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 $-u\cdot W=\Ehat$ 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: Friedmann--Lema\^itre--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 \emph{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.