Manuscript IX-Free@1 · IX-Free.tex · section · 33b20e96

Degree-four relations

F1: concrete presentation-relation kernels › Degree-four relations · lines 592–624

IX-Free@1#eq:pkernel

Let $J^{μ}_{α β} (x, z)$ denote the distribution kernel of $L$ :

w^{μ} (x) = \int J^{μ}_{α β} (x, z) h_{α β} (z) d^{4} z .

Let $e_{00}^{α β} h_{α β} = h_{00}$ . For $v \in E_{L}$ , define

P [v] (x, z_{1}, z_{2}, z_{3}) = Sym_{(z_{i}, α_{i}, β_{i})} \partial_{μ}^{x} [v (x, z_{2}) J^{μ}_{α_{1} β_{1}} (x, z_{1}) \times δ (z_{2} - z_{3}) e_{00}^{α_{2} β_{2}} e_{00}^{α_{3} β_{3}}] .

The symmetrization is the average over the three metric slots. Before smooth multiplication, $J (x, z_{1})$ and $δ (z_{2} - z_{3})$ are tensor factors in disjoint variables. Thus (eq:pkernel) is a defined distribution.

Observation support, finite propagation, and the delta relation give a common compact support for its field variables.

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IX-Free@1#eq:pkernel · line 602
$P [v] (x, z_{1}, z_{2}, z_{3}) = Sym_{(z_{i}, α_{i}, β_{i})} \partial_{μ}^{x} [v (x, z_{2}) J^{μ}_{α_{1} β_{1}} (x, z_{1}) \times δ (z_{2} - z_{3}) e_{00}^{α_{2} β_{2}} e_{00}^{α_{3} β_{3}}] .$
unlabeled · line 596
$w^{μ} (x) = \int J^{μ}_{α β} (x, z) h_{α β} (z) d^{4} z .$

Degree-four relations

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