Manuscript En03-A@1 · En03-A.tex · section · 5398614e

Compact collar and frozen exterior

External conical geometry and certified local moves › Compact collar and frozen exterior · lines 1865–1917

En03-A@1#eq:arms

Frozen exterior means unchanged geometric and affine-time laws, not frozen spacetime time coordinates.

core polygon boundaries. Write

U_{ν} = (a s, 0, ν Δ (- s) / q) (s < 0), Z_{ϵ} = (b s, ϵ Δ s / p, 0) (s > 0),

and

V_{ϵ, ν} = (H + s, ϵ X_{s}, ν Y_{s}), M^{x}_{ϵ} = (H + s, ϵ X_{s}, 0), M^{y}_{ν} = (H + s, 0, ν Y_{s}) .

Counterclockwise incoming boundaries are

E N : U_{-}, V_{+, -}, M_{+}^{x}, V_{+, +}, U_{+}, : U_{+}, V_{+, +}, M_{+}^{y}, V_{-, +} .

The outgoing boundaries are

E N : Z_{+}, V_{+, -}, M_{+}^{x}, V_{+, +}, : Z_{-}, Z_{+}, V_{+, +}, M_{+}^{y}, V_{-, +} .

Reflecting and reversing order gives $W, S$ . At $s = 0$ , the central vertices coincide; remove repeated consecutive vertices in the geometric realization.

The remaining vertices are the four outer corners $(H + s, ϵ R_{x}, ν R_{y})$ and four outer side midpoints $(H + s, ϵ R_{x}, 0)$ , $(H + s, 0, ν R_{y})$ .

Each mixed arm ends at $V_{ϵ, ν}$ , and has length

L_{arm} (s) = ⎩ ⎨ ⎧ \frac{10 + s}{75} 1921, \frac{26 + s}{195} 1921, - 1 \leq s \leq 0, 0 \leq s \leq 1.

The two event values agree.

Displayed formulas in this passage

En03-A@1#eq:arms · line 1907
$L_{arm} (s) = ⎩ ⎨ ⎧ \frac{10 + s}{75} 1921, \frac{26 + s}{195} 1921, - 1 \leq s \leq 0, 0 \leq s \leq 1.$
unlabeled · line 1872
$U_{ν} = (a s, 0, ν Δ (- s) / q) (s < 0), Z_{ϵ} = (b s, ϵ Δ s / p, 0) (s > 0),$
unlabeled · line 1877
$V_{ϵ, ν} = (H + s, ϵ X_{s}, ν Y_{s}), M^{x}_{ϵ} = (H + s, ϵ X_{s}, 0), M^{y}_{ν} = (H + s, 0, ν Y_{s}) .$
unlabeled · line 1885
$E N : U_{-}, V_{+, -}, M_{+}^{x}, V_{+, +}, U_{+}, : U_{+}, V_{+, +}, M_{+}^{y}, V_{-, +} .$
unlabeled · line 1892
$E N : Z_{+}, V_{+, -}, M_{+}^{x}, V_{+, +}, : Z_{-}, Z_{+}, V_{+, +}, M_{+}^{y}, V_{-, +} .$

Compact collar and frozen exterior

Displayed formulas in this passage

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