Scalar, electromagnetic, and gravitational perturbations of Kerr-Newman black holes in the slow-rotation limit

In Einstein-Maxwell theory, according to classic uniqueness theorems, the most general stationary black-hole solution is the axisymmetric Kerr-Newman metric, which is defined by three parameters: mass, spin and electric charge. The radial and angular dependence of gravitational and electromagnetic perturbations in the Kerr-Newman geometry do not seem to be separable. In this paper we circumvent this problem by studying scalar, electromagnetic and gravitational perturbations of Kerr-Newman black holes in the slow-rotation limit. We extend (and provide details of) the analysis presented in a recent Letter [P. Pani, E. Berti, and L. Gualtieri, Phys. Rev. Lett. 110, 241103 (2013)]. Working at linear order in the spin, we present the first detailed derivation of the axial and polar perturbation equations in the gravito-electromagnetic case, and we compute the corresponding quasinormal modes for any value of the electric charge. Our study is the first self-consistent stability analysis of the Kerr-Newman metric, and in principle it can be extended to any order in the small rotation parameter. We find numerical evidence that the axial and polar sectors are isospectral at first order in the spin and speculate on the possible implications of this result.

Figure 1

Left (reproduced from Ref. 49): Zeroth-order (small left panels) and first-order (small right panels) terms of the slow-rotation expansion of the KN QNM frequencies [cf. Eqs. (4, 5)]. 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. Right: The same, but for fundamental modes with $\ell =3$.

Figure 2

Percentage deviation of QNM frequencies in the slow-rotation approximation with respect to the DF equation, for the $\ell =m=2$ fundamental gravitational mode. As expected, the two approximations agree with each other when $Q\to 0$, but they deviate from each other when $\tilde{a}\to 0$. The discrepancy between the two approximations is nearly constant as long as $J\ll J_{\max }$, confirming that the DF equation is not very accurate in that regime when $Q\ne 0$ (cf. 19).