Leading weak-field magnetic corrections to charged scalar quasinormal modes of Kerr black holes in the Melvin--Kerr geometry

We compute the leading magnetic corrections to the charged-scalar quasinormal-mode (QNM) spectrum of a Kerr black hole immersed in a weak external magnetic field, working in the Melvin–Kerr geometry and in the gauge in which the time component of the electromagnetic potential vanishes at large radius. Within the controlled $\mathcal{O}(bq)$ truncation, the charged Klein–Gordon equation separates and the radial problem takes the massive-scalar Kerr form under the effective-mass substitution ${\text{mueff}}^2\equiv {\mu }^2+2qbm$, applied to the asymptotic mass exponent and to the spheroidicity parameter. This gives a parameter-deformed Dolan continued-fraction scheme, with no further finite-radius correction at the order retained. Since the Melvin–Kerr spacetime is not asymptotically flat, the resulting spectrum is not the exact global QNM spectrum of the full magnetized spacetime: the modes are weak-field deviations of Kerr ringdown modes, defined by outgoing boundary conditions in the intermediate Kerr-like region $r_{+}\ll r\ll b^{-1}$. The unmagnetized backbone reproduces Dolan’s tabulated spectra at the ${10}^{-6}$ level for $a\le 0.5M$. For $\mathrm{\ell }=1$, $\mu M\in \{0,0.3\}$, $a\mathrm{/}M\in \{0.3,0.5\}$, $qM=0.1$, and $bM\le {10}^{-2}$, the magnetic shift in $\mathrm{\Re }(\text{Mw})$ is opposite in sign between the two rotating sectors of equal $\mathrm{\mid }m\mathrm{\mid }$: upward for $m=+1$, downward for $m=-1$, and linear in $qb$. The sign and sector-dependent magnitude of each shift are quantitatively reproduced by the unmagnetized slope $\mathrm{\partial }\mathrm{\Re }(\text{Mw})\mathrm{/}\mathrm{\partial }(\mu M{)}^2$ evaluated per sector, confirming that the magnetic effect is fully transmitted through the master substitution. Effective-potential diagnostics and an extension to $\mathrm{\ell }=2$ confirm the picture.