Manuscript Em02-D@4 · Em02-D-v4.tex · lemma · fcf2fac7

Lemma: Cone fundamental solution

The 5z coefficient and complete leading assembly › Distributional inner-tip cancellation · lines 1778–1811 · proof included or linked

Em02-D@4#lem:fundamental

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With normalization (eq:R),

□_{η} R = δ^{(4)} .

The reflected advanced cone measure satisfies the same identity.

Proof. For a compact smooth test $ψ$ , let

h (t, ρ) = \frac{1}{4 π} \int_{S^{2}} ψ (t, ρ m) d m .

Spherical averaging gives

⟨ R, □_{η} ψ ⟩ = \int_{0}^{\infty} [ρ (h_{tt} - h_{ρρ}) - 2 h_{ρ}]_{t = ρ} d ρ .

The integrand is

\frac{d}{d ρ} [ρ (h_{t} - h_{ρ}) - h]_{t = ρ} .

The compact-support boundary at infinity is zero. At $ρ = 0$ , the bracket is $- ψ (0)$ . Therefore the integral is $ψ (0)$ . Reflection preserves $□_{η}$ and $δ^{(4)}$ , proving the advanced assertion.

Displayed formulas in this passage

unlabeled · line 1781
$□_{η} R = δ^{(4)} .$
unlabeled · line 1790
$h (t, ρ) = \frac{1}{4 π} \int_{S^{2}} ψ (t, ρ m) d m .$
unlabeled · line 1795
$⟨ R, □_{η} ψ ⟩ = \int_{0}^{\infty} [ρ (h_{tt} - h_{ρρ}) - 2 h_{ρ}]_{t = ρ} d ρ .$
unlabeled · line 1802
$\frac{d}{d ρ} [ρ (h_{t} - h_{ρ}) - h]_{t = ρ} .$

Nearby in this section

theorem: Theorem: 5a–response coincidence cancellation