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

Frozen source data

The explicit prepared witness › Frozen source data · lines 399–454

XIX-B@1#eq:sigma, XIX-B@1#eq:S, XIX-B@1#eq:Q, XIX-B@1#eq:d1, XIX-B@1#eq:S00

Let $σ \in C^{\infty} (R; R)$ satisfy

σ (t) = 0 (t \leq 0), σ (t) = 1 (t \geq ε) .

No limit in $ε$ is taken. Continuity guarantees a nonempty open interval on which $0 < σ < 1$ .

Fix $λ > 0$ and a Euclidean unit vector $n \in R^{3}$ . The native plus-signature tensor inputs are

k_{1} = (1, n), k_{2} = (1, - n), γ_{a}^{λ} = η_{+} e^{iλ k_{a} \cdot y}, k \cdot y = η_{+, μν} k^{μ} y^{ν} .

Their product phase is $e^{2 iλ t}$ .

We take the following explicit smooth source as frozen data:

S_{λ}^{μν} (t) Q^{00} d_{1}^{μν} = [λ^{2} σ (t) Q^{μν} + λ d_{1}^{μν} (t)] e^{2 iλ t}, = - 2, Q^{0 i} = 0, Q^{ij} = n^{i} n^{j} - 3 δ^{ij}, = - i σ^{'} (t) (η_{+}^{μν} + 2 δ^{μ}_{0} δ^{ν}_{0}) .

Thus

S_{λ}^{00} = [- 2 λ^{2} σ - 3 iλ σ^{'}] e^{2 iλ t}, S_{λ}^{0 i} = 0.

These formulas are the supplied XVI-B/XVI-D witness source CheungXVIB (:pair) CheungXVID (:Slambda). Neither $Q$ nor $d_{1}$ contains an additional frequency factor.

The source definition is sufficient for the present proof. We do not derive the full gravitational source polynomial $J_{D}$ , its Euler origin, or its complete tensor-input domain.

Displayed formulas in this passage

XIX-B@1#eq:sigma · line 402
$σ (t) = 0 (t \leq 0), σ (t) = 1 (t \geq ε) .$
XIX-B@1#eq:S · line 422
$S_{λ}^{μν} (t) Q^{00} d_{1}^{μν} = [λ^{2} σ (t) Q^{μν} + λ d_{1}^{μν} (t)] e^{2 iλ t}, = - 2, Q^{0 i} = 0, Q^{ij} = n^{i} n^{j} - 3 δ^{ij}, = - i σ^{'} (t) (η_{+}^{μν} + 2 δ^{μ}_{0} δ^{ν}_{0}) .$
XIX-B@1#eq:Q · line 422
$S_{λ}^{μν} (t) Q^{00} d_{1}^{μν} = [λ^{2} σ (t) Q^{μν} + λ d_{1}^{μν} (t)] e^{2 iλ t}, = - 2, Q^{0 i} = 0, Q^{ij} = n^{i} n^{j} - 3 δ^{ij}, = - i σ^{'} (t) (η_{+}^{μν} + 2 δ^{μ}_{0} δ^{ν}_{0}) .$
XIX-B@1#eq:d1 · line 422
$S_{λ}^{μν} (t) Q^{00} d_{1}^{μν} = [λ^{2} σ (t) Q^{μν} + λ d_{1}^{μν} (t)] e^{2 iλ t}, = - 2, Q^{0 i} = 0, Q^{ij} = n^{i} n^{j} - 3 δ^{ij}, = - i σ^{'} (t) (η_{+}^{μν} + 2 δ^{μ}_{0} δ^{ν}_{0}) .$
XIX-B@1#eq:S00 · line 437
$S_{λ}^{00} = [- 2 λ^{2} σ - 3 iλ σ^{'}] e^{2 iλ t}, S_{λ}^{0 i} = 0.$
unlabeled · line 412
$k_{1} = (1, n), k_{2} = (1, - n), γ_{a}^{λ} = η_{+} e^{iλ k_{a} \cdot y}, k \cdot y = η_{+, μν} k^{μ} y^{ν} .$