Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

The chart Poisson prescription

The unchanged-product obstruction › The chart Poisson prescription · lines 642–668

III-B@1#eq:poisson

For a compact smooth spatial source $f$ , define

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

Then $Δ N f = f$ , and $N Δ f = f$ for compact smooth $f$ . For the latter assertion, the difference is harmonic and decays at infinity.

Apply $N$ at each time and put

A_{i} [h] = \partial_{j} h_{ij} - \frac{1}{2} \partial_{i} h_{j j}, B [h] = \partial_{i} h_{0 i} - \frac{1}{2} \partial_{s} h_{ii},

w_{P}^{i} [h] = - N A_{i} [h], w_{P}^{0} [h] = N B [h] .

The potentials may have spatial tails. They are evaluated only in the specified chart functional.

Displayed formulas in this passage

III-B@1#eq:poisson · line 660
$w_{P}^{i} [h] = - N A_{i} [h], w_{P}^{0} [h] = N B [h] .$
unlabeled · line 645
$(N f) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{f ( y )}{∣ x - y ∣} d^{3} y .$
unlabeled · line 655
$A_{i} [h] = \partial_{j} h_{ij} - \frac{1}{2} \partial_{i} h_{j j}, B [h] = \partial_{i} h_{0 i} - \frac{1}{2} \partial_{s} h_{ii},$