Brachistochrone-ruled timelike surfaces in Newtonian and relativistic spacetimes

TL;DR

This paper introduces and analyzes brachistochrone-ruled timelike surfaces in Newtonian and relativistic spacetimes, exploring their geometric properties and explicit examples, including in Minkowski and Schwarzschild spacetimes.

Contribution

It generalizes classical brachistochrone problems to relativistic spacetimes, constructing timelike surfaces ruled by time-minimizing trajectories and analyzing their geometric features.

Findings

In Minkowski spacetime, brachistochrone-ruled surfaces are totally geodesic.

In Schwarzschild spacetime, geodesics reduce to Jacobi metric geodesics on spatial slices.

Explicit examples and a numerical scheme for constructing these surfaces are provided.

Abstract

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.