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Optical geometries

Anna Fino, Thomas Leistner, Arman Taghavi-Chabert

TL;DR

This work develops a comprehensive intrinsic-torsion framework for optical geometries on Lorentzian manifolds, i.e., null line distributions reducing the frame bundle to $ extbf{Sim}(n)$, and extends it to conformal and generalised optical geometries. By decomposing the algebraic intrinsic torsion into irreducible $ extbf{P}_0$-modules and analysing their projections, the authors classify non-integrable optical structures into eight geometric classes via expansion, twist, and shear, and connect these to null-geodesic congruences and their leaf-space geometry. The paper then situates these optical structures within conformal geometry, establishing how optical invariants transform under conformal rescalings and detailing special metric subclasses (non-expanding, non-twisting, non-shearing) and Fefferman-type constructions, with emphasis on Walker, Kundt, Robinson–Trautman, and pp-wave spacetimes. In four dimensions, the theory aligns with almost Robinson structures and CR geometry on leaf spaces, while in higher dimensions it provides a robust G-structure viewpoint for generalised optical geometries and their curvature conditions. Overall, the results offer a unified, algebraically transparent approach to congruences of null geodesics and their geometric implications across dimensions and conformal settings, with concrete links to classical exact solutions in relativity.

Abstract

We study the notion of optical geometry, defined to be a Lorentzian manifold equipped with a null line distribution, from the perspective of intrinsic torsion. This is an instance of a non-integrable version of holonomy reduction in Lorentzian geometry. These generate congruences of null curves, which play an important rôle in general relativity. Conformal properties of these are investigated. We also extend this concept to generalised optical geometries as introduced by Robinson and Trautman.

Optical geometries

TL;DR

This work develops a comprehensive intrinsic-torsion framework for optical geometries on Lorentzian manifolds, i.e., null line distributions reducing the frame bundle to , and extends it to conformal and generalised optical geometries. By decomposing the algebraic intrinsic torsion into irreducible -modules and analysing their projections, the authors classify non-integrable optical structures into eight geometric classes via expansion, twist, and shear, and connect these to null-geodesic congruences and their leaf-space geometry. The paper then situates these optical structures within conformal geometry, establishing how optical invariants transform under conformal rescalings and detailing special metric subclasses (non-expanding, non-twisting, non-shearing) and Fefferman-type constructions, with emphasis on Walker, Kundt, Robinson–Trautman, and pp-wave spacetimes. In four dimensions, the theory aligns with almost Robinson structures and CR geometry on leaf spaces, while in higher dimensions it provides a robust G-structure viewpoint for generalised optical geometries and their curvature conditions. Overall, the results offer a unified, algebraically transparent approach to congruences of null geodesics and their geometric implications across dimensions and conformal settings, with concrete links to classical exact solutions in relativity.

Abstract

We study the notion of optical geometry, defined to be a Lorentzian manifold equipped with a null line distribution, from the perspective of intrinsic torsion. This is an instance of a non-integrable version of holonomy reduction in Lorentzian geometry. These generate congruences of null curves, which play an important rôle in general relativity. Conformal properties of these are investigated. We also extend this concept to generalised optical geometries as introduced by Robinson and Trautman.

Paper Structure

This paper contains 38 sections, 39 theorems, 177 equations, 3 tables.

Table of Contents

  1. Introduction
  2. $H$-structures and their intrinsic torsion
  3. Algebraic description
  4. Linear algebra
  5. The space $\mathbb{G}$ of algebraic intrinsic torsions
  6. Optical geometry
  7. Optical structures
  8. Optical structures as $G$-structures
  9. Intrinsic torsion
  10. Congruence of null curves
  11. Congruence of null geodesics
  12. Optical invariants
  13. Non-expanding congruences
  14. Non-twisting congruences
  15. Non-shearing congruence
  16. ...and 23 more sections

Key Result

Lemma 3.1

Any choice of vector $k$ in ${\mathbb K}$ establishes a one-to-one correspondence between elements of $\scaleobj{1.2}{\wedge}^p \mathbb{H}_{\mathbb K}^*$ and elements $\phi$ of $\scaleobj{1.2}{\wedge}^{p+1} {\mathbb V}^*$ such that $k \, \phi = 0$ and $\kappa \wedge \phi = 0$, where $\kappa = g(k,\c for any inclusion $\delta^i_a : \mathbb{H}^*_{\mathbb K} \rightarrow {\mathbb V}^*$. The inverse of

Theorems & Definitions (94)

  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Definition 4.1
  • Remark 4.2
  • Remark 4.3
  • Definition 4.4
  • ...and 84 more