Local Group Velocity Distribution inside Superradiant Condensates

TL;DR

This work analyzes the local group velocity distribution of a superradiant scalar condensate around a Kerr black hole, demonstrating that the observed rotation arises from collective local motions rather than azimuthal phase propagation alone. By solving the Klein-Gordon equation for the dominant unstable mode $\{0,1,1\}$ near the superradiant threshold and examining the energy-momentum tensor, the authors derive explicit asymptotic velocity components: $v^r\sim(1-r/r_p)/(\mu r)\sin(2\omega t-2\varphi)$, $v^\theta\sim(\cot\theta)/(\mu r)\sin(2\omega t-2\varphi)$, and $v^\varphi\sim 2/(\mu r\sin\theta)\sin^2(\omega t-\varphi)$, with $v^\theta$ and $v^\varphi$ decaying as $1/r$ and $v^r$ approaching a finite nonzero value at large $r$. The energy density exhibits a quadrupolar structure with a maximum near $r_p\approx2r_b$, and the zero-flux and rotating patterns persist without superluminal motion. These insights pave the way for future phenomenology, including photon birefringence and dynamical friction within BH-condensate environments.

Abstract

Superradiance enables scalar fields to extract energy and angular momentum from a rotating black hole (BH), leading to the formation of a BH-condensate system. Previous studies mainly focus on the phase velocity, which propagates in the azimuthal direction. In this work, we show that the superradiant scalar condensate presents a nontrivial group velocity distribution. In the region sufficiently far from the BH, the condensate exhibits a radial velocity magnitude that approaches $ (r_gμ/2) \sin (2ωt-2 \varphi)$, while the polar and azimuthal velocity magnitudes asymptotically decline as $\propto 1/r$.

Figures (8)

• Figure 1: Schematic illustration of the BH-condensate system. The blue region represents the distribution of the real scalar field at a certain time. The black object at the center denotes the Kerr BH, which is enlarged by a factor of 10 for illustrative purposes. The system rotates and emits GWs due to the non-axisymmetry of the condensate. The GWs are depicted as gray wavy lines.
• Figure 2: The relative difference $\delta N^{(\mathrm{r})}_{011}$ defined in Eq. \ref{['eq:delta_N']} as a function of the mass coupling $r_g\mu$. Only the $\{0,1,1\}$ mode is considered. $\phi_{011}$ is numerically calculated as described in Sec. \ref{['sec:BHCondensate']} and satisfies the normalization condition \ref{['eq:normalization']}. The BH spin $a_*$ is set to the value that satisfies the superradiant threshold $\omega_{011} = \Omega_\mathrm{H}$ for each value of $r_g \mu$.
• Figure 3: The distribution of the energy density $\rho$ in the $z = 0$ (panel (a)) and $y = 0$ (panel (b)) planes. Only the $\{0,1,1\}$ mode is included in the condensate. The scalar field is normalized such that $M_\mathrm{s} = \omega_{011}$. Here, we choose $r_g\mu = 0.1$ and $a_* \approx 0.38$, to satisfy the superradiant threshold $\omega_{011} = \Omega_\mathrm{H}$.
• Figure 4: Distributions of the scalar field squared $\Phi^2$ (first row) and the energy density difference $\delta\rho \equiv \rho - \rho_\mathrm{ave}$ (second row) at different times in the $z = 0$ plane. The columns from left to right denote the times from $t=0$ to $3T/4$. In panel (e), surfaces with $\delta \rho = 0$ are plotted as dashed gray lines and the red circle, dividing the entire space into eight regions. Other parameters are the same as in Fig. \ref{['fig:rho']}.
• Figure 5: Same as Fig. \ref{['fig:delta_rho_and_Phi_square_xy']}, but in the $y=0$ plane.