09. Hover or orbit?

At the same radial coordinate $r$ outside a Schwarzschild 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: $d\tau/dt = \sqrt{1 - 2GM/rc^2}$ for the hoverer versus $\sqrt{1 - 3GM/rc^2}$ for the circular orbit — the orbiter pays the full gravitational toll plus a velocity toll, with nothing in compensation. As $r$ approaches the photon sphere at $3GM/c^2$, the orbiter’s clock rate slides toward zero while the hoverer’s stays finite.

The resolution is that geodesics are stationary points 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 always contains a conjugate point — tip the orbital plane slightly: the tipped circular orbit has the same $r$, the same period, the same clock rate, and recrosses the original at the antipode, half a revolution later, for every $r$ outside the photon sphere. So the circular orbit between two meetings is a saddle of the proper-time functional: the refocusing family supplies nearby worldlines that age more, while near-null zigzags supply ones that age less. 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 — its apoapsis sits near $2.2\,r$ in the Newtonian limit — which out-ages hoverer and orbiter alike, buying altitude with speed it immediately stops paying for.

Deeper in the notebook: 02. Geodesics, Orbits, the Effective Potential · 02. Conjugate Points and the Second Variation of Arc Length · 01. The Twin Paradox