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Manuscript IV@1 · IV.tex · figure · 351cb2d7

Two classical response maps

Model 3: the classical sourced response › The initial-data comparison is executed · lines 2465–2495

IV@1#fig:response-comparison

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gravitationalscalaroperator / instrumenterror / restrictionsupport / scope
conserved source QQ
Rt0C\mathcal R^{\mathrm C}_{t_0}
jˉ=(a,0,0),jˉ˙=(0,0,Bij)\bar j=(a,0,0),\quad \dot{\bar j}=(0,0,B_{ij})
Rt0NW\mathcal R^{\mathrm{NW}}_{t_0}
jˉ=(a,−Vi,0),jˉ˙=(−∂iVi,0,0)\bar j=(a,-V_i,0),\quad \dot{\bar j}=(-\partial_i V_i,0,0)
both evolve by □jˉ=−2Q\square\bar j=-2Q, D(j)=0D(j)=0, G(1)(j)=QG^{(1)}(j)=Q
RNWQ−RCQ=Kξ\mathcal R^{\mathrm{NW}}Q-\mathcal R^{\mathrm C}Q=K\xi
ξ0=0,ξi=−sin⁡((t−t0)D0)D0Vi\xi_0=0,\quad \xi_i=-\frac{\sin((t-t_0)D_0)}{D_0}V_i
same linearized curvature
same compact conserved-current pairings

Not identical metric maps. ξ\xi is generally noncompact.

Executed comparison of the classical response maps. The two constraint-compatible initial-data prescriptions produce responses differing by the displayed homogeneous pure gauge. Their curvatures and compact conserved-current pairings agree, but arbitrary metric smearings need not. The comparison gauge is not generally compactly supported.

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