Manuscript IX-A-ST@1 · IX-A-ST.tex · section · 6e99d1b2

The exact sharp response and compact polynomial functionals

The exact sharp response and compact polynomial functionals · lines 380–421

IX-A-ST@1#eq:E0, IX-A-ST@1#eq:B, IX-A-ST@1#eq:J

Raise indices with $η$ , and write

h = η^{α β} h_{α β}, (C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} h .

For smooth sources, define the zero-Cauchy response on the supported region by

(E_{0} s) (t) = - \int_{0}^{t} \frac{sin (( t - u ) - Δ )}{- Δ} s (u) d u, t \geq 0.

At zero spatial frequency the quotient equals $t - u$ . Thus

□_{η} E_{0} s = s, E_{0} s ∣_{Σ} = 0, \partial_{t} E_{0} s ∣_{Σ} = 0.

For fields without spatial decay, Lemma lem:saturation defines the local interpretation.

Set

L = E_{0} C, w [h] = L h, B_{f} [h, ϕ] = - \int_{V} f (x) w^{μ} [h] (x) \partial_{μ} ϕ (x) d^{4} x,

and, for a smooth compact covector source $a$ in $V$ ,

J (a) [h] = \int_{V} a_{μ} (x) w^{μ} [h] (x) d^{4} x .

These definitions retain the exact IX-A normalization CheungIXA. No equations are imposed on $h$ or $ϕ$ .

Displayed formulas in this passage

IX-A-ST@1#eq:E0 · line 390
$(E_{0} s) (t) = - \int_{0}^{t} \frac{sin (( t - u ) - Δ )}{- Δ} s (u) d u, t \geq 0.$
IX-A-ST@1#eq:B · line 408
$L = E_{0} C, w [h] = L h, B_{f} [h, ϕ] = - \int_{V} f (x) w^{μ} [h] (x) \partial_{μ} ϕ (x) d^{4} x,$
IX-A-ST@1#eq:J · line 414
$J (a) [h] = \int_{V} a_{μ} (x) w^{μ} [h] (x) d^{4} x .$
unlabeled · line 384
$h = η^{α β} h_{α β}, (C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} h .$
unlabeled · line 399
$□_{η} E_{0} s = s, E_{0} s ∣_{Σ} = 0, \partial_{t} E_{0} s ∣_{Σ} = 0.$