Some Consequences of the Lorentz Equations — Question 2

Question 2

Three events impose intersecting constraints on their observed order

Events A, B and C have $(ct,x)=(0,0)$, $(1,3)$ and $(3,4)$ in common length units. Use $c\Delta t\prime=\gamma(c\Delta t-\beta\Delta x)$, $\beta=v/c$, and $\gamma=(1-\beta^2)^{-1/2}$.

1. Find all boost speeds for which the primed chronology is B, then A, then C.

2. Show that C follows B in every physical boost and test $\beta=0.50$.

Question 2 - Solution

Three events impose intersecting constraints on their observed order

(a) Method and calculation

$$t_B\prime<t_A\prime:\ 1-3\beta<0\Rightarrow\beta>1/3.$$$$t_A\prime<t_C\prime:\ 3-4\beta>0\Rightarrow\boxed{1/3<\beta<3/4}.$$

(b) Interpretation and further calculation

$$c(t_C\prime-t_B\prime)=\gamma(2-\beta)>0\quad(-1<\beta<1).$$$$\beta=1/2:\quad(ct_B\prime,ct_A\prime,ct_C\prime)=\boxed{(-1/\sqrt3,0,2/\sqrt3)}.$$

Physical check

Allowed orderings must satisfy every pairwise constraint. Timelike C-minus-B prevents arbitrary permutations.