Quasinormal modes and stability of the rotating acoustic black hole: Numerical analysis

The study of the quasinormal modes (QNMs) of the $2+1$ dimensional rotating draining bathtub acoustic black hole, the closest analogue found so far to the Kerr black hole, is performed. Both the real and imaginary parts of the quasinormal (QN) frequencies as a function of the rotation parameter $B$ are found through a full nonlinear numerical analysis. Since there is no change in sign in the imaginary part of the frequency as $B$ is increased we conclude that the $2+1$ dimensional rotating draining bathtub acoustic black hole is stable against small perturbations.

Figure 1

The real part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=1$ mode. Here, $r_H=A/c$ is the horizon radius. Note how all the several lowest overtones “coalesce” in the high rotation regime, all growing linearly with $B/A$.

Figure 2

The real part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=2$ mode. Again, all different overtones have the same behavior for high rotation.

Figure 3

The imaginary part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=1$ mode. It is clear from this plot that the imaginary part of the QN frequencies of $m>0$ modes is very insensitive to the rotation of the black hole.

Figure 4

The imaginary part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=2$ mode.

Figure 5

The real part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=-1$ mode. Notice that for each overtone number $n$ there is a critical rotation at which the mode crosses the axis, i.e., there is a critical rotation $B/A$ at which the real part of the QN frequency is zero. Higher overtones cross the axis at a slower rotation. We have not been able to follow the mode beyond this point.

Figure 6

The real part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=-2$ mode.

Figure 7

The imaginary part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=-1$ mode. We have not been able to follow the modes beyond a certain critical point (defined as the rotation $B/A$ for which the real part of the QN frequency is zero- see Figs. 5, 6). Nevertheless, an judging by the modes we did manage to follow, namely, the fundamental mode, it seems that $\mathrm{I}\mathrm{m}[{\omega }_{QN}]$ never crosses the axis, i.e., it is always negative, and therefore the mode is stable.

Figure 8

The imaginary part of the QN frequency as a function of the rotation parameter $B/A$, for several overtones of a $m=-2$ mode.