Black Hole Superradiance and Boson Clouds

How can a spinning black hole amplify a wave and grow a cloud of light particles? A wave that rotates around a spinning black hole more slowly than the horizon turns comes back stronger, because the hole gives up rotational energy. A trapped field of very light particles can then grow into a cloud, but this linear growth says nothing yet about where it stops.

Drag the black hole spin and the wave frequency. A wave with $\omega<m\Omega_H$ is reflected stronger, and a trapped boson cloud grows.

Horizon rotation

m\Omega_H M

$0.434$

Reflected-wave gain

Z_{11}

$0.366\,\%$

Cloud e-fold time $420\,\mathrm{s}$

Boson rest energy

m_b c^2

$5.19\times10^{-12}\,\mathrm{eV}$

Model assumptions

The scene and charts use units with $G=c=M=1$ ; only the e-fold time and the boson rest energy use the selected black-hole mass. The scattering picture solves a massless scalar wave with $l=m=1$ on a Kerr background numerically and checks the energy flux at the horizon and at large radius; the gain chart is that solution, with the low-frequency formula dashed. The cloud is the massive $l=m=1$ ground state. Its growth rate is interpolated from an offline continued-fraction grid computed for this site; the dashed small-coupling hydrogenic formula overestimates the rate near its peak. Only linear growth is computed: no saturation, self-interaction, gravitational-wave emission, accretion or spin-down history. The switch-off spin $a_{\mathrm{stop}}$ is the ideal end point where growth would stop, not a simulated path. The scene is schematic: a logarithmic radial scale, wave fronts drawn from the computed phase, and reflected brightness magnified 100 times. Nothing here is an observation, and no particle limit follows from this Lab alone.

Gain, cloud growth and where clouds grow

Computed: the gain in this browser, the cloud rates from an offline grid. No observations.

Reflected-wave gain

Z_{11}(\omega)

Kerr wave solution
Low-frequency formula
Threshold $m\Omega_H$

Horizontal: $M\omega$ . Vertical: percent of incoming energy gained (positive) or absorbed (negative), massless field.

Cloud growth rate

\log_{10}(M\omega_I)

Continued fraction at $a/M=0.99$
Hydrogenic formula

Horizontal: coupling $\alpha=GMm_b/(\hbar c)$ . The selected cloud has $\alpha=0.389$ . This cloud grows: its frequency is below the horizon rotation.

Where the cloud grows

Growing region, brighter is faster
Ideal switch-off $a_{\mathrm{stop}}/M=0.963$

Vertical: spin $a/M$ . The dotted drop marks where growth would stop; the spin-down path itself is not computed.

One-sentence intuition: A wave pattern that turns more slowly than the horizon, $\omega<m\Omega_H$ , carries negative energy across the horizon, so the reflected wave carries out more than came in.

Core mathematical model

How fast the horizon turns

\Omega_H=\frac{a}{2Mr_+},\qquad r_+=M+\sqrt{M^2-a^2}

Here $a$ is the spin per unit mass. Without spin the horizon does not turn and no wave can gain energy.

Energy flux on both sides

\omega\left(|I|^2-|R|^2\right)=(\omega-m\Omega_H)\,|T|^2

The same conserved Wronskian is evaluated far away, with incoming amplitude $I$ and reflected $R$ , and at the horizon with transmitted $T$ . The sign of $\omega-m\Omega_H$ decides whether $|R|>|I|$ . The Lab checks this balance for every solve.

A cloud that keeps reflecting

M\omega_I\approx\frac{\alpha^9}{48}\left(\frac aM-2\alpha\frac{r_+}M\right),\qquad \alpha=\frac{GMm_b}{\hbar c}

A field with mass $m_b$ is trapped by gravity, like an electron around a proton. This small-coupling formula is drawn dashed; the solid curve comes from an exact continued-fraction solution and is lower near the peak.

Where linear growth would stop

\frac{a_{\mathrm{stop}}}{M}=\frac{4mM\omega_R}{m^2+4M^2\omega_R^2}

Growth stops once the spin has fallen so far that $m\Omega_H=\omega_R$ . This is an ideal end point; the path there, including saturation, is not computed here.

Common difficulties

The energy does not come from inside the hole

Typical misconceptionLight or matter escapes from inside the horizon to make the wave stronger.

Better mental modelNothing crosses outward through the horizon. The transmitted wave carries negative energy inward as measured from far away, so the hole loses rotational energy while its area still grows.

Fast growth is not a particle detection

Typical misconceptionA short e-fold time means such a particle must exist, or that the cloud grows without limit.

Better mental modelThe Lab computes only linear growth for one mode. Bounds on real particles also need measured spins, the age of the system, self-interactions and the cloud’s own radiation, all of which depend on the model.

Run the experiment

01
See the gain
Start from the default and watch the amber outgoing fronts and the flow arrows near the horizon.
What to observe: The gain $Z_{11}$ is positive and the arrows point outward: the reflected wave carries more energy than came in.
02
Push past the threshold
Drag the wavelength marker inward until the frequency exceeds the horizon rotation, or choose the absorbed preset.
What to observe: When $\omega>m\Omega_H$ the gain turns negative and part of the wave is absorbed.
03
Stop the spin
Drag the spin dial back to zero and try several frequencies.
What to observe: A non-rotating hole only absorbs. No frequency gives a positive gain.
04
Compare the cloud curves
Look at the growth chart, then change the black-hole mass in the advanced group.
What to observe: The exact curve and the dashed hydrogenic one agree only at small $\alpha$ . A ten times heavier hole has a ten times longer e-fold time and a ten times lighter boson for the same curve.

Black Hole Superradiance and Boson Clouds

Gain, cloud growth and where clouds grow

How a spinning hole hands energy to a wave

Core mathematical model

How fast the horizon turns

Energy flux on both sides

A cloud that keeps reflecting

Where linear growth would stop

Common difficulties

The energy does not come from inside the hole

Fast growth is not a particle detection

Run the experiment

See the gain

Push past the threshold

Stop the spin

Compare the cloud curves

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From an amplified wave to a growing cloud