Polygons of the Lorentzian plane and spherical simplexes

TL;DR

This paper demonstrates that the moduli space of certain convex polygons in the Lorentzian plane, with fixed normals and a suitable area measure, is isometric to a spherical polyhedron, extending known Euclidean results.

Contribution

It introduces a class of Lorentzian convex polygons whose moduli space forms a spherical polyhedron, revealing a new geometric correspondence in Lorentzian geometry.

Findings

Moduli space of Lorentzian polygons is isometric to a spherical polyhedron.

Polygons are space-like, contained in the future cone, and invariant under linear isometries.

Extends Euclidean polygon moduli space results to Lorentzian setting.

Abstract

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed and endowed with a suitable area, is isometric to a spherical polyhedron. These polygons have an infinite number of vertices, are space-like, contained in the future cone of the origin, and setwise invariant under the action of a linear isometry.