Some Consequences of the Lorentz Equations — Question 4

Question 4

A sweeping light spot can outrun c without carrying a signal along the screen

A laser at the origin aims at a screen a perpendicular distance L away. Its emission angle is $\theta=\omega t_e$; the illuminated screen coordinate is $x=L\tan\theta$. Light arrives at $t_R=t_e+(L/c)\sec\theta$. Ignore material response times.

1. Derive $dx/dt_R$ from the parametric equations.

2. At $\theta=0$ and $L\omega=2c$, find the spot speed and explain why it is not communication between neighboring screen points.

Question 4 - Solution

A sweeping light spot can outrun c without carrying a signal along the screen

(a) Method and calculation

$$\frac{dx}{dt_e}=L\omega\sec^2\theta,\qquad\frac{dt_R}{dt_e}=1+\frac{L\omega}{c}\sec\theta\tan\theta.$$$$\boxed{\frac{dx}{dt_R}=\frac{L\omega\sec^2\theta}{1+(L\omega/c)\sec\theta\tan\theta}}.$$

(b) Interpretation and further calculation

$$\theta=0:\quad\boxed{dx/dt_R=L\omega=2c}.$$$$\text{Each screen event receives a distinct ray from the laser at speed }c.$$

Physical check

The spot is a sequence of illumination events. No object or message propagates from one spot position to the next at the sweep speed.