Manuscript XX-D@1 · XX-D.tex · section · c83a526d

Complete scalar correction

Explicit one-contraction normalization › Complete scalar correction · lines 1097–1163

XX-D@1#eq:gammaloc, XX-D@1#eq:drod, XX-D@1#eq:Gamma

For components, let $D_{00, 00}$ denote the $h_{00}, h_{00}$ entry, and similarly for the other displayed field components. A prime indicates the second slot.

The local one-contraction correction is

γ_{loc} (x) = - \frac{t}{4 A} [D_{00, 00}]_{diag} + \frac{t}{A a ^{2}} δ^{ij} [D_{0 i, 0 j}]_{diag} + \frac{t ^{2}}{A ^{2}} [\partial_{t^{'}} D_{00, φ}]_{diag} + \frac{2 t ^{3}}{A ^{3}} [\partial_{t} \partial_{t^{'}} D_{φ, φ}]_{diag} + \frac{2 t ^{2}}{A ^{2} a ^{2}} [\partial_{i^{'}} D_{0 i, φ}]_{diag} + \frac{t ^{3}}{A ^{3} a ^{2}} δ^{ij} [\partial_{i} \partial_{j^{'}} D_{φ, φ}]_{diag} + \frac{t ^{2}}{A ^{2}} [\partial_{t^{'}} D_{φ, 00}]_{diag} + \frac{t}{A ^{2}} [D_{φ, 00}]_{diag} + \frac{2 t ^{3}}{A ^{3}} [\partial_{t^{'}}^{2} D_{φ, φ}]_{diag} + \frac{6 t ^{2}}{A ^{3}} [\partial_{t^{'}} D_{φ, φ}]_{diag} + \frac{t}{A ^{3}} [D_{φ, φ}]_{diag} .

These are all the monomials of (eq:qloc); no state positivity is needed to form these smooth diagonal values.

Define the differentiated equal-time matrix

d^{ia} (t, y, x) = [C_{y}^{ia} \partial_{x^{a}} ℓ_{x} D (y, x)]_{t_{y} = t_{x} = t},

with the index $a$ summed when paired with $P_{i}$ . The rod correction is

γ_{rod, R} (t, x) Γ_{f, R} = - 2 \int_{R^{3}} P_{i} (x - y) χ_{R} (y) d^{ia} (t, y, x) d^{3} y, = \int f (γ_{loc} + γ_{rod, R}) d V_{g ˉ} .

The integrand is locally integrable at $y = x$ , because $d^{ia}$ is smooth there and $P_{i}$ is locally integrable.

Displayed formulas in this passage

XX-D@1#eq:gammaloc · line 1105
$γ_{loc} (x) = - \frac{t}{4 A} [D_{00, 00}]_{diag} + \frac{t}{A a ^{2}} δ^{ij} [D_{0 i, 0 j}]_{diag} + \frac{t ^{2}}{A ^{2}} [\partial_{t^{'}} D_{00, φ}]_{diag} + \frac{2 t ^{3}}{A ^{3}} [\partial_{t} \partial_{t^{'}} D_{φ, φ}]_{diag} + \frac{2 t ^{2}}{A ^{2} a ^{2}} [\partial_{i^{'}} D_{0 i, φ}]_{diag} + \frac{t ^{3}}{A ^{3} a ^{2}} δ^{ij} [\partial_{i} \partial_{j^{'}} D_{φ, φ}]_{diag} + \frac{t ^{2}}{A ^{2}} [\partial_{t^{'}} D_{φ, 00}]_{diag} + \frac{t}{A ^{2}} [D_{φ, 00}]_{diag} + \frac{2 t ^{3}}{A ^{3}} [\partial_{t^{'}}^{2} D_{φ, φ}]_{diag} + \frac{6 t ^{2}}{A ^{3}} [\partial_{t^{'}} D_{φ, φ}]_{diag} + \frac{t}{A ^{3}} [D_{φ, φ}]_{diag} .$
XX-D@1#eq:drod · line 1137
$d^{ia} (t, y, x) = [C_{y}^{ia} \partial_{x^{a}} ℓ_{x} D (y, x)]_{t_{y} = t_{x} = t},$
XX-D@1#eq:Gamma · line 1147
$γ_{rod, R} (t, x) Γ_{f, R} = - 2 \int_{R^{3}} P_{i} (x - y) χ_{R} (y) d^{ia} (t, y, x) d^{3} y, = \int f (γ_{loc} + γ_{rod, R}) d V_{g ˉ} .$

Complete scalar correction

Displayed formulas in this passage

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