Manuscript Em01-G@5 · Em01-G-v5.tex · proposition · 3fbfe951

Proposition: Unique supported normal form

Characteristic coordinates and the supported normal form · lines 496–542 · proof included or linked

Em01-G@5#prop:normal, Em01-G@5#eq:normal, Em01-G@5#eq:normal-extraction

The complete supported sector has a unique representation

K_{supp}^{I, J} = ∣ ζ ∣ + j \leq 3 \sum E_{ζ, j} (c_{ζ, j}^{I, J}), c_{ζ, j}^{I, J} \in C^{\infty} .

For the ordered transformed seed operator

P_{supp}^{I, J} = ζ, j, γ \sum a_{ζ, j, γ}^{I, J} (τ, r, u) \partial_{τ}^{γ} \partial_{r}^{ζ} \partial_{u}^{j},

the coefficients are

c_{ζ, j}^{I, J} = (- 1)^{∣ ζ ∣ + j} γ \sum (\partial_{τ}^{γ})^{t} [\frac{β}{2 π} a_{ζ, j, γ}^{I, J} (τ, 0, 0)] .

The transpose uses $d τ$ .

Proof. On the separated future patch, $R (b - a) = δ_{D} (u) / (8 π β)$ . Thus

⟨ R (b - a) δ_{D}^{(4)} (a - z), φ ⟩ = \frac{1}{2 π} \int β φ (τ, 0, 0) d τ .

The supported seed operator has order at most three. Restrict its ordered coefficients and transpose tangential derivatives to obtain (eq:normal-extraction). A test $r^{ζ} u^{j} ω (r, u) ψ (τ) / (ζ! j!)$ , with $ω = 1$ near zero, isolates a selected normal coefficient. This proves uniqueness.

Displayed formulas in this passage

Em01-G@5#eq:normal · line 499
$K_{supp}^{I, J} = ∣ ζ ∣ + j \leq 3 \sum E_{ζ, j} (c_{ζ, j}^{I, J}), c_{ζ, j}^{I, J} \in C^{\infty} .$
Em01-G@5#eq:normal-extraction · line 516
$c_{ζ, j}^{I, J} = (- 1)^{∣ ζ ∣ + j} γ \sum (\partial_{τ}^{γ})^{t} [\frac{β}{2 π} a_{ζ, j, γ}^{I, J} (τ, 0, 0)] .$
unlabeled · line 508
$P_{supp}^{I, J} = ζ, j, γ \sum a_{ζ, j, γ}^{I, J} (τ, r, u) \partial_{τ}^{γ} \partial_{r}^{ζ} \partial_{u}^{j},$
unlabeled · line 531
$⟨ R (b - a) δ_{D}^{(4)} (a - z), φ ⟩ = \frac{1}{2 π} \int β φ (τ, 0, 0) d τ .$

Nearby in this section

section: Characteristic coordinates and the supported normal form