Manuscript Em01-D@2 · Em01-D-v2.tex · proposition · d3b49151

Proposition: Exact local geometry

Characteristic coordinates and actual smoothing › Coordinates and full density · lines 974–1028 · proof included or linked

Em01-D@2#prop:geometry, Em01-D@2#eq:cones, Em01-D@2#eq:density, Em01-D@2#eq:defect

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These are coordinates near a retained point of $N$ , with

N = {r = u = v = ξ = 0} .

Writing $d = β - α > 0$ , one has

q_{R} = 4 β u, q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = 4 d (u - v) - ∣ ξ ∣^{2} .

The ambient flat density is

d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d^{4} a d Ω d α d β d^{4} r d u d v d^{2} ξ .

Along $N$ , the three cone differentials have rank two. Throughout the chart,

\frac{q _{1}}{α} + \frac{q _{2}}{d} = \frac{q _{R}}{β} - \frac{β}{α d} ∣ ξ ∣^{2} .

Proof. For $s = b - a$ , the inverse is

n = \frac{s}{∣ s ∣}, β = \frac{s ^{0} + ∣ s ∣}{2}, u = \frac{s ^{0} - ∣ s ∣}{2} .

For $x = y - a$ ,

α = \frac{x ^{0} + n \cdot x}{2}, v = \frac{x ^{0} - n \cdot x}{2}, ξ^{A} = e_{A} \cdot x .

Also $r = z - a$ . These formulas prove invertibility.

The null-frame identities give (eq:cones). The $(β, u)$ Jacobian contributes $2 (β - u)^{2}$ , and the $(α, v, ξ)$ determinant contributes two, proving (eq:density). Differentiating (eq:cones) at $N$ gives the rank statement. Their divided sum gives (eq:defect).

Displayed formulas in this passage

Em01-D@2#eq:cones · line 981
$q_{R} = 4 β u, q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = 4 d (u - v) - ∣ ξ ∣^{2} .$
Em01-D@2#eq:density · line 988
$d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d^{4} a d Ω d α d β d^{4} r d u d v d^{2} ξ .$
Em01-D@2#eq:defect · line 998
$\frac{q _{1}}{α} + \frac{q _{2}}{d} = \frac{q _{R}}{β} - \frac{β}{α d} ∣ ξ ∣^{2} .$
unlabeled · line 977
$N = {r = u = v = ξ = 0} .$
unlabeled · line 1009
$n = \frac{s}{∣ s ∣}, β = \frac{s ^{0} + ∣ s ∣}{2}, u = \frac{s ^{0} - ∣ s ∣}{2} .$
unlabeled · line 1015
$α = \frac{x ^{0} + n \cdot x}{2}, v = \frac{x ^{0} - n \cdot x}{2}, ξ^{A} = e_{A} \cdot x .$

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section: Coordinates and full density