The traveller going to the cinema

Problem

A traveller leaves the bar at time $0$ and has a screening at the cinema at time $T=20\,000$~s, at a point a distance $L=10\,000$~ls away. Calculate the proper time for three scenarios: (1) uniform straight-line motion all the way to the cinema, arriving on time; (2) a stop halfway (at the tobacconist), then continuing and still arriving at time $T$; (3) covering $L$ in $15\,000$~s, then waiting at rest to read the newspaper. Which of the three maximises the proper time?

Solution

Natural units $c=1$. On each uniform straight segment the proper time is $\Delta\tau=\sqrt{\Delta t^2-\Delta s^2}$; on a segment where the traveller is at rest ($\Delta s=0$) the proper time coincides with the laboratory time $\Delta\tau=\Delta t$.

(1) Uniform straight-line motion to the cinema. Covers $\Delta s=L=10\,000$~ls in $\Delta t=T=20\,000$~s: $\Delta\tau_1=\sqrt{20\,000^2-10\,000^2}=\sqrt{3\cdot 10^8}=10\,000\sqrt{3}\approx 17\,320\ \text{s}$

(2) Stop at the tobacconist (halfway). Covers the first half ($5\,000$~ls) quickly, in $7\,000$~s, then the second half ($5\,000$~ls) more slowly, in $13\,000$~s (still arriving at $T=20\,000$~s): $\Delta\tau_2=\sqrt{7\,000^2-5\,000^2}+\sqrt{13\,000^2-5\,000^2}=\sqrt{24\cdot 10^6}+\sqrt{144\cdot 10^6}$ $\Delta\tau_2\approx 4\,899 + 12\,000 = 16\,898\ \text{s}$

(3) Dash plus wait. Covers $L=10\,000$~ls in $15\,000$~s, then stays at rest for the remaining $20\,000-15\,000=5\,000$~s: $\Delta\tau_3=\sqrt{15\,000^2-10\,000^2}+5\,000=\sqrt{125\cdot 10^6}+5\,000\approx 11\,180 + 5\,000 = 16\,180\ \text{s}$

Comparison and conclusion. $\ev{\Delta\tau_1\approx 17\,320\ \text{s} \;>\; \Delta\tau_2\approx 16\,898\ \text{s} \;>\; \Delta\tau_3\approx 16\,180\ \text{s}}$ Uniform straight-line motion (scenario 1) maximises the proper time. Every change of regime — speeding up/slowing down halfway, or stopping — reduces the overall proper time relative to the straight line.

Links

Topics: Special relativity Concepts: Time dilation · Spacetime invariant Skills: Symbolic setup Methods: Minkowski diagram · Natural units