06. The thrown clock

You must send a clock away from your lab bench and have it back in exactly $T$ seconds of bench time — but you want the clock itself to have ticked off as much time as possible when it returns. Carry it, drive it, fly it, throw it: which round trip wins?

Solution

Throw it straight up, so that it is in free fall the whole time, and gravity brings it back at $T$. (The winning toss peaks at height $gT^2/8$ — about 20 meters for $T = 4\ \mathrm{s}$.)

In the weak-field limit a clock’s rate is $d\tau \approx \left(1 + \tfrac{gh}{c^2} - \tfrac{v^2}{2c^2}\right)dt$: it ticks faster the higher it sits and slower the faster it moves. Maximizing $\int d\tau$ therefore means buying as much altitude as possible while spending as little speed as possible — and the optimal compromise is exactly the free-fall parabola. This is no coincidence: maximizing $\int \left(gh - \tfrac{v^2}{2}\right) dt$ is the same variational problem as extremizing $\int (T_{\mathrm{kin}} - U)\,dt$. Hamilton’s principle of least action, for a projectile, is proper-time maximization read in Newtonian light.

That is the honest content of the geodesic hypothesis: free fall is not a force-driven motion but the worldline of greatest aging. Things fall because falling is how you age the most.

Deeper in the notebook: 03. Local Inertial Frames; the Geodesic Hypothesis · 04. Gravity as Geometry - the Heuristic Argument · 01. The Twin Paradox