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

Abstract

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 $\mueff^{2}\equiv\mu^{2}+2qbm$, 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 $\ell=1$, $\mu M\in\{0,0.3\}$, $a/M\in\{0.3,0.5\}$, $qM=0.1$, and $bM\le 10^{-2}$, the magnetic shift in $\Re(\Mw)$ is opposite in sign between the two rotating sectors of equal $|m|$: 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 $\partial\Re(\Mw)/\partial(\mu M)^{2}$ evaluated per sector, confirming that the magnetic effect is fully transmitted through the master substitution. Effective-potential diagnostics and an extension to $\ell=2$ confirm the picture.