The Postulates — Question 8

Question 8

Round-trip light timing alone cannot isolate directional speeds

A source and mirror are fixed a distance L apart. Consider a hypothetical timing convention with outward speed $c_+=c/(1+\epsilon)$ and return speed $c_-=c/(1-\epsilon)$, where $|\epsilon|<1$.

1. Find the two one-way times and the round-trip average speed.

2. Explain what extra convention makes both assigned one-way speeds equal to c.

Question 8 - Solution

Round-trip light timing alone cannot isolate directional speeds

(a) Method and calculation

$$t_+=L(1+\epsilon)/c,\qquad t_-=L(1-\epsilon)/c.$$$$t_{\mathrm{round}}=2L/c,\qquad\boxed{2L/t_{\mathrm{round}}=c}.$$

(b) Interpretation and further calculation

$$t_+=t_-\quad\Longleftrightarrow\quad\boxed{\epsilon=0}.$$$$t_B=(t_{\mathrm{emit}}+t_{\mathrm{receive}})/2\quad\Rightarrow\quad c_+=c_-=c.$$

Physical check

Einstein synchronization assigns equal outbound and inbound times. A round-trip result alone does not determine that distant-clock convention.