Jump into an evacuated hole drilled straight through a uniform, static Earth-like sphere. Accelerate to 7.9 km/s (or 18 000 m.p.h.) at the center, then decelerate back to zero at the antipodes 42 minutes later! Step out of the hole upside down — or return 84 minutes after you left.
Last fall, as part of his senior thesis, Yuchen Gan ’21 and I used computer simulations to generalize this famous result to uniform spinning planets, where Coriolis and centrifugal effects force the tunnels into arcs curving away from the center and intersecting the surface in multiple places. We discovered many families of periodic tunnel networks that connect multiple surface locations even at non-equatorial latitudes, as in the animation. Such tunnels could ideally provide energy-free communication and transportation for the planets’ inhabitants.
But in January, in a wonderful aha! moment, we were surprised and delighted by a dramatic perspective change: the motion of an object or passenger (a “terranaut”) freely falling through the tunnel system is simultaneously spiky concave arcs with respect to the planet and a smooth convex ellipse with respect to inertial space! We subsequently proved mathematically that the inertial motion is that of a two-dimensional harmonic oscillator, and the ellipses are centered (not focused) on the planet.