Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

Strictly separated characteristic coordinates

Localization and complete-sum verification › Strictly separated characteristic coordinates · lines 1153–1209

Em01-E@2#eq:geometry, Em01-E@2#eq:cones, Em01-E@2#eq:geometrydensity

For

ℓ_{n} = (1, n), k_{n} = (1, - n),

and an orthonormal spatial transverse frame $E_{A} (n)$ , use

z = a + r, b = a + β ℓ_{n} + u k_{n}, y = a + α ℓ_{n} + v k_{n} + ξ^{A} E_{A} .

Work on a sphere subchart with positive lower bounds on $α, β - α, β, β - u$ . The aligned-null locus is

N = {r = u = v = ξ = 0} .

The exact cone equations and density are

q_{R} d^{4} a d^{4} b d^{4} z d^{4} y = 4 β u, = 4 (β - u)^{2} d^{4} a d Ω d α d β d^{4} r d u d v d^{2} ξ . q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},

For $s = b - a$ , the inverse has

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

For $d = y - a$ ,

α = \frac{d ^{0} + n \cdot d}{2}, v = \frac{d ^{0} - n \cdot d}{2}, ξ^{A} = e_{A} \cdot d, r = z - a .

These formulas prove the local coordinate property. The null-frame identities prove (eq:cones). The two endpoint Jacobians are $2 (β - u)^{2}$ and $2$ ; the full map is block triangular. This proves (eq:geometrydensity) and compact-local derivative bounds. The convention agrees with the supplied Em01-C coordinate excerpt Em01C.

Strict separation of $a, y, b$ does not remove $z = a$ . The supported incidence must remain in the complete calculation.

Displayed formulas in this passage

Em01-E@2#eq:geometry · line 1162
$z = a + r, b = a + β ℓ_{n} + u k_{n}, y = a + α ℓ_{n} + v k_{n} + ξ^{A} E_{A} .$
Em01-E@2#eq:cones · line 1174
$q_{R} d^{4} a d^{4} b d^{4} z d^{4} y = 4 β u, = 4 (β - u)^{2} d^{4} a d Ω d α d β d^{4} r d u d v d^{2} ξ . q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},$
Em01-E@2#eq:geometrydensity · line 1174
$q_{R} d^{4} a d^{4} b d^{4} z d^{4} y = 4 β u, = 4 (β - u)^{2} d^{4} a d Ω d α d β d^{4} r d u d v d^{2} ξ . q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},$
unlabeled · line 1156
$ℓ_{n} = (1, n), k_{n} = (1, - n),$
unlabeled · line 1170
$N = {r = u = v = ξ = 0} .$
unlabeled · line 1188
$n = \frac{s}{∣ s ∣}, β = \frac{s ^{0} + ∣ s ∣}{2}, u = \frac{s ^{0} - ∣ s ∣}{2} .$
unlabeled · line 1194
$α = \frac{d ^{0} + n \cdot d}{2}, v = \frac{d ^{0} - n \cdot d}{2}, ξ^{A} = e_{A} \cdot d, r = z - a .$