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

Lemma: Exact rescaled integral

Exact nonphysical eight-variable diagnostic › Coordinates and the exact measure · lines 2051–2099 · proof included or linked

Em01-D@2#lem:diagnosticrescale, Em01-D@2#eq:diagnosticrescale

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For sufficiently small scales,

u_{δ} (Φ) = \frac{i}{2} j = 0 \sum^{2} c_{j} δ^{j - 3} \int d τ \int_{R} d S \int_{0}^{\infty} d T P_{j} (S, T) \times \frac{1}{2 π} \int_{0}^{2 π} F_{j} (τ; \frac{δ T}{2}, \frac{δ S}{2}, 2 B δ T e_{θ}) d θ .

Proof. Expansion of the derivative in (eq:diagnosticdifference) gives

j = 0 \sum^{2} c_{j} δ^{j - 5} p (s / δ) p^{(3 - j)} ((s - t) / δ) \partial_{b^{0}}^{j} Φ_{I_{*}} .

The incidence integrates out $z$ .

The two endpoint coordinate densities contribute

d^{4} b = 2 d B d u d^{2} X, d^{4} y = 2 d A d v d^{2} Y .

The future retarded cone is $(2 π)^{- 1} δ_{D} (4 B u - ∣ X ∣^{2})$ . Its $u$ -integration gives the remaining measure

\frac{1}{2 π B} d^{2} X d v

in addition to $d τ$ .

Write $Q = ∣ X ∣^{2}$ and include angular averaging, so $d^{2} X = π d Q$ . Set $t = Q / (2 B)$ and $s = 2 v$ . The exact resulting factor is $d s d t /2$ . Finally $s = δ S, t = δ T$ supplies $δ^{2}$ . This proves (eq:diagnosticrescale).

Displayed formulas in this passage

Em01-D@2#eq:diagnosticrescale · line 2054
$u_{δ} (Φ) = \frac{i}{2} j = 0 \sum^{2} c_{j} δ^{j - 3} \int d τ \int_{R} d S \int_{0}^{\infty} d T P_{j} (S, T) \times \frac{1}{2 π} \int_{0}^{2 π} F_{j} (τ; \frac{δ T}{2}, \frac{δ S}{2}, 2 B δ T e_{θ}) d θ .$
unlabeled · line 2073
$j = 0 \sum^{2} c_{j} δ^{j - 5} p (s / δ) p^{(3 - j)} ((s - t) / δ) \partial_{b^{0}}^{j} Φ_{I_{*}} .$
unlabeled · line 2081
$d^{4} b = 2 d B d u d^{2} X, d^{4} y = 2 d A d v d^{2} Y .$
unlabeled · line 2088
$\frac{1}{2 π B} d^{2} X d v$

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section: Coordinates and the exact measure