Brachistochrone-ruled timelike surfaces in Newtonian and relativistic spacetimes

We introduce and study \emph{brachistochrone-ruled timelike surfaces} in Newtonian and relativistic spacetimes. Starting from the classical cycloidal brachistochrone in a constant gravitational field, we construct a Newtonian ``brachistochrone-ruled worldsheet’’ whose rulings are time-minimizing trajectories between pairs of endpoints. We then generalize this construction to stationary Lorentzian spacetimes by exploiting the reduction of arrival-time functionals to Finsler- or Jacobi-type length functionals on a spatial manifold. In this framework, relativistic brachistochrones arise as geodesics of an associated Finsler structure, and brachistochrone-ruled timelike surfaces are timelike surfaces ruled by these time-minimizing worldlines. We work out explicit examples in Minkowski spacetime and in the Schwarzschild exterior: in the flat case, for a bounded-speed time functional, the brachistochrones are straight timelike lines and a simple family of brachistochrone-ruled surfaces turns out to be totally geodesic; in the Schwarzschild case, we show how coordinate-time minimization at fixed energy reduces to geodesics of a Jacobi metric on the spatial slice, and outline a numerical scheme for constructing brachistochrone-ruled timelike surfaces. Finally, we discuss basic geometric properties of such surfaces and identify natural Jacobi fields along the rulings.

💡 Research Summary

The paper introduces and systematically studies a new geometric object called “brachistochrone‑ruled timelike surfaces” in both Newtonian and relativistic spacetimes. Starting from the classical cycloidal brachistochrone—the curve of fastest descent in a uniform gravitational field—the authors first construct a “brachistochrone‑ruled worldsheet” in Newtonian spacetime. In this toy model, two smooth spatial curves Γ₀(s) and Γ₁(s) serve as the boundary of the surface; for each parameter value s a unique cycloid connecting the corresponding endpoints is taken as a ruling. The resulting two‑parameter map Σ(s,u) yields a timelike surface whose rulings are precisely the time‑minimizing worldlines of the classical problem. Existence and smoothness are guaranteed by smooth functions a(s) and θ₁(s) that encode the cycloid parameters.

The authors then generalize the construction to stationary Lorentzian spacetimes (M,g) with a timelike Killing field K. In adapted coordinates (t,x) the metric takes the standard stationary form g=−β(dt−θ)²+h_{ij}dx^i dx^j, where β>0 is the lapse, θ the shift 1‑form, and h a Riemannian metric on the spatial slice N. For a future‑directed timelike curve γ connecting two fixed spatial points, the arrival‑time functional Δt