Manuscript III@2 · III.tex · section · 82dad3e5

Spatial Poisson compensation

The unchanged free-product obstruction › Spatial Poisson compensation · lines 611–642

III@2#eq:Newton, III@2#eq:Poisson

For compact smooth or Schwartz spatial sources, define

(N s) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{s ( y )}{∣ x - y ∣} d^{3} y .

The fundamental-solution identity gives

Δ N s = s, N Δ s = s .

For the second equality, the difference is smooth harmonic and tends to zero at infinity, hence vanishes.

For smooth spacelike-compact $h$ , apply $N$ at each time:

A_{i} [h] w_{P}^{i} [h] = \partial_{j} h_{ij} - \frac{1}{2} \partial_{i} h_{j j}, = - N A_{i} [h], B_{P} [h] w_{P}^{0} [h] = \partial_{i} h_{0 i} - \frac{1}{2} \partial_{t} h_{ii}, = N B_{P} [h] .

These potentials are smooth but need not be spatially compact. Their sources have uniformly bounded support on finite time intervals, so time differentiation is legitimate.

Displayed formulas in this passage

III@2#eq:Newton · line 614
$(N s) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{s ( y )}{∣ x - y ∣} d^{3} y .$
III@2#eq:Poisson · line 628
$A_{i} [h] w_{P}^{i} [h] = \partial_{j} h_{ij} - \frac{1}{2} \partial_{i} h_{j j}, = - N A_{i} [h], B_{P} [h] w_{P}^{0} [h] = \partial_{i} h_{0 i} - \frac{1}{2} \partial_{t} h_{ii}, = N B_{P} [h] .$
unlabeled · line 621
$Δ N s = s, N Δ s = s .$

Spatial Poisson compensation

Displayed formulas in this passage

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