Lorentz Atlas

Discrete geometry

Householder reflections

Hᵥ = I − 2vvᵀ/vᵀv is an exact spatial isometry with det = −1. Embedded as block-diag(1, H) it sits in O⁺(1, n), not in SO⁺, and not on any Lie flow. Products of even length rotate; odd length is the spatial-parity component. This is the discrete degree of freedom FlorE tried to interpolate with σ ∈ (−1, 1).

ε_group ‖HᵀJH − J‖

0

H² − I

0

det H

-1.0000

Λ₀₀ time

1.0000

Spatial plane (x₁, x₂). Time x₀ is not drawn because H does not move it — residual 0. Sheet residual 0.

H u + u

0

H v − v, v ⊥ u

0

involution H² = I

0

Why this is the discrete bit

  • H² = I

    No continuous path inside O(n) joins I to a reflection: det cannot go from +1 to −1 without leaving the group. Paths among reflections stay in the det = −1 component.

  • H u = −u

    The normal flips. The mirror line (u⊥) is fixed pointwise.

  • block-diag(1, H) ∈ O⁺(1, n)

    Time is untouched, so Λ₀₀ = 1. Orthochronous. Embeddings stay on x₀ > 0.

  • H₂ H₁ = R_{2ψ}

    Two reflections compose to a rotation by twice the angle of the normals. Even products sit in SO⁺; odd products are the other component of O⁺.

  • D(σ) is a Householder only at σ = −1

    σ = +1 is the identity. Every other value in (−1, 1) fails HᵀH = I.

Lie flows cannot reach a single H. so(1, 2) generators.

FlorE’s D(σ) is not this

min(‖D−H‖, ‖D−I‖)

0.5000

|σ² − 1|

0.7500

Residual vanishes only at σ = ±1. σ = −1 is the last-axis Householder. σ = +1 is no reflection. The printed interval (−1, 1) contains neither.