Manuscript XIV-H@1 · XIV-H.tex · section · 695c7c3a

Its internal variables and exact map

H2b: the coordinate gravitational-response leg › Its internal variables and exact map · lines 850–886

XIV-H@1#eq:K8map

The coordinate leg has two internal variables: $v$ is the actual source vertex and $s$ the intermediate coordinate-response vertex. Its scalar attachment is $b$ . Its untested kernel is

K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) \int \int R (b - s) \partial_{s^{α}} (σ_{ε} (s) A^{α μ}_{L} P^{L}_{M} R (s - v) Q^{M, I, J} (v; a, z)) d^{4} s d^{4} v] .

This is the kernel of

- \int f X^{μ} [r_{h}] \partial_{μ} ψ,

after the scalar-slot derivative is transferred to the kernel.

The original maps are therefore:

(a, z) ⟶ v, v ⟶ s, s ⟶ b .

The first is the regular local incidence of Proposition prop:sourceincidence. The other two are the existing retarded maps.

Displayed formulas in this passage

XIV-H@1#eq:K8map · line 856
$K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) \int \int R (b - s) \partial_{s^{α}} (σ_{ε} (s) A^{α μ}_{L} P^{L}_{M} R (s - v) Q^{M, I, J} (v; a, z)) d^{4} s d^{4} v] .$
unlabeled · line 871
$- \int f X^{μ} [r_{h}] \partial_{μ} ψ,$
unlabeled · line 878
$(a, z) ⟶ v, v ⟶ s, s ⟶ b .$