Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 4C / b / Solution

Again set $c=1$ and writeA boost of velocity $v$ givesDefineThe hypothesis says $D(0)<0$. Put $\xi =x_B-x_A$ and $\tau =t_B-t_A$. Causality gives $\tau \ge |\xi |$, while $T_B\ge |X_B|$, and $T_A=T_B+\tau$, $X_A=X_B+\xi$. ThereforeButHence at least one of $D(1),D(-1)$ is positive. By continuity, $D(v)>0$ for some physical velocity $|v|<1$ sufficiently close to that endpoint. In the corresponding inertial frame,as required.