Saturation and Spin-Down

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Superradiant growth cannot continue indefinitely: it changes the black hole and the cloud, and it competes with radiation, self-interaction, accretion, and environmental perturbations. Which effect terminates growth first is model- and regime-dependent. As of the evidence cutoff 2026-08-12, synchronization robustly identifies the zero-flux threshold, but it does not universally select the final nonlinear state.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon mechanics; Black-Hole Perturbation Theory supplies growth rates; Formulations of General Relativity supplies backreaction; Relativistic Matter and Compact Objects supplies field interactions and environments; Computational Relativity supplies nonlinear evidence.

Superradiant growth and competing losses

For a chosen mode and matter Lagrangian, compare:

\tau_{\rm SR,E}=(2\Gamma)^{-1}, \qquad \tau_{\rm GW}=\frac{E_c}{P_{\rm GW}}, \qquad \tau_{\rm self}, \qquad \tau_{\rm acc}, \qquad \tau_{\rm env}.

Here $\Gamma=\operatorname{Im}\omega\gt0$ is the amplitude growth rate, so the cloud energy grows as $e^{2\Gamma t}$ and $\tau_{\rm SR,E}$ is its energy e-folding time. Comparing this energy time with $\tau_{\rm GW}=E_c/P_{\rm GW}$ avoids a factor-of-two mismatch. The remaining times describe self-interaction, accretion, and environmental change. A channel with a competitive timescale can alter the adiabatic track.

Regime	Leading termination candidate	Main uncertainty
weakly coupled complex field, dilute cloud	spin-down to $\omega=m\Omega_H$	stability of the synchronized nonlinear configuration
real field with efficient annihilation	balance between extraction and radiation	multilevel and relativistic corrections
attractive self-interaction	bosenova-like collapse or ejection	nonlinear threshold and repeated cycles
vector field with strong growth	rapid backreaction near synchronization	generic perturbations and real-field endpoint
strong accretion or companion	environmental detuning or replenished spin	environmental model and timescale

The relevant linear and nonlinear regimes are reviewed in Brito, Cardoso, and Pani 2020, §§5.6.5–5.6.8, pp. 114–128, and §6.1, pp. 160–167, PDF.

Spin-down and the threshold line

In the ideal adiabatic picture,

\dot E_c=-\dot M_H, \qquad \dot J_c=-\dot J_H, \qquad \frac{\dot E_c}{\dot J_c}=\frac{\omega_R}{m}.

The black hole moves toward lower $\Omega_H$ until $\omega_R=m\Omega_H$ . At that line the horizon flux of the selected mode vanishes. If the cloud remains bound and other loss channels are slow, the system can linger near synchronization.

The threshold has a useful explicit form. For a co-rotating Kerr hole with $0\lt\chi\lt1$ , define $q=GM_H\omega_R/m$ . Then

q=GM_H\Omega_H =\frac{\chi}{2(1+\sqrt{1-\chi^2})} \quad\Longleftrightarrow\quad \chi_{\rm sync}=\frac{4q}{1+4q^2}, \qquad 0\lt q\lt\frac12.

The domain matters: the right-hand algebraic expression continued to $q\gt1/2$ is not a physical inversion of the Kerr angular velocity. For a fixed frequency, $q$ decreases as the black-hole mass decreases, so this threshold must be intersected with the mass–angular-momentum trajectory rather than evaluated only at the initial mass.

Several qualifications are essential. A different mode may remain superradiant after the first saturates. The cloud’s self-gravity can shift $\omega_R$ . Radiation can remove cloud energy and angular momentum in a ratio different from $\omega_R/m$ . Accretion can spin the black hole back across the threshold. Therefore the threshold is a local balance condition for a mode, not a universal absorbing state.

Evidence for different outcomes

Complex Proca evolutions in full general relativity show substantial extraction followed by a black hole with a nearly synchronized vector cloud East and Pretorius 2017, Eqs. (1)–(4) and Figs. 1–4, pp. 1–5, PDF. The $m=1$ field ansatz keeps the stress tensor axisymmetric and excludes higher- $m$ competition. The authors explicitly note that higher- $m$ modes can remain unstable after $m=1$ saturation. This is evidence for smooth saturation of the selected mode, not stability against all perturbations.

For axion-like scalars with attractive self-interaction, nonlinear calculations find collapse and outward bursts once the cloud reaches a coupling-dependent amplitude Yoshino and Kodama 2012, §§3–4, pp. 8–23, PDF. The calculation uses a specified scalar potential and approximations to gravitational backreaction; it does not imply that every bosonic cloud undergoes a bosenova.

Stationary scalar-haired black holes demonstrate that synchronized regular solutions exist for a complex scalar model Herdeiro and Radu 2014, Eqs. (1)–(4) and Figs. 1–2, pp. 1–5, PDF. Existence is not dynamical selection, and selection is not nonlinear stability.

Which assumptions change the endpoint?

Conclusions depend on field spin, real versus complex field, mass coupling $GM\mu$ , self-interaction, cloud occupation, black-hole spin, asymptotics, symmetry restrictions, and environmental timescales. The open question is not whether synchronization exists, but which channel wins in each specified model and whether the resulting state is stable to generic perturbations.

Decisive progress would be a theorem or independently converged evolution that follows a generic perturbation through growth, backreaction, and long-time relaxation with a closed energy–angular-momentum budget. No simulation was rerun for this page.

This assessment should be revisited after a broader nonlinear theorem, an independently reproduced long-duration evolution, a stable counterexample to a proposed endpoint, or a correction affecting a cited computation.

Formulation-choice diagnostic

A complex Proca simulation settles near $\omega=m\Omega_H$ with a persistent cloud. What may be concluded, and what remains open?

Solution

Within that complex-vector model and simulated data family, the evolution supports backreacted saturation near synchronization. It does not prove that the configuration is stable for all perturbations, that a real Proca field behaves identically, that scalar self-interactions are irrelevant, or that the same endpoint occurs throughout parameter space. Those are separate evidence axes.

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.