Some Consequences of the Lorentz Equations — Question 3

Question 3

A relay detour cannot beat the direct vacuum causal bound

In a laboratory plane, A is at $(0,0)$, relay R at $(3,4)\,\mathrm m$ and B at $(6,0)\,\mathrm m$. A sends a light message at $t=0$. The relay retransmits immediately. Use $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find the earliest relayed arrival and compare with the direct arrival.

2. Express the result as a triangle-inequality bound valid for any relay position.

Question 3 - Solution

A relay detour cannot beat the direct vacuum causal bound

(a) Method and calculation

$$|AR|=|RB|=5\,\mathrm m,\quad\boxed{t_{\mathrm{relay}}=10/c=33.333\,\mathrm{ns}}.$$$$\boxed{t_{\mathrm{direct}}=6/c=20.000\,\mathrm{ns}},\quad\Delta t=13.333\,\mathrm{ns}.$$

(b) Interpretation and further calculation

$$|AR|+|RB|\ge|AB|.$$$$\boxed{t_{\mathrm{relay}}\ge |AB|/c},\quad\text{equality for R on segment AB}.$$

Physical check

Every leg is causal. A sequence of local retransmissions cannot connect spacelike-separated endpoints.