Atlas III goal: local equivalence of the Kerr–Newman perturbation systems (1) and (7), both spins

Fix subextremal Kerr–Newman parameters $M>0$, $Q\neq 0$, $a^2+Q^2<M^2$, a frequency $\omega\in\mathbb R$ and an azimuthal number $m$ (kept real), and let $U=\{r>r_+,\ 0<\theta<\pi\}$. The statement is the conjunction of the two spin cases.

Positive spin. For every quadruple of $C^\infty$ fields $(\psi_2,\psi_1,k,s)$ on $U$ and every $x\in U$: the four first-order equations (1a)–(1d) of Hintz hold at $x$ if and only if the transformed system (7),

holds at $x$ for $z_+=\operatorname{adj}(X)\,y_+^{(+)}$, $z_-=X^*\,y_-^{(+)}$, where $y_\pm^{(+)}=\Delta^{5/4}\sqrt{\sin\theta}\,(\psi_2\pm i\Delta^{-1/2}\psi_1,\,-2Q(k\pm i\Delta^{-1/2}s))$ are the chiral variables of eq. (3), $X$ is the matrix integrating factor of eq. (5), and $C=\det X\cdot X^{-1}(c^{(+)}I-B^{(+)})(X^*)^{-1}$ with the coefficients of eq. (4).

Negative spin. The same equivalence for $C^\infty$ fields $(\psi_{-2},\psi_{-1},n,\ell)$, the negative-spin system obtained from (1a)–(1d) by the paper's replacement rules, the chiral variables $y_\pm^{(-)}=\Delta^{-3/4}\sqrt{\sin\theta}\,(\psi_{-2}\mp i\Delta^{1/2}\psi_{-1},\,2Q(n\mp i\Delta^{1/2}\ell))$ and the coefficients with $\varsigma=-1$.

The transformation $(\psi)\mapsto(y_+,y_-)\mapsto(z_+,z_-)$ is explicit and invertible on $U$ (inverse maps invPos, invNeg, yplus_of_zplus, yminus_of_zminus), so this is a two-sided local correspondence of solutions, proved for arbitrary $C^\infty$ fields rather than for known solutions. It is the algebraic core of Hintz's separation of variables; nothing is asserted about the angular eigenvalue problem, the radial reduction, boundary behaviour or mode stability.