Gravitational train in the interior Schwarzschild metric: NLO and NNLO perturbative expansions

We extend the gravitational train problem---a test particle sliding without friction through a tunnel inside a spherical body---to the interior Schwarzschild spacetime. Working in the equatorial plane, we show that the coordinate travel-time functional is the length of a positive-definite Riemannian optical (Jacobi) metric on the spatial slice. The cyclic symmetry yields a relativistic Snell law; expanding its integrand in the compactness parameter μ = rₛ/R and integrating in closed form, we obtain the angular half-opening as a power series in μ through next-to-next-to-leading order.Perturbative inversion gives explicit NLO and NNLO corrections to the Newtonian relation q = 1 - Δ/π, both positive for 0 < Δ < π: the optimal tunnel becomes progressively shallower with increasing compactness. The perturbative results are validated against direct numerical integration of the exact geodesic equation, and agree to four significant figures for μ 0.3. Near the Buchdahl bound μ → 8/9, the optical index diverges at the centre as (8/9 - μ)⁻¹, creating a central optical barrier that forces the brachistochrone toward the surface---the optical signature of extreme gravitational redshift.