Quasinormal modes of Kerr–de Sitter black holes

We calculate the fundamental quasinormal modes of the Kerr–de Sitter black hole for the first time. In order to calculate the quasinormal modes, we employ the master equations derived by Suzuki, Takasugi, and Umetsu, who transform the Teukolsky equations for the Kerr–de Sitter black hole into the standard form of the Heun’s equation. The transformed functions are expanded around the outer horizon of the black hole or the symmetric axis in the Fröbenius series whose coefficients satisfy a three-term recurrence relation. These three-term recurrence relations allow us to use Leaver’s continued fraction method to calculate the angular separation constant and the quasinormal mode frequency. Any unstable fundamental quasinormal mode is not found in this paper. It is also observed that for some black holes characterized by a large mass parameter, some retrograde modes in the slow rotation limit become prograde as the black hole spin increases. This phenomenon does not occur for the fundamental modes of the Kerr black hole.

Figure 1

Parameter space for the black hole solution in the $a-M$ plane. $M$ and $a$ are shown in units of $\Lambda /3=1$. Black hole solutions are allowed for the parameters $a$ and $M$ in the region between the solid curve and the dashed curve.

Figure 2

Real and imaginary parts of the quasinormal mode frequencies, $\omega$, given as a function of the spin parameter $a$ for the gravitational perturbations having $l=2$. The frequencies of the positive $m$ modes, the axisymmetric mode $(m=0)$, and the negative $m$ modes are shown by the solid, long-dashed, and dashed curves, respectively. The solid (dashed) curves correspond to $m=2$, 1 ($m=-2$, $-1$) modes from top (bottom) to bottom (top). The mass parameter of the black hole $M$ is taken to be $M=0.01290$. The maximum spin parameter of the black hole is given by $a_{\max }=0.01290$.

Figure 3

Same as Fig. 2, but for the black hole with $M=0.1205$ and $a_{\max }=0.1224$.

Figure 4

Same as Fig. 2, but for the black hole with $M=0.1582$ and $a_{\max }=0.1628$.

Figure 5

Same as Fig. 2, but for the black hole with $M=0.1789$ and $a_{\max }=0.1859$.

Figure 6

Same as Fig. 2, but for the black hole with $M=0.1893$ and $a_{\max }=0.1979$.

Figure 7

Same as Fig. 2, but for the black hole with $M=0.1924$ and $a_{\max }=0.2015$.

Figure 8

Same as Fig. 2, but for the black hole with $M=0.19824$ and $a_{\max }=0.2083$.

Figure 9

Values of the surface gravity $r_i\kappa (r_i)$ on the outer and cosmological horizons $r_i=r_{+}$ or $r_{+}^{\prime }$, given as functions of the spin parameter $a$ for four models of the black holes characterized by $M=0.01290$ ($r_{+}\kappa (r_{+})[a=0]=0.499$), $M=0.1582$ ($r_{+}\kappa (r_{+})[a=0]=0.3$), $M=0.1924$ ($r_{+}\kappa (r_{+})[a=0]={10}^{-3}$), and $M=0.1982$. The surface gravities on the outer and the cosmological horizons are indicated by the solid and dashed curves, respectively.

Figure 10

Real and imaginary parts of the quasinormal mode frequencies $\omega$, given as a function of the spin parameter $a$ for the electromagnetic perturbations having $l=1$. The frequencies of the positive $m$ mode, the axisymmetric mode ($m=0$), and the negative $m$ mode are shown by the solid, long-dashed, and dashed curves, respectively. The mass parameter $M$ of the black hole $M$ is taken to be $M=0.01290$. The maximum spin parameter of the black hole is given by $a_{\max }=0.01290$.

Figure 11

Same as Fig. 10, but for the black hole with $M=0.1205$ and $a_{\max }=0.1224$.

Figure 12

Same as Fig. 10, but for the black hole with $M=0.1582$ and $a_{\max }=0.1628$.

Figure 13

Same as Fig. 10, but for the black hole with $M=0.1789$ and $a_{\max }=0.1859$.

Figure 14

Same as Fig. 10, but for the black hole with $M=0.1893$ and $a_{\max }=0.1979$.

Figure 15

Same as Fig. 10, but for the black hole with $M=0.1924$ and $a_{\max }=0.2015$.

Figure 16

Same as Fig. 10, but for the black hole with $M=0.19824$ and $a_{\max }=0.2083$.