Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Option A and its constraints

A separate Newton–wave cubic form › Option A and its constraints · lines 1317–1351

V-B@1#eq:OptionA

For a smooth rapidly decreasing source, prescribe

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

The multiplier is locally integrable in three dimensions. No Fourier distribution supported at $q = 0$ is added.

Let $Q_{μν}$ be a conserved symmetric source: $\partial^{μ} Q_{μν} = 0$ . At time $0$ , set

ρ = Q_{00} ∣_{0}, J_{i} = Q_{0 i} ∣_{0}, a = - 2 N ρ, V_{i} = 2 N J_{i},

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

Define trace-reversed initial data $A, B$ by

A_{00} = a, A_{0 i} = A_{ij} = 0, B_{ij} as above, B_{00} = B_{0 i} = 0,

and

R^{A} Q = r [cos (t ∣ D ∣) A + S_{w} (t) B + 2 \int_{0}^{t} S_{w} (t - s) Q (s) d s] .

Displayed formulas in this passage

V-B@1#eq:OptionA · line 1344
$R^{A} Q = r [cos (t ∣ D ∣) A + S_{w} (t) B + 2 \int_{0}^{t} S_{w} (t - s) Q (s) d s] .$
unlabeled · line 1320
$N b (x) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d^{3} y, N b (q) = - ∣ q ∣^{- 2} b (q) .$
unlabeled · line 1331
$ρ = Q_{00} ∣_{0}, J_{i} = Q_{0 i} ∣_{0}, a = - 2 N ρ, V_{i} = 2 N J_{i},$
unlabeled · line 1335
$B_{ij} = \partial_{i} V_{j} + \partial_{j} V_{i} - δ_{ij} \partial_{ℓ} V_{ℓ} .$
unlabeled · line 1339
$A_{00} = a, A_{0 i} = A_{ij} = 0, B_{ij} as above, B_{00} = B_{0 i} = 0,$