Bosonic Clouds Around Rotating Black Holes

A rotating horizon can amplify a bosonic wave only when the wave carries angular momentum with the appropriate sign. A long-lived cloud additionally needs confinement: a mass term or another effective barrier must keep the amplified field from escaping to infinity. A bound eigenmode in the superradiant range can grow; a stationary threshold cloud must also satisfy the horizon and infinity boundary conditions. Two inequalities alone do not construct that eigenmode.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr horizons; Black-Hole Perturbation Theory supplies separated modes and boundary conditions; Formulations of General Relativity supplies field evolution; Relativistic Matter and Compact Objects supplies bosonic matter models; Computational Relativity supplies nonlinear validation.

Horizon flux and the superradiant inequality

First take a real positive frequency $\omega$ and an integer azimuthal number $m$ in $e^{-i\omega t+im\phi}$. In four-dimensional asymptotically flat Kerr, the future horizon is generated by

$$\chi=\partial_t+\Omega_H\partial_\phi, \qquad \Omega_H=\frac{a}{r_+^2+a^2}$$

where $a=J/M$ and $r_+=GM+\sqrt{(GM)^2-a^2}$, with physical mass $M$, $c=1$, and $|a|\lt GM$. Acting on a scalar mode gives

$$\chi\Phi=-i(\omega-m\Omega_H)\Phi.$$

For a positive-frequency bosonic wave, the horizon energy flux associated with $\partial_t$ has the schematic sign

$$\dot E_H\propto \omega(\omega-m\Omega_H)|\mathcal A_H|^2.$$

Thus

$$0\lt \omega\lt m\Omega_H$$

gives negative Killing-energy flux into the horizon and positive amplification outside. This sign and its scalar, vector, and tensor realizations are derived in Brito, Cardoso, and Pani 2020, §4.4, pp. 50–51, and §4.7.2, pp. 58–59, PDF. The normalization of $\mathcal A_H$ varies, but the flux sign does not.

The field extracts angular momentum and energy in the ratio

$$\frac{dE}{dJ}=\frac{\omega}{m}$$

for a monochromatic mode. Axisymmetric $m=0$ modes cannot satisfy the positive-frequency superradiant inequality.

Confinement by a boson mass

For a scalar of mass $\mu$, a quasibound mode has complex frequency $\omega=\omega_R+i\omega_I$, is ingoing at the future horizon, and decays at infinity. The asymptotic exponential is $e^{-kr}$ with $k=\sqrt{\mu^2-\omega^2}$ on the branch $\operatorname{Re}k\gt0$. In the weak-growth or weak-damping regime $|\omega_I|\ll\omega_R$, the usual confinement condition is

$$\omega_R^2\lt \mu^2,$$

with positive $\omega_R$. This is the regime in which the monochromatic flux argument applies with $\omega$ replaced by $\omega_R$. The dimensionless gravitational coupling is

$$\alpha=GM\mu$$

with $\hbar=c=1$. For $\alpha\ll1$, the cloud is hydrogenic:

$$\omega_R\simeq\mu \left[ 1-\frac{\alpha^2}{2(n+\ell+1)^2} \right], \qquad r_{\rm cloud}\sim\frac{(n+\ell+1)^2GM}{\alpha^2}.$$

Here $n=0,1,\ldots$ counts radial overtones, $\ell$ is the orbital angular number, and $N=n+\ell+1$ is the principal quantum number. The length scale follows by balancing nonrelativistic kinetic energy $1/(2\mu r^2)$ against gravitational binding $GM\mu/r$: the gravitational Bohr radius is $1/(GM\mu^2)=GM/\alpha^2$. Excited hydrogenic states are larger by order $N^2$.

The imaginary part $\omega_I$ determines growth or decay. On the bound-state branch it crosses zero at synchronization. Scalar, Proca, and massive tensor theories have different spectra and fastest modes; the tensor case additionally requires its own consistent field equations. “A bosonic cloud” is not one universal rate.

The small-coupling scalar result originates in Detweiler 1980, pp. 2323–2326; the massive-field equations, hydrogenic limit, and unified instability picture are reviewed in Brito, Cardoso, and Pani 2020, §§5.6.2–5.6.5, pp. 105–115, PDF.

Clouds at the synchronization threshold

At

$$\omega_R=m\Omega_H,$$

the horizon flux vanishes. A real-frequency bound solution satisfying regular horizon and decaying infinity conditions is a stationary cloud with respect to the horizon generator. The field itself can have harmonic phase dependence, while its stress tensor is stationary and axisymmetric for a complex field.

Clouds occur on discrete existence lines in Kerr parameter space because both boundary conditions must be satisfied. They are linear configurations: their stress does not yet backreact on the metric. Herdeiro and Radu showed that scalar clouds mark a bifurcation to fully nonlinear stationary black holes with synchronized complex scalar hair Herdeiro and Radu 2014, Eqs. (1)–(4), Figs. 1–2, and pp. 1–5, PDF. Existence of that branch does not prove it is dynamically selected or stable.

Real fields require care. A single real harmonic generally gives an explicitly time-dependent stress tensor, and nonlinear evolutions can radiate rather than settle to the same stationary complex-field branch. Vector clouds and Proca hair are distinct matter models.

Independent checks and limitations

• Set $a\to0$, so $\Omega_H\to0$: no positive-frequency mode is superradiant.
• Set $\omega=m\Omega_H$: the horizon flux vanishes.
• Require both $\omega_R\lt \mu$ and $\omega_R\lt m\Omega_H$ for a growing massive bound state.
• Track dimensions through $\alpha=GM\mu$; writing $M\mu$ silently assumes $G=1$.
• Distinguish a linear cloud, a nonlinear stationary hairy solution, and a dynamical endpoint.

Environmental absorption, accretion, companions, plasma effects, or boson self-interactions can compete with the vacuum growth rate. Those effects do not alter the kinematic flux sign, but they can determine whether an astrophysical cloud becomes macroscopic.

Check your understanding

A candidate scalar mode has $m=1$, $\omega_R=0.4/(GM)$, $\mu=0.45/(GM)$, and $\Omega_H=0.45/(GM)$. Do these values meet the weak-growth confinement and superradiance conditions? What changes at $\Omega_H=0.4/(GM)$?

Solution

Because $\omega_R\lt\mu$, the mode is eligible to decay at infinity. Initially $\omega_R\lt m\Omega_H=0.45/(GM)$, so the leading monochromatic horizon energy flux is negative. A growing quasibound eigenmode is established only after solving the radial and angular boundary-value problem and finding $\omega_I\gt0$. At $\Omega_H=0.4/(GM)$ the synchronization condition holds; an actual real-frequency bound eigenfunction then has zero horizon flux and is a threshold cloud.

References

• Brito, Richard, Vitor Cardoso, and Paolo Pani, Superradiance, Lecture Notes in Physics 971, second edition, Springer, 2020. Open PDF.
• Detweiler, Steven, “Klein–Gordon Equation and Rotating Black Holes”, Physical Review D 22, 2323–2326, 1980.
• Herdeiro, Carlos A. R., and Eugen Radu, “Kerr Black Holes with Scalar Hair”, Physical Review Letters 112, 221101, 2014. Open PDF.