Past exam of the short options course of the University of Oxford / 2021 / 8 / a

Differentiate the worldline with respect to $\theta$:Since $|\tan \theta |\le 1$ and $0<\kappa <1$, this is a future-directed timelike curve. The proper time element satisfies $c^{2}d{\tau }^2=c^{2}d{t}^2-d{x}^2$, henceAlice's total proper time is thereforeThis definite integral is an exact answer; it is an elliptic integral of the second kind. In notation $E(\alpha \mid m)={\int }_0^{\alpha }\sqrt{1-m{\sin }^{2}u}du$, it is $2\sqrt{2}TE(\pi /4\mid 1+{\kappa }^2)$. Although this parameter exceeds one, the integrand is real on the stated interval because its minimum squared value is $(1-{\kappa }^2)/2>0$.

The endpoint events have $(t,x)=(0,0)$ and $(2T,0)$. Sam stays at $x=0$ in his inertial frame, so his elapsed proper time is $2T$. For $\kappa >0$, the integrand is strictly less than $\cos \theta$ except at $\theta =0$, and ${\tau }_A<2T$.

On the fixed parameter interval the integrand tends to $\cos \theta$ as $\kappa \to 0$ and is bounded by $\cos \theta$. The dominated convergence theorem, or uniform convergence here, givesThis equals Sam's time between departure and reunion. Physically, $dx/dt=-c\kappa \tan \theta$ tends uniformly to zero, so Alice's worldline approaches Sam's, and the time dilation disappears.