Lemma: The inverse needed by the tree

For $p^2\ne 0$ and $p^{\mu }{J}_{\mu \nu }=0$,

Also $K_{0}G(k,\xi )=0$.

Proof. Write $J_{\mathrm{tr}}={\eta }^{\mu \nu }{J}_{\mu \nu }$. For $h=DJ$,

Substitution into the Fourier transform of (eq:K0) cancels the trace and longitudinal terms. The remainder is $J$. Direct substitution of (eq:G) proves the second identity.