Manuscript V@1 · V.tex · section · e7ddf37d

Option A inverse and its constraints

Contrast: the Newton–wave cubic form › Option A inverse and its constraints · lines 3080–3141

V@1#eq:RA

For $P ϕ = 0$ , use the stress from Section sec:external and its polarization

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

Differentiation gives

\partial^{μ} T_{μν} (v, w) = \frac{1}{2} ((P v) \partial_{ν} w + (P w) \partial_{ν} v) .

Thus it is conserved for free scalar solutions. Take these solutions to have Schwartz Cauchy data. On bounded time intervals the polarized sources are smooth Schwartz functions in space.

Fix the Newton inverse

Δ^{- 1} b (x) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d^{3} y, Δ^{- 1} b (q) = - \frac{b ( q )}{∣ q ∣ ^{2}},

with its locally integrable multiplier and no added zero-supported Fourier distribution.

For a conserved source $Q_{μν}$ , let

ρ = Q_{00} (t_{0}), J_{i} = Q_{0 i} (t_{0}), a = - 2 Δ^{- 1} ρ, V_{i} = 2 Δ^{- 1} J_{i},

B_{ij} = \partial_{i} V_{j} + \partial_{j} V_{i} - δ_{ij} \partial_{ℓ} V_{ℓ} .

Specify trace-reversed metric data by

a_{00} = a, a_{0 i} = a_{ij} = 0, b_{ij} = B_{ij}, b_{00} = b_{0 i} = 0.

Let $r$ denote trace reversal. Define

R^{A, init}_{t_{0}} Q (t) R^{A, src}_{t_{0}} Q (t) = r [cos ((t - t_{0}) ∣ D ∣) a + S_{w} (t - t_{0}) b], = 2 r \int_{t_{0}}^{t} S_{w} (t - s) Q (s) d s, j = R^{A}_{t_{0}} Q .

Then $□ j ˉ = - 2 Q$ .

Displayed formulas in this passage

V@1#eq:RA · line 3128
$R^{A, init}_{t_{0}} Q (t) R^{A, src}_{t_{0}} Q (t) = r [cos ((t - t_{0}) ∣ D ∣) a + S_{w} (t - t_{0}) b], = 2 r \int_{t_{0}}^{t} S_{w} (t - s) Q (s) d s, j = R^{A}_{t_{0}} Q .$
unlabeled · line 3084
$T_{μν} (v, w) = \partial_{(μ} v \partial_{ν)} w - \frac{1}{2} η_{μν} (\partial_{α} v \partial^{α} w + m^{2} v w) .$
unlabeled · line 3091
$\partial^{μ} T_{μν} (v, w) = \frac{1}{2} ((P v) \partial_{ν} w + (P w) \partial_{ν} v) .$
unlabeled · line 3101
$Δ^{- 1} b (x) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d^{3} y, Δ^{- 1} b (q) = - \frac{b ( q )}{∣ q ∣ ^{2}},$
unlabeled · line 3112
$ρ = Q_{00} (t_{0}), J_{i} = Q_{0 i} (t_{0}), a = - 2 Δ^{- 1} ρ, V_{i} = 2 Δ^{- 1} J_{i},$
unlabeled · line 3116
$B_{ij} = \partial_{i} V_{j} + \partial_{j} V_{i} - δ_{ij} \partial_{ℓ} V_{ℓ} .$
unlabeled · line 3121
$a_{00} = a, a_{0 i} = a_{ij} = 0, b_{ij} = B_{ij}, b_{00} = b_{0 i} = 0.$

Option A inverse and its constraints

Displayed formulas in this passage

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