Manuscript Em01-F@3 · Em01-F-v3.tex · proposition · 225e59af

Proposition: Coordinates, cone equations, and density

Tensor conventions and separated characteristic geometry › Exact characteristic coordinates · lines 526–573 · proof included or linked

Em01-F@3#prop:geometry, Em01-F@3#eq:qR, Em01-F@3#eq:q1, Em01-F@3#eq:q2, Em01-F@3#eq:coordinate-density

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Near $r = u = v = ξ = 0$ , $0 < α < β$ , these are smooth coordinates, and

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

The 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 \times d^{4} r d u d v d^{2} ξ .

The map and inverse have bounded derivatives on compact subcharts with positive lower bounds for $α, β - α, β, β - u$ .

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 $d = y - a$ ,

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 .

The null-frame identities give the cone equations.

The absolute Jacobian from $(β, u)$ to $(s^{0}, ∣ s ∣)$ is two; its spatial radial factor is $(β - u)^{2}$ . At fixed $n$ , the map $(α, v, ξ) \mapsto d$ has absolute determinant two. The full map is block triangular after choosing $a, s$ . This proves the density and the stated bounds.

Displayed formulas in this passage

Em01-F@3#eq:qR · line 530
$q_{R} q_{1} q_{2} = (b - a)^{2} = 4 β u, = (y - a)^{2} = 4 α v - ∣ ξ ∣^{2}, = (b - y)^{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .$
Em01-F@3#eq:q1 · line 530
$q_{R} q_{1} q_{2} = (b - a)^{2} = 4 β u, = (y - a)^{2} = 4 α v - ∣ ξ ∣^{2}, = (b - y)^{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .$
Em01-F@3#eq:q2 · line 530
$q_{R} q_{1} q_{2} = (b - a)^{2} = 4 β u, = (y - a)^{2} = 4 α v - ∣ ξ ∣^{2}, = (b - y)^{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .$
Em01-F@3#eq:coordinate-density · line 537
$d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d^{4} a d Ω d α d β \times d^{4} r d u d v d^{2} ξ .$
unlabeled · line 554
$n = \frac{s}{∣ s ∣}, β = \frac{s ^{0} + ∣ s ∣}{2}, u = \frac{s ^{0} - ∣ s ∣}{2} .$
unlabeled · line 560
$α = \frac{d ^{0} + n \cdot d}{2}, v = \frac{d ^{0} - n \cdot d}{2}, ξ^{A} = e_{A} \cdot d, r = z - a .$

Proposition: Coordinates, cone equations, and density

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