Manuscript X-E-ST@1 · X-E-ST.tex · theorem · 44614221

Theorem: Retarded response normalization

E4: the isolated matter-induced tree › Classical normalization and the switch-divergence tensor · lines 1619–1668 · proof included or linked

X-E-ST@1#thm:normalization, X-E-ST@1#eq:minus2, X-E-ST@1#eq:classicaltree

For a smooth free solution $P_{m} u = 0$ , define

k_{1} [u] = - 2 P_{tr} E_{0}^{R} (g T_{0} [u]) .

Then its coordinate response is

w^{μ} [k_{1} [u]] = - 2 (E_{0}^{R})^{2} ((\partial_{ν} g) T_{0}^{ν μ} [u]) .

The classical normal-symbol value of the tree is

τ [u] = \int f (2 \partial_{μ} u (E_{0}^{R})^{2} ((\partial_{ν} g) T_{0}^{ν μ} [u]) - E_{m}^{R} L_{1} [k_{1} [u]] u) d^{4} x .

Thus the normalization constant is $- 2$ . The scalar-response term is part of the same tree.

Proof. The response vanishes before the source onset, so $σ k_{1} [u] = k_{1} [u]$ on the coordinate source. Since $a_{1} [l] = - P_{tr} l$ ,

w [k_{1} [u]] = E_{0}^{R} C [k_{1} [u]] .

Theorem thm:free and (eq:stress-div) give

C^{μ} [k_{1} [u]] = - 2 E_{0}^{R} \partial_{ν} (g T_{0}^{ν μ} [u]) = - 2 E_{0}^{R} ((\partial_{ν} g) T_{0}^{ν μ} [u]) .

This proves (eq:minus2).

Finally, self-adjointness of $L_{1}$ on compact pairings and retarded–advanced transposition give

⟨ d_{1} [l], u ⟩ = - \int f w^{μ} [l] \partial_{μ} u d^{4} x - \int f E_{m}^{R} L_{1} [l] u d^{4} x .

Substitute $l = k_{1} [u]$ .

Displayed formulas in this passage

X-E-ST@1#eq:minus2 · line 1626
$w^{μ} [k_{1} [u]] = - 2 (E_{0}^{R})^{2} ((\partial_{ν} g) T_{0}^{ν μ} [u]) .$
X-E-ST@1#eq:classicaltree · line 1633
$τ [u] = \int f (2 \partial_{μ} u (E_{0}^{R})^{2} ((\partial_{ν} g) T_{0}^{ν μ} [u]) - E_{m}^{R} L_{1} [k_{1} [u]] u) d^{4} x .$
unlabeled · line 1622
$k_{1} [u] = - 2 P_{tr} E_{0}^{R} (g T_{0} [u]) .$
unlabeled · line 1649
$w [k_{1} [u]] = E_{0}^{R} C [k_{1} [u]] .$
unlabeled · line 1653
$C^{μ} [k_{1} [u]] = - 2 E_{0}^{R} \partial_{ν} (g T_{0}^{ν μ} [u]) = - 2 E_{0}^{R} ((\partial_{ν} g) T_{0}^{ν μ} [u]) .$
unlabeled · line 1662
$⟨ d_{1} [l], u ⟩ = - \int f w^{μ} [l] \partial_{μ} u d^{4} x - \int f E_{m}^{R} L_{1} [l] u d^{4} x .$