Gravitoelectromagnetic Perturbations of Kerr-Newman Black Holes: Stability and Isospectrality in the Slow-Rotation Limit

The most general stationary black-hole solution of Einstein-Maxwell theory in vacuum is the Kerr-Newman metric, specified by three parameters: mass $M$, spin $J$, and charge $Q$. Within classical general relativity, one of the most important and challenging open problems in black-hole perturbation theory is the study of gravitational and electromagnetic fields in the Kerr-Newman geometry, because of the indissoluble coupling of the perturbation functions. Here we circumvent this long-standing problem by working in the slow-rotation limit. We compute the quasinormal modes up to linear order in $J$ for any value of $Q$ and provide the first, fully consistent stability analysis of the Kerr-Newman metric. For scalar perturbations the quasinormal modes can be computed exactly, and we demonstrate that the method is accurate within 3% for spins $J/J_{\max }\lesssim 0.5$, where $J_{\max }$ is the maximum allowed spin for any value of $Q$. Quite remarkably, we find numerical evidence that the axial and polar sectors of the gravitoelectromagnetic perturbations are isospectral to linear order in the spin. The extension of our results to nonasymptotically flat space-times could be useful in the context of gauge-gravity dualities and string theory.

Figure 1

Top panel: percentage error $\Delta \equiv {10}^2|1-{\omega }_{\mathrm{slow}}/{\omega }_{\mathrm{exact}}|$ induced by the slow-rotation approximation in the real part of the fundamental $\ell =m=1$ scalar mode. Bottom panel: percentage error for the imaginary part of the same mode. The errors are only mildly sensitive to $Q$ if plotted as functions of $J/J_{\max }$, where $J_{\max }$ is defined below Eq. (6). Similar results hold for other scalar modes [30].

Figure 2

Zeroth-order (left panels) and first-order (right panels) terms of the slow-rotation expansion of the KN QNM frequencies [cf. Eq. (7)]. All quantities are plotted as a function of $Q/M$, and they refer to the fundamental mode ($n=0$) with $\ell =2$. The lower part of each panel shows the percentage difference between axial and polar quantities: our results are consistent with isospectrality to $\mathcal{O}(0.1\%)$ for the real part and to $\mathcal{O}(1\%)$ for the imaginary part of these modes.