At the same Schwarzschild radius outside a black hole, one observer hovers on rockets while another passes by in a circular orbit, meeting the hoverer once per revolution. Problema VI said free fall maximizes aging — so the orbiter, who is in free fall, ages more between meetings. Correct?
Solution
No: the hoverer ages more — and Problema VI is not contradicted, only read too generously.
The rates are unambiguous: for the hoverer versus for the circular orbit — the orbiter pays the full gravitational toll plus a velocity toll, with nothing in compensation. As approaches the photon sphere at , the orbiter’s clock rate slides toward zero while the hoverer’s stays finite.
The resolution is that geodesics are local extrema of proper time: maximal only against nearby competing worldlines, and only up to the first conjugate point. A full revolution is not a small variation away from hovering, and it contains a conjugate point (neighboring orbits refocus within an epicyclic period), so the circular orbit between two meetings is a saddle of the proper-time functional, not a maximum. The true champion between those same two events is a third worldline: the radial toss of Problema VI, thrown upward hard enough to fall back exactly one revolution later, which out-ages hoverer and orbiter alike.
Deeper in the notebook: 02. Geodesics, Orbits, the Effective Potential · 02. Conjugate Points and the Second Variation of Arc Length · 01. The Twin Paradox