Manuscript Em01-C@3 · Em01-C-v3.tex · theorem · f0953702

Theorem: Characteristic coordinate normal form

Exact characteristic normal coordinates · lines 934–1009 · proof included or linked

Em01-C@3#thm:coordinates, Em01-C@3#eq:coordinate-map, Em01-C@3#eq:qR, Em01-C@3#eq:q1, Em01-C@3#eq:q2, Em01-C@3#eq:coordinate-density

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Near $w_{0}$ , the map

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

is a smooth coordinate map, with

τ = (a, n, α, β), ν = (r, u, v, ξ), ξ \in R^{2} .

There are eight tangential and eight normal variables:

N = {ν = 0}, S_{resp} = {r = 0, u = 0} .

The cone equations are

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

The map and inverse have bounded derivatives on compact chart sets where $α, β - α, β, β - u$ have positive lower bounds and the sphere coordinate remains in a compact subchart.

Proof. For $s = b - a$ , with $∣ s ∣ > 0$ , the inverse coordinates are

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 invertibility.

The null-frame identities give the three exact cone equations. They also identify $N$ and $S_{resp}$ .

The spatial radius of $s$ is $β - u$ , and $s^{0} = β + u$ . The absolute Jacobian of $(β, u) \mapsto (s^{0}, ∣ s ∣)$ is two:

d^{4} s = 2 (β - u)^{2} d β d u d Ω.

At fixed $n$ , the map $(α, v, ξ) \mapsto d$ has absolute determinant two. The full transformation is block triangular after selecting $a, s$ , so the determinants multiply. Uniformity follows from the explicit smooth inverse.

Displayed formulas in this passage

Em01-C@3#eq:coordinate-map · line 937
$z b y = a + r, = a + β ℓ_{n} + u k_{n}, = a + α ℓ_{n} + v k_{n} + ξ^{A} E_{A} (n)$
Em01-C@3#eq:qR · line 956
$q_{R} q_{1} q_{2} = 4 β u, = 4 α v - ∣ ξ ∣^{2}, = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .$
Em01-C@3#eq:q1 · line 956
$q_{R} q_{1} q_{2} = 4 β u, = 4 α v - ∣ ξ ∣^{2}, = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .$
Em01-C@3#eq:q2 · line 956
$q_{R} q_{1} q_{2} = 4 β u, = 4 α v - ∣ ξ ∣^{2}, = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .$
Em01-C@3#eq:coordinate-density · line 965
$d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d^{4} a d Ω d α d β \cdot d^{4} r d u d v d^{2} ξ .$
unlabeled · line 947
$τ = (a, n, α, β), ν = (r, u, v, ξ), ξ \in R^{2} .$
unlabeled · line 952
$N = {ν = 0}, S_{resp} = {r = 0, u = 0} .$
unlabeled · line 982
$n = \frac{s}{∣ s ∣}, β = \frac{s ^{0} + ∣ s ∣}{2}, u = \frac{s ^{0} - ∣ s ∣}{2} .$
unlabeled · line 988
$α = \frac{d ^{0} + n \cdot d}{2}, v = \frac{d ^{0} - n \cdot d}{2}, ξ^{A} = e_{A} \cdot d, r = z - a .$
unlabeled · line 1001
$d^{4} s = 2 (β - u)^{2} d β d u d Ω.$

Theorem: Characteristic coordinate normal form

Displayed formulas in this passage

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