Rotating black holes in a draining bathtub: Superradiant scattering of gravity waves

In a draining rotating fluid flow background, surface perturbations behave as a scalar field on a rotating effective black hole spacetime. We propose a new model for the background flow which takes into account the varying depth of the water. Numerical integration of the associated Klein-Gordon equation using accessible experimental parameters shows that gravity waves in an appropriate frequency range are amplified through the mechanism of superradiance. Our numerical results suggest that the observation of this phenomenon in a common fluid mechanical system is within experimental reach. Unlike the case of wave scattering around Kerr black holes, which depends only on one dimensionless background parameter (the ratio $a/M$ between the specific angular momentum and the mass of the black hole), our system depends on two dimensionless background parameters, namely the normalized angular velocity and surface gravity at the effective black hole horizon.

Figure 1

Lines of constant ${\overline{v}}_{\varphi }(r_{\mathrm{H}})$ (dashed blue lines) and $\overline{\kappa }(r_{\mathrm{H}})$ (solid red lines) are plotted over the cartesian $\overline{A},\overline{B}$ plane for $\overline{A}=[0,8]$ and $\overline{B}=[0,22]$. The grey scale represents the value of $h^{\prime }({\overline{r}}_{\mathrm{H}})$ for the background configuration at each point in parameter space (we have artificially cut off the grey scale at 1 to improve visualization). The intersections of the ${\overline{v}}_{\varphi }(r_{\mathrm{H}})$ and $\overline{\kappa }(r_{\mathrm{H}})$ contour lines (indicated by small black dots) are used in our numerical simulations of superradiant scattering.

Figure 2

The water profiles (solid lines) and the corresponding location of the event horizon (asterisks), as computed in Eqs. (41) and (45), for several values of the (a) normalized angular velocity at the horizon (with $\overline{\kappa }(r_{\mathrm{H}})=1.2$ fixed) and the (b) normalized surface gravity at the horizon (with ${\overline{v}}_{\varphi }(r_{\mathrm{H}})=0.4$ fixed).

Figure 3

The spectra of reflection coefficients in the superradiant regime for several values of the (a) normalized angular velocity at the horizon (with $\overline{\kappa }(r_{\mathrm{H}})=1.2$ fixed) and the (b) normalized surface gravity at the horizon (with ${\overline{v}}_{\varphi }(r_{\mathrm{H}})=0.4$ fixed).

Figure 4

(a) The maximum possible amplification ${\mathcal{R}}_{\max }$ as a function of ${\overline{v}}_{\varphi }(r_{\mathrm{H}})$ and $\overline{\kappa }(r_{\mathrm{H}})$. (b) The corresponding scaled frequency ${\overline{\omega }}_{\text{peak}}$ at which the maximum is attained.