Manuscript X-D-ST@1 · X-D-ST.tex · proposition · e0cf8fdb

Proposition: A possible jump, but no hypersurface delta

Identification: continuous coordinates and normal flux › The smeared linear background kernel · lines 1278–1334 · proof included or linked

X-D-ST@1#prop:kernel, X-D-ST@1#eq:sharp-kernel

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For a compact smooth covector test $ℓ_{ρ}$ supported in the open slab, set $b_{ρ} = E_{0, A} ℓ_{ρ}$ . Then

\int ℓ_{ρ} w_{0}^{ρ} d^{4} x = \int h_{α β} θ (t) (C^{*} b)^{α β} d^{4} x .

Thus the observation-smeared linear background kernel is

q_{ℓ, 0} = θ C^{*} (E_{0, A} ℓ) .

It is compactly supported, smooth on each side of $Σ$ , and locally bounded. It can jump at $Σ$ , but it has no hypersurface-delta summand.

The finite-width kernels satisfy

q_{ℓ, ε} = σ_{ε} C^{*} (E_{0, A} ℓ), ∥ q_{ℓ, ε} - q_{ℓ, 0} ∥_{L^{1}} = O (ε) .

Proof. For positive width, Green transposition and compact integration by parts give

\int ℓ_{ρ} w_{ε}^{ρ} d^{4} x = - \int b_{ρ} \partial_{μ} (σ_{ε} a^{μ ρ}) d^{4} x = \int σ_{ε} (\partial_{μ} b_{ρ}) a^{μ ρ} d^{4} x = \int h_{α β} σ_{ε} (C^{*} b)^{α β} d^{4} x .

The source is past-trivial and the ramp is flat at zero, so there is no initial boundary term. The advanced solution vanishes after the observation source.

After restricting to $t \geq 0$ , its advanced support has a compact truncated causal past. The kernel difference is bounded and supported in a layer of thickness $ε$ , with one fixed compact spatial reserve. This proves the $L^{1}$ estimate. Theorem thm:convergence, applied to the coordinate pairing, and dominated convergence then give (eq:sharp-kernel).

The limiting kernel is an explicitly locally integrable function. Its formula has a step, not a surface delta.

Displayed formulas in this passage

X-D-ST@1#eq:sharp-kernel · line 1283
$\int ℓ_{ρ} w_{0}^{ρ} d^{4} x = \int h_{α β} θ (t) (C^{*} b)^{α β} d^{4} x .$
unlabeled · line 1309
$\int ℓ_{ρ} w_{ε}^{ρ} d^{4} x = - \int b_{ρ} \partial_{μ} (σ_{ε} a^{μ ρ}) d^{4} x = \int σ_{ε} (\partial_{μ} b_{ρ}) a^{μ ρ} d^{4} x = \int h_{α β} σ_{ε} (C^{*} b)^{α β} d^{4} x .$
unlabeled · line 1290
$q_{ℓ, 0} = θ C^{*} (E_{0, A} ℓ) .$
unlabeled · line 1298
$q_{ℓ, ε} = σ_{ε} C^{*} (E_{0, A} ℓ), ∥ q_{ℓ, ε} - q_{ℓ, 0} ∥_{L^{1}} = O (ε) .$

Proposition: A possible jump, but no hypersurface delta

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