Manuscript XVI-D@1 · XVI-D.tex · section · b73c6f4a

Prepared source and witness

Setting, source input, and signs › Prepared source and witness · lines 368–435

XVI-D@1#eq:preparedsource, XVI-D@1#eq:sourcepolarization, XVI-D@1#eq:witness, XVI-D@1#eq:Slambda, XVI-D@1#eq:q, XVI-D@1#eq:d1

Write $J_{D} = J_{D}$ for the designated prepared source. The imported source relation is

J_{D} [γ] = σ J_{3} [γ] + D_{σ} [γ] .

It is a real homogeneous quadratic differential polynomial. Its real polarization is

J_{D} (a, c) = \frac{1}{2} (J_{D} [a + c] - J_{D} [a] - J_{D} [c]) .

The complex extension is bilinear, not sesquilinear.

Let $∣ n ∣ = 1$ and, for $λ > 0$ , set

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

Both $k_{a}$ are null, so $□ γ_{a}^{λ} = 0$ .

The source-level result imported from XVI-B is

S_{λ}^{μν} := J_{D}^{C} (γ_{1}^{λ}, γ_{2}^{λ}) = [λ^{2} σ (t) q^{μν} + λ d_{1}^{μν} (t)] e^{2 i λ t},

where

q^{00} d_{1}^{μν} = - 2, = - i σ^{'} (η^{μν} + 2 δ^{μ}_{0} δ^{ν}_{0}) . q^{0 i} = 0, q^{ij} = n^{i} n^{j} - 3 δ^{ij},

In particular,

d_{1}^{00} = - 3 i σ^{'}, d_{1}^{0 i} = 0, d_{1}^{ij} = i σ^{'} δ^{ij} .

Neither $q$ nor $d_{1}$ contains an additional factor of $λ$ XVIB.

Equation (eq:Slambda) defines an ambient smooth tensor. It is not merely a slab formula extended discontinuously at an endpoint.

Displayed formulas in this passage

XVI-D@1#eq:preparedsource · line 372
$J_{D} [γ] = σ J_{3} [γ] + D_{σ} [γ] .$
XVI-D@1#eq:sourcepolarization · line 378
$J_{D} (a, c) = \frac{1}{2} (J_{D} [a + c] - J_{D} [a] - J_{D} [c]) .$
XVI-D@1#eq:witness · line 386
$k_{1} = (1, n), k_{2} = (1, - n), γ_{a}^{λ} = η e^{i λ k_{a} \cdot y}, k \cdot y = η_{μν} k^{μ} y^{ν} .$
XVI-D@1#eq:Slambda · line 397
$S_{λ}^{μν} := J_{D}^{C} (γ_{1}^{λ}, γ_{2}^{λ}) = [λ^{2} σ (t) q^{μν} + λ d_{1}^{μν} (t)] e^{2 i λ t},$
XVI-D@1#eq:q · line 409
$q^{00} d_{1}^{μν} = - 2, = - i σ^{'} (η^{μν} + 2 δ^{μ}_{0} δ^{ν}_{0}) . q^{0 i} = 0, q^{ij} = n^{i} n^{j} - 3 δ^{ij},$
XVI-D@1#eq:d1 · line 409
$q^{00} d_{1}^{μν} = - 2, = - i σ^{'} (η^{μν} + 2 δ^{μ}_{0} δ^{ν}_{0}) . q^{0 i} = 0, q^{ij} = n^{i} n^{j} - 3 δ^{ij},$
unlabeled · line 422
$d_{1}^{00} = - 3 i σ^{'}, d_{1}^{0 i} = 0, d_{1}^{ij} = i σ^{'} δ^{ij} .$