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On the space of cone geodesics and positive paths of contactomorphisms

Jakob Hedicke

TL;DR

The paper extends Penrose–Low ideas by showing that the space of cone geodesics $\mathcal{N}_C$ associated to a smooth strongly convex cone structure can be endowed with a natural contact structure, and in the globally hyperbolic setting it is canonically contactomorphic to the spherical cotangent bundle $ST^{\ast}\Sigma$ of a Cauchy hypersurface $\Sigma$. It then establishes a bidirectional correspondence between globally hyperbolic cone structures with a Cauchy time function and positive paths of contactomorphisms on $ST^{\ast}\Sigma$, via explicit constructions: cone geodesics arise as $t\mapsto (t,\pi(\varphi_t(v)))$, and conversely a positive path yields a Lorentz–Finsler cone $C_f$ with a time-dependent family of Finsler metrics $F_t$ satisfying $L=dt^2-F_t^2$. These results generalize light-ray geometry to cone structures and Lorentz–Finsler spacetimes, linking causality, Reeb flows, and Legendrian geometry within a unified contact-geometric framework. The constructions provide tools to translate dynamical data on $ST^{\ast}\Sigma$ into cone-structure causality, enabling new approaches to study global hyperbolicity, positivity, and geodesic behavior in generalized spacetimes.

Abstract

Often it is possible to equip the space of all cone geodesics of a strongly convex cone structure with the structure of a smooth contact manifold. This generalizes the analogous notions for the space of light rays of a Lorentzian spacetime. After reviewing these constructions on the space of cone geodesics, with a focus on the natural contact structure, we establish a correspondence between positive paths of contactomorphisms in spherical cotangent bundles and certain globally hyperbolic cone structures.

On the space of cone geodesics and positive paths of contactomorphisms

TL;DR

The paper extends Penrose–Low ideas by showing that the space of cone geodesics associated to a smooth strongly convex cone structure can be endowed with a natural contact structure, and in the globally hyperbolic setting it is canonically contactomorphic to the spherical cotangent bundle of a Cauchy hypersurface . It then establishes a bidirectional correspondence between globally hyperbolic cone structures with a Cauchy time function and positive paths of contactomorphisms on , via explicit constructions: cone geodesics arise as , and conversely a positive path yields a Lorentz–Finsler cone with a time-dependent family of Finsler metrics satisfying . These results generalize light-ray geometry to cone structures and Lorentz–Finsler spacetimes, linking causality, Reeb flows, and Legendrian geometry within a unified contact-geometric framework. The constructions provide tools to translate dynamical data on into cone-structure causality, enabling new approaches to study global hyperbolicity, positivity, and geodesic behavior in generalized spacetimes.

Abstract

Often it is possible to equip the space of all cone geodesics of a strongly convex cone structure with the structure of a smooth contact manifold. This generalizes the analogous notions for the space of light rays of a Lorentzian spacetime. After reviewing these constructions on the space of cone geodesics, with a focus on the natural contact structure, we establish a correspondence between positive paths of contactomorphisms in spherical cotangent bundles and certain globally hyperbolic cone structures.
Paper Structure (10 sections, 14 theorems, 40 equations)

This paper contains 10 sections, 14 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cone structures and causality
  4. Lorentz-Finsler metrics and cone geodesics
  5. The space of cone geodesics
  6. Contact manifolds
  7. The contact structure on the space of cone geodesics
  8. Positive paths and cone structures
  9. From cone structures to positive paths
  10. The cone structure induced by a positive path

Key Result

Theorem 1.1

Let $(\mathbb R\times\Sigma,C)$ be a globally hyperbolic strongly convex cone structure such that the projection to the first coordinate is a Cauchy time function. Then there exists a positive path of contactomorphism $(\varphi_t^C)_{t\in\mathbb R}$ of the spherical cotangent bundle $ST^{\ast}\Sigma

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Proposition 2.7
  • Remark 2.8
  • ...and 39 more