Boomeranging through the Earth: When free fall doesn't maximize proper time

In teaching special relativity, John Bell emphasized the value of pre-relativistic intuitions. His famous spaceship puzzle showed that, in certain cases, the older heuristics of Lorentz and FitzGerald get the right answer to a genuinely relativistic problem more reliably than the principles of special relativity themselves. In teaching general relativity, John Wheeler emphasized the value of simplicity, illustrating curved-spacetime geometry with capsules boomeranging through a tunnel along the Earth's diameter. Here, I put the two ideas together. I share a simple thought experiment, involving genuinely general-relativistic effects, in which the standard GR heuristic—“freely falling observers maximize proper time”—gives the wrong answer, while the older heuristic of velocity and gravitational time dilation, applied in a single frame, gets it right immediately. The resolution involves conjugate points and the distinction between local and global extremality of geodesics, but the moral is the same as Bell's: do not let a newer slogan overwrite a perfectly sound older intuition.