Superradiant Evolution, Saturation, and Hair Formation

Superradiance begins with a horizon-flux sign and ends with a nonlinear selection problem. Massive bosonic fields can be both amplified by a rotating horizon and confined outside it, producing growing clouds. Backreaction then changes the black-hole spin, while self-interaction, radiation, mode competition, and environmental forcing decide whether the evolution synchronizes, forms persistent hair, collapses, or disperses.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr mechanics; Black-Hole Perturbation Theory supplies mode spectra; Formulations of General Relativity supplies nonlinear backreaction; Relativistic Matter and Compact Objects supplies bosonic matter models.

From a growing mode to an endpoint

Bosonic Clouds Around Rotating Black Holes combines the two necessary inequalities: $\omega_R\lt m\Omega_H$ for superradiant horizon flux of a quasibound mode and $\omega_R\lt \mu$ for massive confinement. It identifies threshold clouds at synchronization.

Adiabatic Superradiant Growth follows energy and angular momentum as the cloud grows and the black hole spins down. Saturation and Spin-Down compares synchronization with radiation, self-interaction, accretion, and environmental timescales.

Hairy Black-Hole Branches explains how a linear cloud can bifurcate into a regular synchronized family while separating existence, stability, and dynamical selection. Bosenovae and Nonlinear Cloud Collapse treats the computational evidence and validation requirements for attractive self-interaction. Radiative and Hairy End States assembles the conditional endpoint map.

Three statements to keep separate

Use the mode convention $e^{-i\omega t+im\phi}$, where $m\in\mathbb Z$ is the azimuthal number, not a mass; $\omega=\omega_R+i\omega_I$ and positive $\omega_I$ means exponential growth. Here $\Omega_H$ is the angular velocity of the future horizon and $\mu$ is the boson mass, expressed as an angular frequency because $\hbar=1$.

1. Flux sign: for a quasibound mode $e^{-i\omega t+im\phi}$, the horizon energy flux is superradiant when $0\lt \omega_R\lt m\Omega_H$; for a strictly monochromatic scattering wave, $\omega$ is real and the same condition is written with $\omega$.
2. Bound-state instability: a confined eigenmode has $\omega_I\gt 0$ and grows exponentially.
3. Nonlinear endpoint: the coupled geometry and field approach a radiative, synchronized, hairy, collapsing, or recurrent state.

The first is kinematic, the second is spectral, and the third is dynamical. Evidence for one does not automatically establish the next.

For weakly growing modes these distinctions can be tracked with three observables: the sign of the horizon energy flux, the fitted imaginary part of the mode frequency, and the long-time energy–angular-momentum budget. The second is not established merely by substituting candidate real frequencies into the first inequality.

The synchronization relation

$$\omega_R=m\Omega_H$$

appears in two roles. It is the zero-flux boundary of the linear instability and the zero-flux equilibrium condition at a regular horizon with stationary complex-field hair. That connection explains the bifurcation, but it does not prove that every evolution follows the branch.

Model-dependence map

Choice	Consequence
real field	generally time-dependent stress; radiation natural
complex field	harmonic phase can yield stationary stress and hair
weak self-interaction	adiabatic spin-down may dominate
attractive strong self-interaction	collapse or bosenova cycles possible
scalar versus vector	spectra, growth rates, constraints, and branches differ
isolated asymptotically flat system	energy can escape
confining boundary or environment	recurrent growth and detuning can occur

The standard review gives the linear condition, massive spectra, nonlinear evidence, clouds, and hair in a unified notation Brito, Cardoso, and Pani 2020, §§4.4–4.7 and 5.6, pp. 50–128, PDF. Primary studies support different branches of this model-dependent map: vector-field backreaction approaching synchronization East and Pretorius 2017, Eqs. (1)–(4), Figs. 1–4, pp. 1–5, PDF, stationary synchronized scalar hair Herdeiro and Radu 2014, Eqs. (1)–(4), Figs. 1–2, pp. 1–5, PDF, and fixed-background axion-cloud collapse Yoshino and Kodama 2012, §§3–4, pp. 8–23, PDF. None alone supplies a universal endpoint.

Chapter diagnostic

A mode satisfies $\omega_R\lt \mu$ and $\omega_R\lt m\Omega_H$. A stationary synchronized hairy family also exists in the same complex-field theory. Which conclusions follow immediately?

Solution

The candidate frequency satisfies the weak-growth confinement and superradiance inequalities. Solving the boundary-value problem is still necessary to establish an actual eigenmode; if its eigenvalue has $\omega_I\gt0$, it grows linearly. The hairy family provides a possible nonlinear configuration connected at $\omega_R=m\Omega_H$. It does not follow that the evolution reaches that branch, that the branch is stable, or that no radiation, competing mode, or nonlinear collapse intervenes. Those require dynamical evidence.

Scope and continuation

This chapter uses linear spectra as inputs but does not replace their derivation. It treats matter models only far enough to understand spacetime backreaction and endpoints. No page claims a newly rerun simulation. The next chapter turns from bosonic-field evolution to externally driven finite-amplitude tides and encounters.

References

• Brito, Richard, Vitor Cardoso, and Paolo Pani, Superradiance, Lecture Notes in Physics 971, second edition, Springer, 2020. Open PDF.
• East, William E., and Frans Pretorius, “Superradiant Instability and Backreaction of Massive Vector Fields around Kerr Black Holes”, Physical Review Letters 119, 041101, 2017. Open PDF.
• Herdeiro, Carlos A. R., and Eugen Radu, “Kerr Black Holes with Scalar Hair”, Physical Review Letters 112, 221101, 2014. Open PDF.
• Yoshino, Hirotaka, and Hideo Kodama, “Bosenova Collapse of Axion Cloud around a Rotating Black Hole”, Progress of Theoretical and Experimental Physics 2012, 043E02, 2012. Open PDF.

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• Superradiant Saturation and Hair Formation