---
title: "Black Hole Superradiance and Boson Clouds Interactive Lab | Sandbox Physics"
description: "How can a spinning black hole amplify a wave and grow a cloud of light particles?"
canonical_url: "https://sandboxphysics.com/labs/black-hole-superradiance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/black-hole-superradiance/"
last_modified: "2026-11-14"
---

# 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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

## How a spinning hole hands energy to a wave

**Background** A rotating black hole drags space around with it. Its horizon turns at the angular velocity $\Omega_H$. A wave with azimuthal number $m$ and frequency $\omega$ has a pattern that turns at $\omega/m$.

**Why it matters** Change the spin and the frequency and watch whether the reflected wave comes back brighter or dimmer than the wave that went in.

- **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 misconception** Light or matter escapes from inside the horizon to make the wave stronger.

**Better mental model** Nothing 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 misconception** A short e-fold time means such a particle must exist, or that the cloud grows without limit.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Superradiance needs a rotating horizon and a wave that turns more slowly than it. A trapped massive field can turn that gain into exponential growth, but how far a cloud grows and what it says about real particles needs more physics than this linear model.

## Related experiments and articles

Spinning black holes and boson clouds · Reading path

### From an amplified wave to a growing cloud

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why can a spinning black hole make the waves around it ever stronger?**](https://sandboxphysics.com/stories/black-hole-superradiance/why-a-spinning-black-hole-amplifies-waves/)
2. [Article **Waves, spin and the scales of a Kerr black hole**](https://sandboxphysics.com/stories/black-hole-superradiance/waves-spin-and-kerr-scales/)
3. [Article **Conserved flux, physical units and a first growth estimate**](https://sandboxphysics.com/stories/black-hole-superradiance/flux-units-and-a-first-growth-estimate/)
4. [Article **Solving for the bound cloud's growth rate reproducibly**](https://sandboxphysics.com/stories/black-hole-superradiance/solving-the-bound-cloud-reproducibly/)
5. [Article **Recomputing a published superradiance amplification table**](https://sandboxphysics.com/stories/black-hole-superradiance/recomputing-a-published-amplification-table/)
6. [Article **How self-interactions change boson-cloud limits from spins and gravitational waves**](https://sandboxphysics.com/stories/black-hole-superradiance/reading-self-interacting-boson-cloud-limits/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)