Brachistochrones in coordinate time and in proper time in the Kerr geometry

Building on Perlick's formulation of the brachistochrone problem in stationary spacetimes~{Perlick1991} and on the optical Randers--Finsler framework for stationary metrics~{GibbonsHerdeiroWarnickWerner2009,Randers1941}, we give a unified treatment of the two brachistochrones of a test particle in the Kerr field: the one minimizing the coordinate time t and the one minimizing the proper time τ. On the equatorial plane Tₜ is the length of a Randers metric, due to a linear gravitomagnetic term, whereas T_τ is purely Riemannian. We derive the first integrals, exact orbit equations, travel times, and Hamiltonian formulations for both branches. From the equatorial Hamiltonians we obtain a new analytical result: for common initial phase-space data the two branches carry different on-shell energies, H₀^τ/H₀ᵗ→ E²/f₀², and this, combined with the gravitomagnetic enhancement of the centrifugal barrier, produces an inversion of the radial plunge across a critical angular momentum Jᵢₙᵥ(a). The resulting closed-form criterion is validated numerically to four significant digits. Off the equator the brachistochrone Hamilton--Jacobi equation is non-separable: the conformal factor n_τ²=F/DE obstructs Carter separation at finite energy, while separability is recovered in the Fermat limit. We therefore construct approximate quasi-invariants by perturbative Poisson-bracket closure: for τ a single O(a²) term reduces the drift, while for t the bare Carter form is already optimal. We also verify the a→0 Schwarzschild limit, develop spin and weak-field expansions, and extend the construction toward the ergoregion using ZAMO variables. These results complement the companion interior treatments~{Rosignoli2026Schw,Rosignoli2026TOV}.