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

Poisson and Coulomb kernels

Compensator choice and classical kernel comparison › Poisson and Coulomb kernels · lines 1525–1568

III@2#eq:VC0, III@2#eq:VCi, III@2#eq:DC0, III@2#eq:DCi

At fixed time let

d = y - x, d = ∣ d ∣, d = d / d .

For spatial Schwartz fields define the Coulomb kernels and their deformations as in the version-specific formulas of Donnelly and Giddings DG (Eq. (52), Eqs. (73)–(74)):

V_{C}^{0} (x) V_{C}^{i} (x) D_{C}^{0} (x) D_{C}^{i} (x) = - \frac{κ}{4 π} \int_{R^{3}} (\frac{d ^{i} d ^{j} h ˙ _{ij}}{2 d} + \frac{d ^{i} h _{0 i}}{d ^{2}}) d^{3} y, = \frac{3 κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} d ^{k}}{d ^{2}} h_{j k} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} + δ ^{ij}}{d} h ˙_{ij} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{3 d ^{i} d ^{j} d ^{k} + d ^{i} δ ^{j k} - 2 δ ^{ij} d ^{k}}{d ^{2}} h_{j k} d^{3} y .

Fields in the integrals are evaluated at $(t, y)$ .

Our active convention is $δ h = K ξ$ . The gauge parameter in the corresponding convention of DG has the opposite sign. The displayed kernel definitions themselves are unchanged.

Displayed formulas in this passage

III@2#eq:VC0 · line 1535
$V_{C}^{0} (x) V_{C}^{i} (x) D_{C}^{0} (x) D_{C}^{i} (x) = - \frac{κ}{4 π} \int_{R^{3}} (\frac{d ^{i} d ^{j} h ˙ _{ij}}{2 d} + \frac{d ^{i} h _{0 i}}{d ^{2}}) d^{3} y, = \frac{3 κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} d ^{k}}{d ^{2}} h_{j k} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} + δ ^{ij}}{d} h ˙_{ij} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{3 d ^{i} d ^{j} d ^{k} + d ^{i} δ ^{j k} - 2 δ ^{ij} d ^{k}}{d ^{2}} h_{j k} d^{3} y .$
III@2#eq:VCi · line 1535
$V_{C}^{0} (x) V_{C}^{i} (x) D_{C}^{0} (x) D_{C}^{i} (x) = - \frac{κ}{4 π} \int_{R^{3}} (\frac{d ^{i} d ^{j} h ˙ _{ij}}{2 d} + \frac{d ^{i} h _{0 i}}{d ^{2}}) d^{3} y, = \frac{3 κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} d ^{k}}{d ^{2}} h_{j k} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} + δ ^{ij}}{d} h ˙_{ij} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{3 d ^{i} d ^{j} d ^{k} + d ^{i} δ ^{j k} - 2 δ ^{ij} d ^{k}}{d ^{2}} h_{j k} d^{3} y .$
III@2#eq:DC0 · line 1535
$V_{C}^{0} (x) V_{C}^{i} (x) D_{C}^{0} (x) D_{C}^{i} (x) = - \frac{κ}{4 π} \int_{R^{3}} (\frac{d ^{i} d ^{j} h ˙ _{ij}}{2 d} + \frac{d ^{i} h _{0 i}}{d ^{2}}) d^{3} y, = \frac{3 κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} d ^{k}}{d ^{2}} h_{j k} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} + δ ^{ij}}{d} h ˙_{ij} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{3 d ^{i} d ^{j} d ^{k} + d ^{i} δ ^{j k} - 2 δ ^{ij} d ^{k}}{d ^{2}} h_{j k} d^{3} y .$
III@2#eq:DCi · line 1535
$V_{C}^{0} (x) V_{C}^{i} (x) D_{C}^{0} (x) D_{C}^{i} (x) = - \frac{κ}{4 π} \int_{R^{3}} (\frac{d ^{i} d ^{j} h ˙ _{ij}}{2 d} + \frac{d ^{i} h _{0 i}}{d ^{2}}) d^{3} y, = \frac{3 κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} d ^{k}}{d ^{2}} h_{j k} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{d ^{i} d ^{j} + δ ^{ij}}{d} h ˙_{ij} d^{3} y, = \frac{κ}{8 π} \int_{R^{3}} \frac{3 d ^{i} d ^{j} d ^{k} + d ^{i} δ ^{j k} - 2 δ ^{ij} d ^{k}}{d ^{2}} h_{j k} d^{3} y .$
unlabeled · line 1528
$d = y - x, d = ∣ d ∣, d = d / d .$

Poisson and Coulomb kernels

Displayed formulas in this passage

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