Lemma: Inverse on conserved currents

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

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

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

Substitution in the Fourier transform of (eq:K0) cancels the $p_{\mu }{p}_{\nu }{J}_{\mathrm{tr}}$ terms and the remaining trace terms, leaving $J_{\mu \nu }$. Direct substitution of (eq:gauge-tensor) proves the second identity.