Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

Cauchy pullbacks

Cauchy covariances and the full tensor audit › Cauchy pullbacks · lines 608–639

VII-C@1#eq:Cauchymatrix

Define

H_{Σ}^{ij} = (γ_{i} \otimes γ_{j}) H_{hh}, i, j \in {0, 1} .

These pullbacks exist because the conormals of the spacelike Cauchy surface are timelike, whereas the nonzero Hadamard covectors are null. The same argument permits the bulk–boundary blocks. This is an application of the pullback criterion, not a multiplication of boundary delta distributions BDH.

For the scalar massless factor, spatial Fourier transformation gives

H_{Σ} (p) = \frac{1}{2 r} (1 - i r i r r^{2}), r = ∣ p ∣,

multiplied by $P_{α β ρ σ}$ in the graviton sector. The entries are locally integrable at $r = 0$ in three spatial dimensions. The full matrix, including the off-diagonal phases, is retained.

Displayed formulas in this passage

VII-C@1#eq:Cauchymatrix · line 624
$H_{Σ} (p) = \frac{1}{2 r} (1 - i r i r r^{2}), r = ∣ p ∣,$
unlabeled · line 611
$H_{Σ}^{ij} = (γ_{i} \otimes γ_{j}) H_{hh}, i, j \in {0, 1} .$