Preprint shows free-fall geodesics fail to maximize proper time in Earth-diameter tunnel

The paper constructs a thought experiment contrasting two heuristics: the GR slogan that geodesics maximize proper time versus older velocity-plus-gravitational time-dilation calculations performed in a single frame. It shows the latter correctly identifies a non-maximizing free-fall trajectory that boomerangs through an Earth-diameter tunnel while the geodesic heuristic misleads without additional global analysis. Methods rely entirely on analytic comparison of proper-time integrals along radial and perturbed paths in Schwarzschild geometry.

Contextually this revives John Bell's emphasis on pre-relativistic intuitions for relativity teaching and John Wheeler's capsule-through-Earth illustration, but exposes their tension. The resolution turns on the distinction between locally extremal and globally maximal geodesics, a subtlety rarely quantified in introductory GR. No numerical simulations or observational data are presented.

The work has no immediate engineering implications yet sharpens conceptual tools for interpreting clock comparisons in strong fields. Future analytic or numerical extensions could test whether similar failures appear in Kerr or cosmological spacetimes. Replications would require independent verification of the conjugate-point locations along the chosen paths.