Linear Mode Stability of the Kerr-Newman Black Hole and Its Quasinormal Modes

We provide strong evidence that, up to 99.999% of extremality, Kerr-Newman black holes (KNBHs) are linear mode stable within Einstein-Maxwell theory. We derive and solve, numerically, a coupled system of two partial differential equations for two gauge invariant fields that describe the most general linear perturbations of a KNBH. We determine the quasinormal mode (QNM) spectrum of the KNBH as a function of its three parameters and find no unstable modes. In addition, we find that the lowest radial overtone QNMs that are connected continuously to the gravitational $\ell =m=2$ Schwarzschild QNM dominate the spectrum for all values of the parameter space ($m$ is the azimuthal number of the wave function and $\ell$ measures the number of nodes along the polar direction). Furthermore, the (lowest radial overtone) QNMs with $\ell =m$ approach $\mathrm{Re}\omega =m{\mathrm{\Omega }}_H^{\text{ext}}$ and $\mathrm{Im}\omega =0$ at extremality; this is a universal property for any field of arbitrary spin $|s|\le 2$ propagating on a KNBH background ($\omega$ is the wave frequency and ${\mathrm{\Omega }}_H^{\text{ext}}$ the black hole angular velocity at extremality). We compare our results with available perturbative results in the small charge or small rotation regimes and find good agreement.

Figure 1

All lowest radial overtone QNMs of $Q=a$ KNBHs that start at the $\ell =3$ SCHW gravitational QNM (red disc). Note that the family of axisymmetric QNMs ($m=0$) form pairs of $\{\omega ,-{\omega }^{*}\}$.

Figure 2

Real part of the QNM frequencies with $\ell =m=2$ of KNBHs with fixed $Q/r_{+}=0.0,0.2,\cdots ,0.9$ that start at the SCHW gravitational QNM with $\ell =2$ (red disc). We also show the $\ell =m=2$ QNMs of $Q=a$ KNBHs.

Figure 3

Similar to Fig. 2 but now for $\mathrm{Im}(\omega M)$.

Figure 4

Comparison (for $\mathrm{Re}\omega$) between the exact $\ell =m=2$ $Z_2$ QNMs of KN with $Q=a$ (blue disks) with the small $a$ approximations of [27] (red diamonds with their 1% error bar) and with the small $Q$ approximations of [26] (green circles).

Figure 5

Similar comparison to that in Fig. 4 but now for $\mathrm{Im}\omega$.