Black Hole Quasibound States from a Draining Bathtub Vortex Flow

Quasinormal modes are a set of damped resonances that describe how an excited open system is driven back to equilibrium. In gravitational physics these modes characterize the ringdown of a perturbed black hole, e.g., following a binary black hole merger. A careful analysis of the ringdown spectrum reveals the properties of the black hole, such as its angular momentum and mass. In more complex gravitational systems, the spectrum might depend on more parameters and hence allows us to search for new physics. We present a hydrodynamic analog of a rotating black hole that illustrates how the presence of extra structure affects the quasinormal mode spectrum. The analogy is obtained by considering wave scattering on a draining bathtub vortex flow. We show that due to vorticity of the background flow, the resulting field theory corresponds to a scalar field on an effective curved spacetime which acquires a local mass in the vortex core. The obtained quasinormal mode spectrum exhibits long-lived trapped modes, commonly known as quasibound states. Our findings can be tested in future experiments building upon recent successful implementations of analog rotating black holes.

Figure 1

We plot the functions ${\omega }_{+}(r)$ (black curve) and ${\omega }_{-}(r)$ (blue curve) defined in Eq. (10) for the dimensionless parameters $C/c{r}_0=3$, $D/c{r}_0=0.1$ and for the $m=-4$ mode. The region between the dashed lines is shown in the inset to highlight the presence of the smaller peak. The QNM frequencies are approximately those satisfying ${\omega }_{\pm }=\omega$, ${{\omega }_{\pm }}^{\prime }=0$. A QBS frequency is shown for illustrative purposes. The turning points of the QBS are the $r_j$ values with $j=1,\dots ,4$ moving from left to right in the figure.

Figure 2

Results obtained for the fluid flow parameters $C/c{r}_0=3$, $D/c{r}_0=0.1$. (a) The ringdown obtained from the MOL simulation of a initial Gaussian wave packet after it has passed the vortex. (b1)–(b6) The decomposition of the ringdown signal onto an azimuthal basis for $m\in \{-6,-5,-4,-3,-1,+2\}$. We display only the values of $m$ for which the signal is above the noise level. To show the relative amplitudes of each component, we refer to a common color bar rescaled by a factor $q$ [indicated above each of the panels (b1)–(b6)], normalized to 1 for the overall signal [panel (a)]. (c1),(c2) The real and imaginary parts of the complex frequency spectrum obtained by the three methods: MOL (red crosses), CFM (blue diamonds), and WKB (green dots for QBSs and black dots for the usual QNMs). The MOL and the CFM are in excellent agreement. Moreover, by comparing the MOL and the CFM with the WKB results, we see that only $m=-1$ is a usual QNM, while $m\in \{-6,-4,-3,+2\}$ indicates QBSs. Note that the distinction between the usual QNMs and QBSs close to the peaks in the scattering potential is not sharply defined, as in the case of the $m=-5$ mode here.