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

Sourced tails and polynomial degree

Model 3: the classical sourced response › Coulomb obstruction and scalar compatibility · lines 2554–2591

IV@1#fig:Coulomb-and-degree

gravitationalscalaroperator / instrumenterror / restrictionsupport / scope

(a)Sourced constraint and tail

compact scalar data,

E_\phi>0

boundary flux of G^{(1)}_{00}(j)

j_{ij}(t_0,x)=\frac{E_\phi}{4\pi|x|}\delta_{ij}+O(|x|^{-2})

j_{00}(t_0,x)=\frac{E_\phi}{4\pi|x|}+O(|x|^{-2})

Asymptotic label, not a numerical profile. Finite derivative energy; the metric need not lie in $L^2$ .

(b)Polynomial degree

\phi

→

T(\phi)

→

j=\mathcal R T(\phi)

quadratic

\int J j

quadratic response pairing: Model 3

S_{t_0}(j)\phi

cubic scalar correction: not represented

$L_j\phi$ is supported where $\phi$ is supported, so the classical scalar response still exists on finite slabs.

Sourced tails and polynomial degree. Nonzero scalar energy requires a noncompact constraint-compatible metric response, although its derivatives have finite energy. The reconstructed metric and its curvature pairing are quadratic in the scalar preparation. Inserting that metric into a scalar correction produces a different cubic functional, not quantized in Model 3.

Nearby in this section

section: Coulomb obstruction and scalar compatibility