Instability for massive scalar fields in Kerr-Newman spacetime

It is known that a massive charged scalar field can trigger a superradiant instability in the background of a Kerr-Newman black hole. In this paper, we present a numerical study of such an instability by using the continued-fraction method. It is shown that for given a black hole, the unstable scalar mode with a specific azimuthal index $m$ only occurs in a finite region in the parameter space of the scalar field. The maximum mass of the scalar cloud is exactly the upper bound of the mass of the unstable modes. We show that due to the electromagnetic interaction between the scalar field and the Kerr-Newman black hole, the growth rate of the instability can be 15.7% larger than that of a scalar field in Kerr spacetime of the same rotation parameter. In addition, we find a maximum value of the growth rate ${\tau }^{-1}=1.788\times {10}^{-7}{M}^{-1}$, which is about 4% larger than that in the Kerr case.

Figure 1

Bound-state spectrum of a neutral scalar field ($l=m=1$) for different values of $a$ and $Q$. The rotation parameters of the black hole are $a=0.9$, 0.99, $0.997M$, from left to right. The plot clearly shows that the instability ${\omega }_I>0$ occurs when the superradiant condition $0<{\omega }_R<{\omega }_c$ is satisfied.

Figure 2

The dependence of the mass $\mu$ and charge $q$ on the imaginary part ${\omega }_I$ of the quasibound-state frequency, with $Q=0.2M$, $a=0.9M$, and $l=m=1$. The plot confirms that superradiant instability only happens in the region determined by inequalities (18) and (19). The plus symbol represents the location of the most unstable mode, i.e., $(\mu M,qM)\approx (0.282,-0.264)$, with ${\omega }_{I}M\approx 2.243\times {10}^{-8}$.

Figure 3

Frequency spectrum of the bound state in the unstable region. The parameters in this plot are the same as those in Fig. 2. Left panel: The dependence of the dimensionless mass $\mu M$ and charge $qM$ on $({\omega }_c-{\omega }_R)M$. Right panel: The imaginary part ${\omega }_I$ as a function of $\mu M$ and $qM$. This plot shows that inside the unstable region the superradiance condition ${\omega }_R<{\omega }_c$ is satisfied, and ${\omega }_I>0$.

Figure 4

The most unstable modes for $l=m=1$. Left panel: The imaginary part of ${\omega }_I$ of the most unstable modes as a function of $Q$. Right panel: The dimensionless coupling $qQ$ (top) and $\mu M$ (bottom) of the most unstable modes as a function of $Q$.

Figure 5

Comparison of numerical and analytical results of the imaginary part of the frequency as a function of $qM$ for given $M\mu$ with $a=0.9$, $Q=0.2$, $l=m=1$, and $n=0$. In this plot, the dimensionless mass is fixed as $M\mu =0.282$, which is expected to give the maximum growth rate, as shown in Fig. 2. The vertical dashed line gives the charge of the stationary cloud, while the vertical solid line corresponds to $qQ=M\mu$.