Crooked halfspaces

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.Date: November 7, 2018. 2000 Mathematics Subject Classification. 53B30 (Lorentz metrics, indefinite metrics), 53C50 (Lorentz manifolds, manifolds with indefinite metrics).

Foliations of Minkowski 2 + 1 spacetime by crooked planes

“…Let V denote R 3 endowed with a scalar product of signature (2,1). To fix ideas, we will usually assume that it takes the following form in the standard basis :   Say that vectors u, v ∈ V are Lorentz-orthogonal if u · v = 0.…”

Visualization of Minkowski Patch

“…7 we can see the wings are half planes tangent to and the corresponding crooked plane is its boundary ∂H(v, p) [7].…”

“…The bilinear form on inner product space [7]: Where a point on a line p 0 ϵ E n,1 , a vector u ϵ R space-like, time-like, or null and a real parame Then there are three types of curves in Minkowski space [3]:…”

Margulis spacetimes via the arc complex

“…A complete disjointness criterion for crooked planes was given by Drumm-Goldman in [18]. More recently, the geometry of crooked planes and crooked half-spaces was studied in [5]. We now recall a sufficient condition due to Drumm.…”