Controlled-rail brachistochrones in non-stationary spacetimes: conformal symmetry, Kodama energy, and Vaidya dynamics

What is the fastest constrained worldline in a spacetime that is itself evolving? In a stationary spacetime the relativistic brachistochrone reduces to Fermat/Randers optics with a conserved rail energy supplied by a timelike Killing vector; in a dynamical spacetime no such conservation law exists and the variational problem becomes non-autonomous. We address this by formulating the brachistochrone as a controlled-rail optimal-control problem, in which the invariant $-u\cdot W=\hat{E}$ is actively maintained along the worldline by a selector $W$ that follows a Killing $\to$ conformal-Killing $\to$ Kodama hierarchy and reduces to the Kodama vector when no timelike Killing field survives. We show that this construction is a legitimate Pontryagin problem—establishing, under stated hypotheses on a regular timelike-selector domain, existence, normality of interior minimizers, and a non-autonomous Hamilton–Jacobi verification criterion (global minimisation being conditional, not automatic)—and then derive the extremal equations and their closed representations in two spherically symmetric non-stationary cases: Friedmann–Lemaître–Robertson–Walker (FLRW) at arbitrary spatial curvature and the ingoing Vaidya spacetime. The domain is delimited at the outset: $g(K,K)=-(1-2m(v)/r)$ for the Kodama selector, so the admissible-velocity set is compact only for $r>2m(v)$ and the compact-control problem ends at $r=2m(v)$; what the Vaidya calculation establishes is an exterior approach and contact threshold rather than an interior optimal trajectory, and $m'(v)<0$ in the ingoing metric is a formal continuation, physical evaporation requiring the outgoing problem. For Vaidya we obtain the approach phenomenology and a complete first-order adiabatic correction, verified against the true non-autonomous flow, that separates the homogeneous-expansion effects of the FLRW base from the radial spatial-gradient and mass-flow effects of the trapping boundary, which is a dynamical horizon in the sense of Ashtekar and Krishnan precisely when $m'(v)>0$. The rotating, axisymmetric conformal-Kerr (Thakurta–Kerr) application is developed in a companion paper. v8 — references brought into line with the companion and the code archive. No change to the text, the results or the figures. The companion paper is now cited at its version v7 (DOI 10.5281/zenodo.23071752), deposited alongside this one, and the reproducibility package at release v1.7.3 (DOI 10.5281/zenodo.23071786), which contains every script v7 relied on plus one check for the companion paper.