Exact Regions of Superradiant Instability of Kerr-Newman Black Holes and Massive Scalar Fields

TL;DR

This work delivers an exact, parameter-space–level characterization of superradiant instabilities for Kerr–Newman black holes in the presence of a massive charged scalar field by solving the exact VBK resonance condition for quasibound states. It advances beyond the hydrogenic approximation by employing confluent Heun functions to obtain a quartic resonance equation for the frequency $\omega$, enabling analytic expressions for instability boundaries and growth rates, including the exact boundary $\mu_1$ and its linear/quadratic approximations. The study reveals how the instability domain shifts from Kerr-like to highly charged near-extremal Kerr–Newman black holes as the scalar mass $\mu$ increases, while confirming the absence of instability in the Reissner–Nordström limit. By contrasting with previous numerical works that relied on hydrogenic potentials, the paper underscores the accuracy and applicability of the VBK method for probing black-hole superradiance and suggests several avenues for extension, including dyonic spacetimes and AdS boundaries.

Abstract

We investigate the superradiant instability of Kerr-Newman black holes in the presence of a massive, charged scalar field using the Vieira-Bezerra-Kokkotas (VBK) method. We study the solutions of the exact polynomial condition for quasibound state frequencies and determine the domain of superradiant instability in parameter space without relying on the hydrogenic approximation or numerics. We derive the minimum scalar mass needed for quasibound states to exist, and identify the precise overlap region between the quasibound and superradiant conditions where instability can occur. We obtain perturbative and exact analytic expressions for the instability boundaries and growth rates, and clarify their relation to previous numerical results. Our analysis reveals how the instability region shifts from nearly neutral Kerr black holes for light fields to highly charged near-extremal Kerr-Newman black holes for heavier fields, while remaining absent in the Reissner-Nordstrom limit.

Figures (5)

• Figure 1: Plots of $\text{Re} \bigl[ \sqrt{\mu^2 - \omega^2} \bigr]$ versus $\mu$ for (a) Kerr with various different values of $a$, (b) Reissner-Nordström with various different values of $Q$, and for (c)-(f) Kerr-Newman with different values of $a$ and $Q$. The linear approximation \ref{['eq:Reslope']} at the nontrivial root is plotted as red dashed lines. For Kerr, the plots vanish exactly at $\mu = 0$ and are flat, as predicted by \ref{['eq:Reslope']}.
• Figure 2: Plots of the superradiant region for (a) Kerr, (b) Reissner-Nordström, (c) Kerr-Newman as a function of $a$ for various values of $Q$, and (d) Kerr-Newman as a function of $Q$ for various values of $a$.
• Figure 3: Regions of superradiant instability with $M = 1,\, q = 1,\, m =1, \, N=0$.
• Figure 4: Plots of $\text{Im}\, \omega$ vs. $\mu$ with $a=0.98 M$, $Q=0.01M$, $M = 1$, $m=1$, $N = 0$ and $q$ as the tuning parameter. The left figure shows the points where $\text{Im} \, \omega$ turns negative for different values of $qQ$. The right figure shows portions of the plots zoomed into the region near $\mu = \mu_0 = qQ/M$ for several positive values of $qQ$.
• Figure 5: Plot showing the regions in the $\mu M$ and $qQ$ plane (parameter values $a = 0.98 M$, $Q = 0.01 M$, $m=1$, $N=1$) where there are quasibound states and superradiance (shaded gray). The dotdashed line corresponds to $\mu M = q Q$ and is the boundary of the region with no quasibound states. The boundary of the region with no superradiance is the dotted line according to the numerical analysis of Furuhashi and Nambu furuhashi2004instability, the dashed line according to our first-order analytic result \ref{['eq:mu11']}, the solid black line according to our second-order analytic result \ref{['eq:mu12']}, and the solid red line according to our exact analytic result \ref{['eq:mu1']}. The circle points are exact results from solving the resonance equation of Vieira, Bezerra, and Muniz vieira2022instability. The plot on the right zooms in on the square region outlined in the plot on the left in order to show the difference between the 2nd-order result and the exact result.