Manuscript XIV-H@1 · XIV-H.tex · proposition · 695c7c3a
Proposition: Exact reduced coordinate-leg kernel
H2b: the coordinate gravitational-response leg › The existing retarded convolution · lines 974–1016 · proof included or linked
XIV-H@1#prop:K8exact, XIV-H@1#eq:K8reduced, XIV-H@1#eq:K8seed
On the retained patch, (eq:K8map) equals
Since and , it is a finite sum of derivatives of total order at most three of smooth tensor coefficients times
\delta((b-a)^2)\delta^{(4)}(a-z).
\label{eq:K8seed}
\end{proposition}
\begin{proof}
After Lemma~\ref{lem:weight}, constant fiber maps
commute with the scalar propagation.
Causal convolution and differentiation give
\[
\int R(b-s)
\partial_{s^\alpha}R(s-v)\,\,\mathrm d^4s
=\partial_{b^\alpha}B(b-v),which proves (eq:K8reduced).
On the separated chronological patch,
The source incidence contributes at most two external derivatives, and the scalar attachment contributes one more. Derivatives of the source coefficients are smooth coefficients of lower terms. This gives the finite representation (eq:K8seed).
Displayed formulas in this passage
XIV-H@1#eq:K8reduced · line 977
XIV-H@1#eq:K8seed · line 990
\delta((b-a)^2)\delta^{(4)}(a-z). \label{eq:K8seed} \end{proposition} \begin{proof} After Lemma~\ref{lem:weight}, constant fiber maps commute with the scalar propagation. Causal convolution and differentiation give \[ \int R(b-s) \partial_{s^\alpha}R(s-v)\,\,\mathrm d^4s =\partial_{b^\alpha}B(b-v),unlabeled · line 1007