# Electrons faster than 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?

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You could also just consider a person shining a laser pointer at a distant wall. As you spin around, the spot of the laser pointer moves on the wall with a speed dependent on the distance to the wall. In principle, the wall could be so far away that the spot moves faster than the speed of light. But the light is still moving at the speed of light (in air, or whatever). The spot is not really an object - unless you are the inmate trying to escape from the insane asylum on a beam of light! – Greg P Dec 28 '10 at 22:17
@Greg oh! move across the screen... so is it talking about the picture itself? I thought it was saying the beam from the electron gun was moving faster than light – wrongusername Dec 28 '10 at 22:21

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Yes. It is something I remember from an intro relativity book. It means the actual spot (yes, the image) moving across the screen. Otherwise, I don't get the point of the question. The electrons themselves don't move faster than light. It is just an illusion of something moving faster than the speed of light. – Greg P Dec 28 '10 at 22:31
There were some other 'paradoxes' where objects seem to move at superluminal speeds. Particularly one from astrophysics which seemed interesting...perhaps someone can remember it for me. – Greg P Dec 28 '10 at 22:35

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– David Zaslavsky♦ Dec 28 '10 at 23:47
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## 3 Answers

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.

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This is a conflation of phase velocity, and group velocity. The beam can be seen to move from say left to right at higher than c, but no information or particles are traveling that fast. Information is being transmitted from the electron gun to the phosphor at well under the speed of light.

It has nothing to do with the media it is embedded in. The information is going from the electron gun to the screen, not from one location on the screen to another.

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How come the beam is a wave? – wrongusername Dec 28 '10 at 22:12
@wrongusername: A beam of electrons behaves like a wave beam, the same is behaviour is verified even in much larger particles (such as small atoms). The reason lies in Quantum Mechanics, and I can't get into it here. – Bruce Connor Dec 28 '10 at 22:47

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All of this is mostly irrelevant though, as the wave nature of the beam has nothing to do with your question. This could happen with virtually anything that moves. – Bruce Connor Dec 28 '10 at 22:48

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Replace the electron beam with a marshmallow gun. People think of quantum theory and waves when they hear "electron" but not with "marshmallows". Of course, it may be hard to actually create a series of marshmallow collisions on a distant wall appearing to move faster than c, but heck this is only a thought experiment, so imagine if you have a powerful enough gun... – DarenW Mar 5 '11 at 21:17

Here's another example from Griffith's book "Introduction to Electrodynamics" which illustrates phenomena where what we see is not what we observe. The apparent speed can be much greater than the speed of light. This speed is just what we see, an illusion, and it's the result of our inability sometimes to see the actual direction of movement of an distant object w.r.t. us and the fact that the light needs some finite time to get to our eyes.

Problem 12.6 Every 2 years, more or less, The New York Times publishes

an article in which some astronomer claims to have found an object

traveling faster than the speed of light. Many of these reports

result from a failure to distinguish what is seen from what is

observed--that is, from a failure to account for light travel time.

Here's an example: A star is traveling with speed $v$ at an angle $\theta$ to

the line of sight (Fig. 12.6). What is its apparent speed across the

sky'?

(Suppose the light signal from $b$ reaches the earth at a time At after

the signal from a, and the star has meanwhile advanced a distance $\Delta s$

across the celestial sphere; by "apparent speed" I mean $\Delta s/\Delta t$.) What

angle $\theta$ gives the maximum apparent speed? Show that the apparent

speed can be much greater than $c$, even if $v$ itself is less than $c$.

It can be easily shown that the apparent speed in this example is:

$u_{app}=\frac{v\sin\theta}{1-\frac{v}{c}\cos\theta}$

To find the angle $\theta$ that gives the maximum apparent speed we just differentiate and solve, for $\theta$, the equation:

$\frac{d u_{app}}{d\theta}=0 \Leftrightarrow \theta_{max}=\cos^{-1}(\frac{v}{c})$

At this angle, $u_{app}=\frac{v}{\sqrt{1-v^2/c^2}}=\gamma v$

This result shows that when $v\to c$, $u_{app}\to \infty$, even though $v<c$.

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This is not quite the same issue as in the question, though. – David Zaslavsky♦ Jul 17 '11 at 23:37
I think this is exactly the same issue. The electrons gun changes its direction, lets say by $\theta=\phi$. If we suppose that the electrons are emitted once at $\theta=0$ and once at $\theta=\phi$, we get an triangle. While the beam at $\theta=0$ travels to the screen, the electron gun rotates by $\phi$ and emits the second beam. So the time difference between the arrival of two beams can be very small. – Andyk Jul 17 '11 at 23:55
Yes, but in the case of the electron gun (and the light bulbs, and the laser pointer, etc.), the light is being emitted by two completely different objects. Nothing actually moves even close to the speed of light. In fact, nothing has to move at all, in the case of the light bulbs. But your example with the star involves light being emitted by the same object at two separate points. Without motion, there is no superluminal effect. That's why they're different phenomena. – David Zaslavsky♦ Jul 18 '11 at 0:05
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