Slowly decaying resonances of charged massive scalar fields in the Reissner-Nordström black-hole spacetime

Abstract: We determine the characteristic timescales associated with the linearized relaxation dynamics of the composed Reissner-Nordstr\"om-black-hole-charged-massive-scalar-field system. To that end, the quasinormal resonant frequencies ${\omega_n(\mu,q,M,Q)}{n=0}^{n=\infty}$ which characterize the dynamics of a charged scalar field of mass $\mu$ and charge coupling constant $q$ in the charged Reissner-Nordstr\"om black-hole spacetime of mass $M$ and electric charge $Q$ are determined {\it analytically} in the eikonal regime $1\ll M\mu<qQ$. Interestingly, we find that, for a given value of the dimensionless black-hole electric charge $Q/M$, the imaginary part of the resonant oscillation frequency is a monotonically {\it decreasing} function of the dimensionless ratio $\mu/q$. In particular, it is shown that the quasinormal resonance spectrum is characterized by the asymptotic behavior $\Im\omega\to0$ in the limiting case $M\mu\to qQ$. This intriguing finding implies that the composed Reissner-Nordstr\"om-black-hole-charged-massive-scalar-field system is characterized by extremely long relaxation times $\tau{\text{relax}}\equiv 1/\Im\omega\to\infty$ in the $M\mu/qQ\to 1^-$ limit.