Can a dot of light travel faster than the speed of light?

Say I have a laser. If I spin the laser so that the beam sweeps in an arc along a very distant object, could that dot travel faster than the speed of light?

In Diagram form:

Answer

Yes, the dot would travel at a speed faster than light. Firstly, note that this wouldn't be immediate. if $r$ is the radius of the arc, it would take $\frac r c$ time before the dot started moving (in the meantime, the beam of light, if "frozen" in time¹, would look bent from most reference frames)

This doesn't violate causality. Why? Because the dot cannot relay information along the arc. The motion of the dot is controlled by the person at the center, and he is the only one who can send information by moving it. Since there is a delay of $\frac r c$ seconds before the dot moves, this is perfectly in line with causality.

Remember, the dot isn't a physical object. It's a certain area having some properties (point of intersection of distant object and laser beam)

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A simple argument to justify that the dot will go FTL:

At time $t_0$, I shine my laser pointer at point A, on the left of the arc. I sweep it around, and at time $t_1$ it points towards B on the opposite end of the arc. Note that in my reference frame, $t_0$ and $t_1$ can be quite close, while $r$ can be arbitrarily large.

In my reference frame, the photons from my initial point reach A (remember, a photon does not change its direction in special relativity) at time $t_0+\frac r c$, and the photons from my final position reach B at $t_1+\frac r c$

So now, the dot has crossed a distance of $r\theta$ in a time $t_1-t_0$. Note that $r$ can be arbitrarily large, and $t_1-t_0$ is only constrained by the rate of production of photons. Which means that we can cover an arbitrary distance in a fixed time interval, so there will be some $r$ for which $\frac{r\theta}{t_1-t_0}>c$.

Note that since the events of the dot reaching the two points are spacelike separated (assuming we choose $r$ such that $\frac{r\theta}{t_1-t_0}>c$) in my reference frame, they are spacelike separated in all reference frames, and the dot moves faster than light for all.

^{1. To specify "frozen in time" rigorously: let there be an array of photon detectors. They are synchronized such that they will all record the existence of photons simultaneously in some given reference frame (the frame of the laser pointer or the frame of somebody on some given point on the arc). Now, if that person checks which detectors have recorded a photon, he will get a curved locus.}