors on the screen can be lit in rapid sequence, but the electrons in the beam do not ever need to move at (or even near) the speed of light.

More generally, there are loads of examples of some imaginary or conceptual "object" moving faster than light, but in all these cases there is nothing actually moving at all. A classic example is the intersection point of two nearly parallel lines, which moves very rapidly as the angle between the lines changes. In this case it is obvious that the moving "object" isn't moving at all, but its still a good example of a case where you can discuss something moving faster than light without there being any violation of physical law.