Relating black hole shadow to quasinormal modes for rotating black holes

In this work, we explore the connection between the critical curves (“shadows”) and the quasinormal mode frequencies (in the eikonal limit) of Kerr black holes. This mapping has been previously established for nonrotating black holes. We show that the shadow seen by an distant observer at a given inclination angle can be mapped to a family of quasinormal modes with $m/(\ell +1/2)$ bounded within a certain range, where $m$ is azimuthal node number and $\ell$ is the angular node number. We discuss the possibility of testing such relation with spaceborne gravitational wave detectors and the next-generation Event Horizon Telescope.

Figure 1

The impact parameter $R(\mu )$ as a function of $\mu$ for various black hole spins: $a/M=0.9$ (solid red), $a/M=0.5$ (solid blue), $a/M=0.3$ (dashed purple), $a/M=0.1$ (dotted black).

Figure 2

The product $\tilde{L}{\mathrm{\Omega }}_{\mathrm{R}}$ as a function of $\mu$ for various black hole spins: $a/M=0.9$ (solid red), $a/M=0.5$ (solid blue), $a/M=0.3$ (dashed purple), $a/M=0.1$ (dotted black).

Figure 3

The critical curve of a Kerr black hole with $a/M=0.9$ and $\iota =\pi /4$. The red dots on the y axis represent the points with $\mu =0$. The left/right dots on the x axis correspond to the maximum/minimum $\mu =0.683/-0.725$. For other points in the critical curve, the corresponding $\mu$ can be found using its radius and Fig. 1.