Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 4C (Dynamics and Relativity)

Set $c=1$. If $B$ is in the future lightcone of $A$ and $C$ in that of $B$, thenAdding and using the triangle inequality givesThus the statement is true.

False. Takewith coordinates written as $(x,t)$. Both $B$ and $C$ lie on the future lightcone of $A$, but $B$ and $C$ are simultaneous and spatially separated, so neither is in the future lightcone of the other.

True. The assumptions giveAdding and applying the triangle inequality provesso $B$ is in the future lightcone of $C$.

False. With $(x,t)$ coordinates, takeBoth separations from $A$ are lightlike, but $B-C=(2,0)$ is spacelike.

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.