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Manuscript V@1 · V.tex · figure · e7ddf37d

Three gauge tests

Raw current and complete gauge defect › Gauge transformation and initial data · lines 900–929

V@1#fig:three-gauge-tests

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metricscalarrepresented quantumdefect / unresolvedsupport / scope
H=K∗Jraw=γ+δ(t)a+δ′(t)b\mathbf H=K^*J_{\mathrm{raw}}=\gamma+\delta(t)a+\delta'(t)bγν=f∂νϕ,aν=u˙0∂νϕ∣0−u0∂t∂νϕ∣0,bν=u0∂νϕ∣0\gamma_\nu=f\partial_\nu\phi,\quad a_\nu=\dot u_0\partial_\nu\phi|_0-u_0\partial_t\partial_\nu\phi|_0,\quad b_\nu=u_0\partial_\nu\phi|_0
B_raw(Kξ) = ⟨H, ξ⟩
compact parameters strictly after the initial surfaceγ=0\gamma=0
all compact parametersH=0  ⟺  γ=a=b=0\mathbf H=0\iff\gamma=a=b=0
spacelike-compact residual vector wavesE1H=0\mathsf E_1\mathbf H=0
S0(Kξ)ϕ=Xξϕ−Hm(Data0(Xξϕ))S_0(K\xi)\phi=X_\xi\phi-\mathsf H_m(\mathrm{Data}_0(X_\xi\phi))

The three conditions are not identified. The homogeneous subtraction records prescribed zero Cauchy data.

Three gauge tests for the fixed-time response. Future-supported compact parameters test the bulk density, arbitrary compact parameters also test the initial-surface coefficients, and residual vector waves test the propagated defect. The homogeneous scalar subtraction records the response's prescribed zero Cauchy data.

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