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Brachistochrone-ruled timelike surfaces in Newtonian and relativistic spacetimes

Ferhat Taş

TL;DR

This work introduces brachistochrone-ruled timelike surfaces, a new geometric object where each ruling is a time-minimizing trajectory between endpoint families in Newtonian and relativistic spacetimes. By reducing arrival-time functionals to Finsler (Randers-type) or Jacobi metrics on a spatial slice, the authors connect time-optimal transport to geodesic geometry on $N$ and define a two-parameter ruled surface $oldsymbol extSigma(s,u)$. They provide explicit Newtonian toy models, complete Minkowski-space constructions with planar totally geodesic surfaces, and a Schwarzschild exterior example with a detailed numerical scheme for generating such surfaces. The results illuminate how time-optimal propagation imprints intrinsic and extrinsic geometric features on brachistochrone-ruled surfaces, suggesting rich connections to conjugate points, caustics, and stability in curved spacetimes.

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.

Brachistochrone-ruled timelike surfaces in Newtonian and relativistic spacetimes

TL;DR

This work introduces brachistochrone-ruled timelike surfaces, a new geometric object where each ruling is a time-minimizing trajectory between endpoint families in Newtonian and relativistic spacetimes. By reducing arrival-time functionals to Finsler (Randers-type) or Jacobi metrics on a spatial slice, the authors connect time-optimal transport to geodesic geometry on and define a two-parameter ruled surface . They provide explicit Newtonian toy models, complete Minkowski-space constructions with planar totally geodesic surfaces, and a Schwarzschild exterior example with a detailed numerical scheme for generating such surfaces. The results illuminate how time-optimal propagation imprints intrinsic and extrinsic geometric features on brachistochrone-ruled surfaces, suggesting rich connections to conjugate points, caustics, and stability in curved spacetimes.

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.

Paper Structure

This paper contains 38 sections, 7 theorems, 186 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A Toy Model: Brachistochrone-Ruled Worldsheets in Newtonian Spacetime
  3. Classical brachistochrones revisited
  4. From a family of endpoint pairs to a brachistochrone-ruled worldsheet
  5. Relativistic brachistochrone-ruled timelike surfaces
  6. Stationary spacetimes and reduction to Finsler length functionals
  7. Derivation for fixed energy
  8. Families of endpoints and brachistochrone-ruled timelike surfaces
  9. An explicit example in Minkowski spacetime
  10. Minkowski spacetime and stationary observers
  11. A time functional with bounded coordinate speed
  12. A brachistochrone-ruled timelike surface between two families of observers
  13. A simple planar brachistochrone-ruled surface
  14. Choice of boundary curves and parametrization
  15. Induced metric and causal character
  16. ...and 23 more sections

Key Result

Proposition 1

Let $(M,g)$ be a stationary spacetime with metric of the form (25), and let $C$ be a class of future-directed timelike curves $\gamma(\lambda)=(t(\lambda),x(\lambda))$ satisfying a prescribed normalization condition (for example, proper-time parametrization $g(\dot{\gamma},\dot{\gamma})=-1$, or fixe smooth away from the zero section and strictly convex in each fibre, such that: In the proper-time

Figures (5)

  • Figure 1: Standard brachistochrone curve connecting two points in a uniform gravitational field.
  • Figure 2: Brachistochrone-ruled worldsheet in Newtonian spacetime.
  • Figure 3: Brachistochrone-ruled worldsheet in the Schwarzschild exterior.
  • Figure 4: Equatorial slice of the Schwarzschild exterior with two boundary curves $\alpha_0(s)$ and $\alpha_1(s)$ at radii $r_0$ and $r_1$. The time-minimizing timelike geodesics at fixed energy project to geodesics of the Jacobi metric $h^{\mathrm{J}}$, here shown schematically as a family of curved arcs connecting the two boundaries. Their lifts generate a brachistochrone-ruled timelike surface in spacetime.
  • Figure 5: Family of Jacobi geodesics in Schwarzschild exterior

Theorems & Definitions (27)

  • Definition 1: Brachistochrone-ruled worldsheet in Newtonian spacetime
  • Remark 1
  • Proposition 1: Reduction to a Finsler length functional
  • Remark 2
  • Remark 3
  • Definition 2: Relativistic brachistochrone
  • Definition 3: Relativistic brachistochrone-ruled timelike surface
  • Remark 4
  • Remark 5
  • Remark 6: Physical interpretation of the instantaneous-velocity-change idealization
  • ...and 17 more