Relativistic Brachistochrones in Conformal Kerr Spacetimes: Adiabatic Theory, Ergosphere Transitions and Hyperelliptic Closed Forms

Abstract

We develop the relativistic controlled-rail brachistochrone — the fastest constrained worldline reaching a fixed spatial target at a free arrival clock — in the rotating, axisymmetric Thakurta–Kerr spacetime, the conformal Kerr geometry \(g = A(\eta)^2 g_{Kerr}\). The controlled-rail principle, its existence/normality and non-autonomous Hamilton–Jacobi verification, and the Killing, conformal-Killing and Kodama selector hierarchy are established in the companion paper (with the spherically symmetric FLRW and Vaidya cases); here frame dragging, an ergosphere and genus-two spectral curves appear. We derive the breathing-indicatrix Hamiltonians and the conformal-time (\(\eta\)) brachistochrone — the arrival branch \(t \equiv \eta\), distinct from proper time \(\tau\); closed-form equatorial separatrices in Weierstrass functions, continued through the ergosphere in Doran coordinates, with a trichotomy and an analytic cusp theorem; the rotational and conformal plunge inversion, and an existence theorem for the conformal inversion, under a same-launch comparison; and the first-order adiabatic correction, on-shell and off-shell, canonically re-derived from the extended Hamiltonian and verified against the true non-autonomous flow to \(O(\varepsilon^2)\), its weight-two content a length-two iterated Abelian integral on the genus-two curve. The closed-form separatrices and cusp theorem and the on-shell reduction are proved; the fixed-endpoint no-inversion is established in the stated analytic domain \((r_0 \le R_*(E,a))\) and machine-certified at representative \(r_0\), while its unrestricted extension remains conjectural; interior-ergosphere optimality and a rigorous higher-genus polylogarithm framework remain open.