Manuscript XIX-B@1 · XIX-B.tex · section · b5670709

Response, response sign, and coordinate candidate

The explicit prepared witness › Response, response sign, and coordinate candidate · lines 455–524

XIX-B@1#eq:V, XIX-B@1#eq:a1-witness, XIX-B@1#eq:Fco, XIX-B@1#eq:Xcand, XIX-B@1#eq:candidate-PDE

Define

V_{λ}^{μν} (t) = \int_{0}^{t} (t - s) S_{λ}^{μν} (s) d s, R_{λ}^{μν} = - (P_{+} V_{λ})^{μν} .

Then

V_{λ}^{''} = S_{λ}, V_{λ} (0) = V_{λ}^{'} (0) = 0, V_{λ}^{0 i} = 0.

Because the source is ambient smooth and flat at $t = 0$ , extension of $V_{λ}$ by zero for $t \leq 0$ is smooth.

The sign $R_{λ} = - P_{+} V_{λ}$ is retained. It is the polarized response $U_{D}$ on the witness, also explicitly recorded in XVI-B's account of XVI-D CheungXVIB (:later-response) CheungXVID. The separate action-equation coefficient $r_{D} = - u_{D}$ is not silently substituted for the response argument.

For a contravariant response argument $R$ , its native first density coefficient is

a_{1, +}^{μν} [R] = \frac{1}{2} η_{+}^{μν} η_{+, α β} R^{α β} - R^{μν} = - (P_{+} R)^{μν} .

Consequently,

a_{1, +} [R_{λ}] = V_{λ} .

The prepared coordinate forcing and candidate are

F_{co}^{μ} X_{cand}^{μ} (t) = \partial_{α} (σ a_{1, +}^{α μ} [R_{λ}]) = \partial_{t} (σ V_{λ}^{0 μ}), = - \int_{0}^{t} σ (s) V_{λ}^{0 μ} (s) d s .

These are the supplied completion interface and candidate CheungXVIB (:coordinate-leg) CheungXVID (:Xcand).

Direct differentiation, with spatial homogeneity, proves

□_{+} X_{cand}^{μ} = - F_{co}^{μ}, X_{cand}^{μ} (0) = \partial_{t} X_{cand}^{μ} (0) = 0, X_{cand}^{i} = 0.

This calculation does not identify an inherited Green symbol outside its declared domain.

Displayed formulas in this passage

XIX-B@1#eq:V · line 458
$V_{λ}^{μν} (t) = \int_{0}^{t} (t - s) S_{λ}^{μν} (s) d s, R_{λ}^{μν} = - (P_{+} V_{λ})^{μν} .$
XIX-B@1#eq:a1-witness · line 493
$a_{1, +} [R_{λ}] = V_{λ} .$
XIX-B@1#eq:Fco · line 498
$F_{co}^{μ} X_{cand}^{μ} (t) = \partial_{α} (σ a_{1, +}^{α μ} [R_{λ}]) = \partial_{t} (σ V_{λ}^{0 μ}), = - \int_{0}^{t} σ (s) V_{λ}^{0 μ} (s) d s .$
XIX-B@1#eq:Xcand · line 498
$F_{co}^{μ} X_{cand}^{μ} (t) = \partial_{α} (σ a_{1, +}^{α μ} [R_{λ}]) = \partial_{t} (σ V_{λ}^{0 μ}), = - \int_{0}^{t} σ (s) V_{λ}^{0 μ} (s) d s .$
XIX-B@1#eq:candidate-PDE · line 515
$□_{+} X_{cand}^{μ} = - F_{co}^{μ}, X_{cand}^{μ} (0) = \partial_{t} X_{cand}^{μ} (0) = 0, X_{cand}^{i} = 0.$
unlabeled · line 466
$V_{λ}^{''} = S_{λ}, V_{λ} (0) = V_{λ}^{'} (0) = 0, V_{λ}^{0 i} = 0.$
unlabeled · line 485
$a_{1, +}^{μν} [R] = \frac{1}{2} η_{+}^{μν} η_{+, α β} R^{α β} - R^{μν} = - (P_{+} R)^{μν} .$