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

Proposition: Classical kernel identity

Compensator choice and classical kernel comparison › Poisson and Coulomb kernels · lines 1569–1599 · proof included or linked

III@2#prop:DG

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For $V_{P}^{μ} = κ w_{P}^{μ}$ ,

V_{P}^{0} = V_{C}^{0} + D_{C}^{0}, V_{P}^{i} = V_{C}^{i} - D_{C}^{i} .

Proof. Integrating by parts in the Poisson formula gives

V_{P}^{0} = - \frac{κ}{4 π} \int (\frac{d ^{i} h _{0 i}}{d ^{2}} - \frac{h ˙ _{ii}}{2 d}) d^{3} y,

V_{P}^{i} = \frac{κ}{8 π} \int \frac{2 d ^{j} h _{ij} - d ^{i} h _{j j}}{d ^{2}} d^{3} y .

The kernels are locally integrable. Boundary terms from a shrinking inner sphere vanish; Schwartz decay removes the outer terms. Adding and subtracting the displayed Coulomb deformation kernels gives exactly these expressions.

Displayed formulas in this passage

unlabeled · line 1572
$V_{P}^{0} = V_{C}^{0} + D_{C}^{0}, V_{P}^{i} = V_{C}^{i} - D_{C}^{i} .$
unlabeled · line 1580
$V_{P}^{0} = - \frac{κ}{4 π} \int (\frac{d ^{i} h _{0 i}}{d ^{2}} - \frac{h ˙ _{ii}}{2 d}) d^{3} y,$
unlabeled · line 1588
$V_{P}^{i} = \frac{κ}{8 π} \int \frac{2 d ^{j} h _{ij} - d ^{i} h _{j j}}{d ^{2}} d^{3} y .$

Proposition: Classical kernel identity

Displayed formulas in this passage

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