The Postulates — Question 9

Question 9

A rotating receiver is outside the global inertial-frame assumption

In an inertial laboratory, two light pulses circulate in opposite directions along an ideal circular vacuum guide of circumference C at speed c. A receiver moves along the guide at constant tangential speed v. At $t=0$ both pulses leave it; neglect all material optical effects.

1. Find the first return times for the pulse chasing the receiver and the counter-propagating pulse.

2. Find their laboratory time difference and its small-v limit.

Question 9 - Solution

A rotating receiver is outside the global inertial-frame assumption

(a) Method and calculation

$$(c-v)t_+=C,\quad(c+v)t_-=C\quad\Rightarrow\quad t_\pm=\frac{C}{c\mp v}.$$$$t_+>t_-\quad(v>0).$$

(b) Interpretation and further calculation

$$\boxed{\Delta t=\frac{2Cv}{c^2-v^2}}.$$$$v\ll c:\quad\boxed{\Delta t\simeq2Cv/c^2}.$$

Physical check

The receiver changes direction continuously. This laboratory calculation does not assign a single global inertial rest frame to the rotating receiver.