Crooked Halfspaces

TL;DR

This paper explores the geometry of crooked halfspaces in 2+1-dimensional Minkowski space, analyzing their automorphisms, stratification, and relationships with hyperbolic geometry, and applying these insights to foliations by crooked planes.

Contribution

It introduces a detailed geometric and automorphism analysis of crooked halfspaces, connecting them with hyperbolic geometry and foliations in Minkowski space, providing new structural insights.

Findings

Automorphism groups of crooked halfspaces are explicitly calculated.

A correspondence between crooked halfspaces and hyperbolic halfplanes is established.

Conditions for particles to lie within crooked halfspaces are identified.

Abstract

We develop the Lorentzian geometry of a crooked halfspace in 2+1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parallelism classes of timelike lines, or particles, in a crooked halfspace is a geodesic halfplane in the hyperbolic plane. Every point in an open crooked halfspace lies on a particle. The correspondence between crooked halfspaces and halfplanes in hyperbolic 2-space preserves the partial order defined by inclusion, and the involution defined by complementarity. We find conditions for when a particle lies completely in a crooked half space. We revisit the disjointness criterion for crooked planes developed by Drumm and Goldman in terms of the semigroup of translations preserving a crooked halfspace. These ideas are then applied to describe foliations of Minkowski space by crooked planes.