Manuscript VII-I@1 · VII-I.tex · theorem · 907e96ab

Theorem: First-order transmission law

Route B1b: a divergence-form density preparation › The sharp limit changes the data · lines 973–1031 · proof included or linked

VII-I@1#thm:transmission, VII-I@1#eq:transmission

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Let $h$ be smooth near the closed future half-space. On every compact subset of $t > 0$ ,

w_{ϵ} [h] ⟶ w_{tr} [h] in C^{\infty},

where

w_{tr}^{μ} [h] = E_{0} C^{μ} h - \frac{sin ( t - Δ )}{- Δ} a_{1}^{0 μ} [h] (0, \cdot) .

In particular,

w_{tr} ∣_{Σ} = 0, \partial_{t} w_{tr}^{μ} ∣_{Σ^{+}} = - a_{1}^{0 μ} [h] ∣_{Σ} .

The prepared limit is generally not the sharp prescription $w = E_{0} C h$ .

Proof. Equation (eq:layer-source) gives

w_{ϵ}^{μ} = E^{R}_{0} (σ_{ε} C^{μ} h) - E^{R}_{0} (σ_{ε}^{'} a_{1}^{0 μ} [h]) .

On bounded causal regions, $σ_{ε} C h \to θ C h$ . The first response converges to $E_{0} C h$ away from the initial surface.

For the second response use the smooth Cauchy formula

E^{R}_{0} (σ_{ε}^{'} a_{1}^{0 μ} [h]) (t) = \int_{0}^{ε} \frac{sin (( t - s ) - Δ )}{- Δ} σ_{ε}^{'} (s) a_{1}^{0 μ} [h] (s) d s .

The positive smooth weight $σ_{ε}^{'}$ has integral one. For $t$ bounded away from zero, continuous smooth Cauchy dependence, with spatial localization on the fixed causal compact, permits passage to the limit in every derivative. The limit is the homogeneous solution displayed in (eq:transmission). A Taylor estimate gives an $O (ε)$ rate on each such smooth seminorm, with a constant depending on finitely many derivatives of $h$ .

That homogeneous solution has initial displacement zero and initial velocity $a_{1}^{0 μ} [h] (0)$ .

Displayed formulas in this passage

VII-I@1#eq:transmission · line 982
$w_{tr}^{μ} [h] = E_{0} C^{μ} h - \frac{sin ( t - Δ )}{- Δ} a_{1}^{0 μ} [h] (0, \cdot) .$
unlabeled · line 977
$w_{ϵ} [h] ⟶ w_{tr} [h] in C^{\infty},$
unlabeled · line 991
$w_{tr} ∣_{Σ} = 0, \partial_{t} w_{tr}^{μ} ∣_{Σ^{+}} = - a_{1}^{0 μ} [h] ∣_{Σ} .$
unlabeled · line 1002
$w_{ϵ}^{μ} = E^{R}_{0} (σ_{ε} C^{μ} h) - E^{R}_{0} (σ_{ε}^{'} a_{1}^{0 μ} [h]) .$
unlabeled · line 1013
$E^{R}_{0} (σ_{ε}^{'} a_{1}^{0 μ} [h]) (t) = \int_{0}^{ε} \frac{sin (( t - s ) - Δ )}{- Δ} σ_{ε}^{'} (s) a_{1}^{0 μ} [h] (s) d s .$

Nearby in this section

section: The sharp limit changes the data