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 $\mu = r_s/R$ and integrating in closed form, we obtain the angular half-opening as a power series in $\mu$ through next-to-next-to-leading order.
Perturbative inversion gives explicit NLO and NNLO corrections to the Newtonian relation $q = 1 - \Delta/\pi$, both positive for $0 < \Delta < \pi$: 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 $\mu \lesssim 0.3$.

Near the Buchdahl bound $\mu \to 8/9$, the optical index diverges at the centre as $(8/9 - \mu)^{-1}$, creating a central optical barrier that forces the brachistochrone toward the surface---the optical signature of extreme gravitational redshift.