Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

A form, not an independent metric field

Model 3: a specified sourced form and an exterior pairing › A form, not an independent metric field · lines 1163–1194

IV-B@1#eq:metric-form

H_{3} = Γ_{s} (L^{2} (R^{3})), D_{3} = Γ_{s}^{alg} (S (R^{3})),

use the massive free scalar and the fixed vacuum-normal-ordered stress

: T_{μν} :=: \partial_{μ} φ \partial_{ν} φ : - \frac{1}{2} η_{μν} : (\partial^{α} φ \partial_{α} φ + m^{2} φ^{2}) : .

For $Ψ, Ξ \in D_{3}$ , its matrix element $t_{μν}^{Ψ, Ξ} (x)$ is a finite sum of products of Schwartz wave packets and derivatives. Polarized Klein–Gordon conservation shows that it is conserved.

For real compact symmetric $J^{μν}$ , define

j^{NW} (J) [Ψ, Ξ] = \int J^{μν} (R^{NW} t^{Ψ, Ξ})_{μν} d^{4} x .

The preceding seminorm estimates prove that this is a well-defined sesquilinear form, symmetric for real $J$ . Its conical patch interpretation is (eq:completed-response) and Lemma lem:patch.

Displayed formulas in this passage

IV-B@1#eq:metric-form · line 1184
$j^{NW} (J) [Ψ, Ξ] = \int J^{μν} (R^{NW} t^{Ψ, Ξ})_{μν} d^{4} x .$
unlabeled · line 1166
$H_{3} = Γ_{s} (L^{2} (R^{3})), D_{3} = Γ_{s}^{alg} (S (R^{3})),$
unlabeled · line 1171
$: T_{μν} :=: \partial_{μ} φ \partial_{ν} φ : - \frac{1}{2} η_{μν} : (\partial^{α} φ \partial_{α} φ + m^{2} φ^{2}) : .$