Manuscript XIX-A@1 · XIX-A.tex · proposition · e30d8062

Proposition: Plateau response and transition-region defect

Third refusal: the density ramp is a named non-clock › The ramp changes the first harmonic response · lines 1116–1177 · proof included or linked

XIX-A@1#prop:ramp-defect, XIX-A@1#eq:ramp-defect

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The unregulated first coordinate correction satisfies $W_{1} [K ξ] = ξ$ . The ramped first correction satisfies

W_{1, ε}^{μ} [K ξ] - ξ^{μ} = G_{C} [(σ_{ε} - 1) □_{η} ξ^{μ} - (\partial_{α} σ_{ε}) A_{1}^{α μ} [K ξ]] .

It returns $ξ$ when its support lies strictly inside the $σ_{ε} = 1$ plateau. There is a compact pure-gauge input in the transition region for which it does not return $ξ$ .

Proof. For $h = K ξ$ ,

A_{1}^{α μ} = (\partial_{ρ} ξ^{ρ}) η^{α μ} - \partial^{α} ξ^{μ} - \partial^{μ} ξ^{α} .

Taking the divergence proves (eq:pure-gauge-density). The unregulated equation is $□_{η} W_{1} = □_{η} ξ$ with the same zero data as $ξ$ ; uniqueness proves $W_{1} = ξ$ .

The ramped equation is

□_{η} W_{1, ε}^{μ} = σ_{ε} □_{η} ξ^{μ} - (\partial_{α} σ_{ε}) A_{1}^{α μ} [K ξ] .

Subtract the equation for $ξ$ and apply $G_{C}$ . On the plateau the forcing in (eq:ramp-defect) vanishes.

For a detecting input, choose a transition time $t_{c}$ with $σ_{ε} (t_{c}) < 1$ , a compact spatial bump $ψ$ equal to one near $x_{c}$ , and a compact time bump $a$ in the transition region such that $a^{'} (t_{c}) = 0$ and $a^{''} (t_{c}) \neq = 0$ . Set

ξ^{0} = a (t) ψ (x), ξ^{i} = 0.

Near the detecting point,

A_{1}^{00} = a^{'} ψ, □_{η} ξ^{0} = - a^{''} ψ .

The forcing for $W_{1, ε}^{0} - ξ^{0}$ there equals

(1 - σ_{ε} (t_{c})) a^{''} (t_{c}) \neq = 0.

If the difference field vanished identically, its wave operator would vanish identically, contradicting this forcing.

Displayed formulas in this passage

XIX-A@1#eq:ramp-defect · line 1121
$W_{1, ε}^{μ} [K ξ] - ξ^{μ} = G_{C} [(σ_{ε} - 1) □_{η} ξ^{μ} - (\partial_{α} σ_{ε}) A_{1}^{α μ} [K ξ]] .$
unlabeled · line 1137
$A_{1}^{α μ} = (\partial_{ρ} ξ^{ρ}) η^{α μ} - \partial^{α} ξ^{μ} - \partial^{μ} ξ^{α} .$
unlabeled · line 1148
$□_{η} W_{1, ε}^{μ} = σ_{ε} □_{η} ξ^{μ} - (\partial_{α} σ_{ε}) A_{1}^{α μ} [K ξ] .$
unlabeled · line 1163
$ξ^{0} = a (t) ψ (x), ξ^{i} = 0.$
unlabeled · line 1167
$A_{1}^{00} = a^{'} ψ, □_{η} ξ^{0} = - a^{''} ψ .$
unlabeled · line 1172
$(1 - σ_{ε} (t_{c})) a^{''} (t_{c}) \neq = 0.$

Proposition: Plateau response and transition-region defect

Displayed formulas in this passage

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