Some Consequences of the Lorentz Equations — Question 9

Question 9

The first event that can combine two messages lies in a shared future

Two emitters at $x=-L$ and $x=+L$ flash at $t=0$. A processing event M at coordinate x must have received both messages, each traveling at speed no greater than c.

1. Find the earliest allowed time of M as a function of x.

2. Minimize that time and identify the unique earliest meeting position for $L>0$.

Question 9 - Solution

The first event that can combine two messages lies in a shared future

(a) Method and calculation

$$ct_M\ge |x+L|,\qquad ct_M\ge |x-L|.$$$$\boxed{t_{\min}(x)=\frac{\max(|x+L|,|x-L|)}c=\frac{L+|x|}c}.$$

(b) Interpretation and further calculation

$$|x|\ge0\quad\Rightarrow\quad t_M\ge L/c.$$$$\boxed{x_M=0,\qquad t_M=L/c}.$$

Physical check

Combining information requires the intersection of both future light cones, not merely reaching the future of either source.