Instability of Massive Scalar Fields in Kerr-Newman Spacetime

TL;DR

The study addresses the instability of a charged massive scalar field in Kerr-Newman spacetime due to superradiance, formulating the problem as a Klein-Gordon eigenvalue problem with appropriate boundary conditions. It combines analytic leading-order results from Detweiler's asymptotic matching in the small-parameter regime with a numerical solution of the radial equation for $\mu M \lesssim 1$ to compute the growth rate $\gamma$. The main finding is a maximal growth rate of $\gamma M \simeq 1.13\times 10^{-7}$ occurring near $\mu M \simeq 0.35$, $qQ \simeq -0.08$, but the numerically obtained maximum is about three times larger than the analytic estimate; instability exists where both $P^{(0)}>0$ (superradiance) and $M\mu \gtrsim qQ$ (bound state). The results illuminate the black-hole bomb mechanism in Kerr-Newman geometry and suggest possible implications for primordial black hole evolution, particularly when $\mu$ is large enough to maximize the effect.

Abstract

We investigate the instability of charged massive scalar fields in Kerr-Newman spacetime. Due to the super-radiant effect of the background geometry, the bound state of the scalar field is unstable, and its amplitude grows in time. By solving the Klein-Gordon equation of the scalar field as an eigenvalue problem, we numerically obtain the growth rate of the amplitude of the scalar field. Although the dependence of the scalar field mass and the scalar field charge on this growth rate agrees with the result of the analytic approximation, the maximum value of the growth rate is three times larger than that of the analytic approximation. We also discuss the effect of the electric charge on the instability of the scalar field.

Figures (7)

• Figure 1: The dependence of the mass $\mu$ and the charge $q$ of the scalar field on the growth rate $\gamma$. The parameter values here are $a=0.98M,~Q=0.01M,~l=m=1$, and $n=0$. The solid line corresponds to $P^{(0)}=0$, and the dotted line corresponds to $M\mu-qQ=0$.
• Figure 2: The effective potential $V_\text{eff}(x)$ of the scalar field for $a=0.98M ,~Q=0.01M,~\mu M=0.35,~qQ=-0.08,~l=m=1$.
• Figure 3: The sets of parameter values used in the numerical calculations plotted corresponds to in $(\mu,q)$-space. The solid line corresponds to $P^{(0)}=0$ and the dotted line to $M\mu-qQ=0$. The scalar field is expected to be unstable for sets of parameter values in the grey region.
• Figure 4: The value of $|\mu^2/\omega^2-1|$ as a function of $\mu$ for $q=0$. This value is less than $5.5\times 10^{-2}$ for all values of $\mu M$, and we find that our numerical results are consistent with the assumption $\omega \sim\mu$.
• Figure 5: The numerically obtained value of the growth rate $\gamma$. The spheres represent the obtained values of $\gamma$. The left panel is a birds-eye view of $\gamma(\mu,q)$ for $a=0.98M,~Q=0.01M,~l=m=1$. The right panel is the same function as viewed from above.