Radiative and Hairy End States

The endpoint of superradiant growth is not fixed by the quasibound-mode inequality $\omega_R\lt m\Omega_H$. That inequality determines the sign of a mode’s horizon flux. Endpoint selection additionally depends on whether the field is real or complex, its spin and self-interaction, radiation channels, competing modes, gravitational backreaction, boundary conditions, and environment. As of the evidence cutoff 2026-08-12, several conditional outcomes are supported, but no universal endpoint theorem exists.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon mechanics; Black-Hole Perturbation Theory supplies growth and radiation modes; Formulations of General Relativity supplies backreaction; Relativistic Matter and Compact Objects supplies bosonic models; Computational Relativity supplies nonlinear validation.

Superradiant endpoints and their evidence

Candidate outcome	Supporting evidence	Required qualification
Kerr plus a dilute cloud near synchronization	adiabatic flux balance and nonlinear complex-Proca evolution	model, duration, and perturbation sector
stationary black hole with synchronized hair	nonlinear scalar and Proca solution branches	existence is not stability or selection
radiative depletion toward nearly bald Kerr	annihilation or transition channels for real fields	timescale and mode population dependent
bosenova collapse and repeated growth	nonlinear axion-like fixed-background calculations	potential and backreaction dependent
multilevel or recurrent evolution	several superradiant modes and environmental spin-up	long-time nonlinear competition unresolved
environmental disruption	companion, accretion, plasma, or tidal perturbation	external model dependent

The kinematic endpoint for one isolated mode is synchronization. The horizon flux vanishes at $\omega_R=m\Omega_H$, but flux to infinity can remain. A real cloud can radiate through annihilations or transitions; a complex field can carry an exactly stationary stress tensor. These are different systems, not two descriptions of the same endpoint Brito, Cardoso, and Pani 2020, §§5.6.6–5.6.8 and §6.1, pp. 115–128 and 160–167, PDF.

Competing dynamical pictures

The radiative picture treats the cloud as a long-lived reservoir that can lose energy to gravitational or bosonic waves near or after saturation. For a real harmonic field, products in the stress tensor contain a component oscillating at $2\omega_R$ as well as a stationary part; when its angular structure permits, that component sources gravitational radiation. For a single complex harmonic, products of the field and its conjugate cancel the phase instead. This explains why the reality condition matters without assuming identical radiation rates for different spins or angular modes.

The hairy picture uses synchronized complex fields. Linear clouds bifurcate to nonlinear stationary branches, and an $m=1$ complex-Proca evolution with axisymmetric geometry has approached synchronization Herdeiro and Radu 2014, Eqs. (1)–(4) and Figs. 1–2, pp. 1–5, PDF, East and Pretorius 2017, Eqs. (1)–(4) and Figs. 1–4, pp. 1–5, PDF. This restricted evolution does not test competing azimuthal numbers. Growing higher-$m$ perturbations are explicitly found on sampled scalar-haired backgrounds Ganchev and Santos 2018, “Results,” p. 4, PDF, so persistence of hair over the first saturation timescale is weaker than an asymptotically stable endpoint.

The collapse-cycle picture arises when attractive self-interaction becomes important. A fraction of the cloud falls into the hole or escapes, leaving a remnant that may grow again Yoshino and Kodama 2012, §§3–4, pp. 8–23, PDF. The threshold is not universal because it depends on the potential and occupation.

Selection axes

Any endpoint statement must declare:

• scalar, vector, or tensor field;
• real or complex field;
• $GM\mu$, spin, mode numbers, and initial occupation;
• self-interaction and gravitational coupling;
• asymptotically flat or confining boundary;
• isolated black hole or specified accretion, companion, and plasma environment;
• symmetry restrictions and allowed perturbations;
• duration relative to growth, depletion, and environmental times.

Changing one axis can reverse the conclusion. A stationary complex-scalar solution is not evidence for a stationary real-scalar endpoint. A vacuum asymptotically flat evolution does not decide an AdS cavity problem. A short-lived plateau near synchronization does not establish asymptotic stability.

What would resolve the endpoint

A decisive nonlinear study would evolve constraint-satisfying data through many growth and depletion times, close horizon and infinity budgets, resolve all relevant angular modes, vary generic perturbations, and converge the final invariant state. A theorem would need a basin of attraction and a norm demonstrating convergence modulo gauge and parameters.

The numerical results described here come from the cited studies and have not been independently reproduced for this account. Their conclusions remain conditional on the chosen formulations. A stability theorem for a hairy family, independent long-time replication, or a stable counterexample could clarify the unresolved general endpoint; corrections or withdrawals could change the existing conclusions.

Formulation-choice diagnostic

Two studies disagree: one complex-vector evolution approaches synchronized hair; one real axion calculation undergoes collapse and radiation. Are the results inconsistent?

Solution

No. They use different matter reality conditions, field spin, self-interaction, and approximations. A complex vector can have stationary stress and support synchronized hair; a real attractive scalar can develop nonlinear collapse. The studies would conflict only after matching the Lagrangian, reality condition, parameters, boundary conditions, initial-data family, diagnostics, and numerical validation.

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.
• Ganchev, Bogdan, and Jorge E. Santos, “Scalar Hairy Black Holes in Four Dimensions Are Unstable”, Physical Review Letters 120, 171101, 2018. 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.