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. v7 — the appendix brought into line with section 4, and every cited check re-run. Proper-time blocks relabelled. The mass of ingoing Vaidya runs with the advanced time, so the first-order drift is weighted by $v$ on every branch, as section 4 states since v6. Two appendix passages still presented the proper-time density as the clock of a "proper-time branch" response. The block formula holds for any clock, and that block is now presented as an algebraic instance of it; the $v$-weighted block is the ingoing-Vaidya response. No result changes. Checks that could not reproduce what they were cited for.- The script behind the assembled closed form of section 4 evaluated the tortoise term at the upper limit only. The printed formula takes it between the limits and agrees with the quadrature to $5\times10^{-40}$; the script now says so, and fails without the bracket.- The Jacobi-conjugate check printed a spurious failure for want of the assumption that the scale factor is positive.- The script cited for the elliptic collapse of the letters at the genus degeneration contained only a description of the check. It now runs it, at thirty digits and with a control that must fail. Statements matched to their arguments. Irreducibility of the weight-two letters and non-reduction to classical dilogarithms are stated as conjectures and blocked routes, not as facts. One quoted residual is corrected to the script's output. The archive citation now points to the release that contains these scripts. Every quoted residual traced to a script. Each number the text quotes was traced to the script that produces it and re-run. Three overstated the precision actually reached and now give the reproduced value (the proper-time and advanced-time blocks, $3\times10^{-8}$ and $2\times10^{-7}$; the stationary-limit check, $10^{-11}$ in $d\varphi/dr$); the degeneration-family check is labelled as the proper-time-density instance it is; the check of the three spectral expressions obtained from $H_\tau$ now has an archived script (agreement $10^{-50}$); and the tracked source of the degeneration family is stated to be the same for both weights. One gap in a proof closed. The Hamilton–Jacobi verification criterion (Proposition I.3) now states the target condition a free-arrival problem needs: competitors arrive at their own clock, and the value function must be constant along the arrival line for the calibration to compare them. The condition holds automatically in the stationary case. The inner-end limit in Theorem I.F is now argued correctly. The AI-use statement lists every model used. Corrections after an independent audit of the proofs.- The branch Hamiltonian of Theorem I.1 is the support function minus the running cost, so free-arrival transversality reads $h=1$; the proper-time cost stays inside the maximum.- The boundary form of the first-order source (Theorem I.5) subtracts the elapsed cost clock: the elapsed $v$ on the arrival branch, the elapsed proper time on the proper-time branch. The earlier text had $-\lambda$ on both; the new identities are exact zeros in the verification script.- The claim that the closed-form extremal is certified up to its first axis crossing, and that wider apertures need multi-excursion extremals, is withdrawn: the caustic comes first, and a frozen extremal has a single periapsis.- The elliptic letters of Lemma I.J carry third-kind residues at the points at infinity; the Abel coordinate is based at a branch point and the primitives are anchored at the launch.- The velocity set is an ellipsoid through a projective map (Lemma I.A); the degenerate cases of the reduction system, $J=0$ included, are listed (Lemma I.4); a Killing selector makes the rail need no power, not no force; the conclusions state existence under the full hypotheses and normality for interior minimizers.- The closed-FLRW example with $a=e^{Ht}$ is no longer called de Sitter: its scalar curvature is not constant, and in de Sitter proper the antipode is out of reach. The figure is relabelled.- The claim that a non-Killing selector prevents the optical metric from descending to a quotient is withdrawn (a tuned metric keeps the optical data invariant at one energy); for Vaidya the optical coefficients are shown directly to depend on $v$ whenever $m'\neq0$.- The geometric dictionary of the indicatrix is stated in the charts used, with the memory tied to the explicit time dependence of the Hamiltonian.- The minimum thrust diverges as the freezing margin $\hat{E}^2-|W|^2$ tends to zero, provided the drift numerator stays away from zero, not merely as $|W|\to0$; coercivity is lost in two different ways, at freezing (the indicatrix collapses) and at the stationary limit or a horizon (the selector stops being timelike); the coincidence of the $t$, $\tau$ and $\eta$ brachistochrones in FLRW is derived from the monotonicity of each terminal clock in the final conformal time.