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

Abstract

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 \emph{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 \emph{Kodama} vector when no timelike Killing field survives. We show that this construction is a legitimate Pontryagin problem---establishing existence and normality on a regular timelike-selector domain and a non-autonomous Hamilton--Jacobi \emph{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: spatially flat Friedmann--Lema\^itre--Robertson--Walker (FLRW) and the ingoing Vaidya dynamical black hole. For Vaidya we obtain the plunge 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 dynamical horizon. The rotating, axisymmetric conformal-Kerr (Thakurta--Kerr) application is developed in a companion paper.