Some Consequences of the Lorentz Equations — Question 7

Question 7

The straight inertial path maximizes proper time between fixed events

Two travelers share endpoints separated by $T=10.0\,\mathrm s$ and $\Delta x=6.0c\,\mathrm s$ in one laboratory. One travels at constant velocity; the other spends 5.0 s at $0.40c$ and 5.0 s at $0.80c$. Transitions are negligible.

1. Find both proper durations and check that the endpoints agree.

2. Use concavity of $f(\beta)=\sqrt{1-\beta^2}$ to explain the general comparison.

Question 7 - Solution

The straight inertial path maximizes proper time between fixed events

(a) Method and calculation

$$\bar\beta=\Delta x/(cT)=0.60,\quad\boxed{\tau_{\mathrm{straight}}=10\sqrt{1-.60^2}=8.00\,\mathrm s}.$$$$\tau_{\mathrm{broken}}=5\sqrt{.84}+5\sqrt{.36}=\boxed{7.58258\,\mathrm s}.$$

(b) Interpretation and further calculation

$$\Delta x/c=5(.40)+5(.80)=6.0\,\mathrm s,\quad f\prime\prime(\beta)=-(1-\beta^2)^{-3/2}<0.$$$$\boxed{\int_0^T f(\beta(t))\,dt\le T f(\langle\beta\rangle)}.$$

Physical check

The endpoint comparison differs from finding the smallest coordinate-time interval by changing frames. Here different worldlines connect fixed events.