---
title: "Waves, spin and the scales of a Kerr black hole"
description: "Set up the units, horizon, horizon rotation and wave angular momentum, then derive when a wave can gain energy and when a light field is trapped."
canonical_url: "https://sandboxphysics.com/stories/black-hole-superradiance/waves-spin-and-kerr-scales/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/black-hole-superradiance/waves-spin-and-kerr-scales/"
last_modified: "2026-11-12"
---

[Spinning black holes and boson clouds](https://sandboxphysics.com/stories/black-hole-superradiance/)

# Waves, spin and the scales of a Kerr black hole

Sandbox Physics · 2026-11-12

A wave can draw energy from a spinning black hole only when its pattern turns more slowly than the horizon, $0<\omega<m\Omega_H$. A field with a small mass is held near the hole and can keep drawing energy only when its frequency also sits below that limit. This chapter builds the quantities behind both statements: the hole's natural units, its horizon and rotation rate, the angular momentum a wave carries, and the size of a trapped cloud.

[Open the black hole superradiance Lab](https://sandboxphysics.com/labs/black-hole-superradiance/)

Read [why a spinning black hole can make waves stronger](https://sandboxphysics.com/stories/black-hole-superradiance/why-a-spinning-black-hole-amplifies-waves/) first if the idea is new. Algebra, angular frequency and the energy levels of hydrogen are enough for this chapter.

## Measure everything in units of the hole's mass

A black hole of mass $M$ sets its own length and time. Set Newton's constant and the speed of light to one, $G=c=1$. A length is then counted in units of $GM/c^2$ and a time in units of $GM/c^3$. Writing "$r=2M$" means a radius of $2GM/c^2$.

**Table 1.1** Natural units of two black holes.

| Mass | Length unit $GM/c^2$ | Time unit $GM/c^3$ |
| --- | --- | --- |
| $10\,\mathrm{M}_\odot$ | $14.8\,\mathrm{km}$ | $49.3\,\mu\mathrm{s}$ |
| $4\times10^6\,\mathrm{M}_\odot$ | $5.91\times10^6\,\mathrm{km}$ | $19.7\,\mathrm s$ |

Both columns grow in proportion to the mass. Any dimensionless result, such as a frequency times $M$, holds for every mass at once; only the conversion to kilometres or seconds changes.

## The Kerr horizon and how fast it turns

A rotating, uncharged black hole is described by the Kerr solution. Besides $M$ it has one more number, the spin $a=J/M$, the angular momentum per unit mass. In these units $0\le a\le M$. The horizon sits at the larger root of $r^2-2Mr+a^2=0$,

$$
r_\pm=M\pm\sqrt{M^2-a^2}.
$$

The rotating hole drags space around it. A small object hovering just outside the horizon is forced to circle with the horizon's angular velocity

$$
\Omega_H=\frac{a}{2Mr_+}.
$$

In the equatorial plane the region from $r_+$ out to $r=2M$ is the ergoregion: inside it nothing can stay at rest relative to distant stars.

**Worked example 1.** Take $a=0.99M$. Then $\sqrt{1-0.99^2}=0.141$, so $r_+=1.141M$ and $M\Omega_H=0.99/(2\times1.141)=0.434$. For a hole of ten solar masses, $\Omega_H=0.434/(49.3\,\mu\mathrm s)=8.81\times10^3\,\mathrm{rad\,s^{-1}}$. The horizon turns about $1.40\times10^3$ times per second.

**Table 1.2** Horizon radius and rotation rate.

| Spin $a/M$ | $r_+/M$ | $M\Omega_H$ |
| --- | --- | --- |
| $0.5$ | $1.866$ | $0.134$ |
| $0.7$ | $1.714$ | $0.204$ |
| $0.9$ | $1.436$ | $0.313$ |
| $0.99$ | $1.141$ | $0.434$ |

[Brito, Cardoso and Pani, Superradiance, §4.7](https://arxiv.org/abs/1501.06570v8)

## A wave carries angular momentum with its energy

Write a wave that circles the axis as $e^{-i\omega t+im\varphi}$, where $\varphi$ is the azimuthal angle and $m$ an integer. A crest stays at fixed phase $m\varphi-\omega t$, so the pattern turns at angular velocity $\omega/m$. When the wave delivers energy $\mathrm dE$ to an absorber, it also delivers angular momentum

$$
\mathrm dJ=\frac{m}{\omega}\,\mathrm dE .
$$

The ratio $m/\omega$ is fixed because the wave keeps a single frequency and a single $m$ while it scatters off an axisymmetric body. [Brito, Cardoso and Pani, Superradiance, §3.5.1](https://arxiv.org/abs/1501.06570v8)

## When the horizon must give energy away

First consider an ordinary body turning at rate $\Omega$. In the frame that turns with it, the energy it receives from the wave is $\mathrm dE^\prime=\mathrm dE-\Omega\,\mathrm dJ$. Using the ratio above,

$$
\mathrm dE^\prime=\mathrm dE\left(1-\frac{m\Omega}{\omega}\right).
$$

An absorber turns ordered wave energy into heat, so $\mathrm dE^\prime\ge0$. When $\omega<m\Omega$ the bracket is negative, and $\mathrm dE$ must be negative: the body loses energy to the wave. That is Zel'dovich's rotational superradiance.

A black hole obeys the same bookkeeping, with the horizon area $A$ in the role of entropy. Its first law is $\mathrm dM=\Omega_H\,\mathrm dJ+(\kappa/8\pi)\,\mathrm dA$, where $\kappa$ is the surface gravity, and the area theorem says $\mathrm dA\ge0$. Substituting $\mathrm dJ=(m/\omega)\,\mathrm dM$,

$$
\mathrm dM\left(1-\frac{m\Omega_H}{\omega}\right)=\frac{\kappa}{8\pi}\,\mathrm dA\ge0 .
$$

For $0<\omega<m\Omega_H$ the hole's mass must go down. A wave with $m\le0$ can never satisfy the condition, and neither can any wave when $a=0$.

## A trapped wave: the gravitational atom

Now give the field a mass $m_b$. Its wave cannot escape if its energy is below its rest energy, so gravity holds it in bound states, much as a proton holds an electron. The strength of the binding is set by the dimensionless coupling

$$
\alpha=\frac{GMm_b}{\hbar c},
$$

the ratio of the hole's size $GM/c^2$ to the particle's reduced Compton wavelength $\hbar/(m_bc)$. In units $G=c=\hbar=M=1$ the field mass and the coupling are the same number, and the coupling plays the role of the fine-structure constant.

For small $\alpha$ the states look like hydrogen. The most important one has one unit of angular momentum, $l=m=1$, and no radial node. Its frequency is close to the rest energy,

$$
M\omega_R\approx\alpha\left(1-\frac{\alpha^2}{8}\right),
$$

and its radial profile is $r^\prime e^{-r^\prime/2}$ with $r^\prime=r\alpha^2/M$. The probability of finding the field at radius $r$ peaks at $r=4M/\alpha^2$, so a small coupling gives a large, dilute cloud. [Brito, Cardoso and Pani, Superradiance, §5.6.2](https://arxiv.org/abs/1501.06570v8)

The cloud brushes the horizon on every orbit. It grows when $\omega_R<m\Omega_H$ and is slowly swallowed otherwise. [Dolan, Instability of the massive Klein-Gordon field on the Kerr spacetime, §II](https://arxiv.org/abs/0705.2880v2)

**Worked example 2.** Keep $a=0.99M$, so $M\Omega_H=0.434$. At $\alpha=0.3$, $M\omega_R\approx0.3\times(1-0.01125)=0.297$, below $0.434$: the cloud grows. Its peak sits near $r=4M/0.09=44M$, about $660\,\mathrm{km}$ for ten solar masses. At $\alpha=0.45$, $M\omega_R\approx0.439$, just above the horizon rate: the cloud decays. A full numerical solution gives $M\omega_R=0.436$ there, also above, and a decaying cloud.

## Lab tasks

1. Predict the threshold frequency for $a=0.7M$ from Table 1.2. In the Lab, set the spin to 0.7 and drag the wavelength marker until the reflected-wave gain changes sign. Compare the crossing with your prediction.
2. Keep the frequency fixed and drag the spin dial down to zero. Explain from the first-law argument why the gain never returns above zero.
3. With spin 0.99, find the coupling where the cloud growth curve drops to zero. Compare it with the estimate $\alpha\approx M\Omega_H$ that follows from $\omega_R\approx m_b$.

## Exercises with explained answers

1. Find $r_+$ and $M\Omega_H$ for $a=0.5M$.
2. A wave with $m=2$ and $M\omega=0.6$ meets a hole with $a=0.99M$. Is it amplified?
3. For a hole of ten solar masses, where does the $l=m=1$ cloud peak when $\alpha=0.1$?
4. Why can a wave with $m=-1$ never be amplified by a hole with $a>0$?

Answers

1. $\sqrt{1-0.25}=0.866$, so $r_+=1.866M$ and $M\Omega_H=0.5/(2\times1.866)=0.134$, as in Table 1.2.
2. The condition is $\omega<m\Omega_H=2\times0.434/M=0.868/M$. Since $0.6<0.868$, the wave is amplified. Higher $m$ raises the threshold because the pattern turns at $\omega/m$, half as fast.
3. The peak is at $4M/\alpha^2=400M$. With $GM/c^2=14.8\,\mathrm{km}$ that is about $5.9\times10^3\,\mathrm{km}$.
4. For positive frequency, $m\Omega_H$ is negative when $m=-1$, so $\omega<m\Omega_H$ cannot hold. The pattern turns against the horizon, and the first law then forces $\mathrm dM>0$: the hole only absorbs.

Next, prove the gain condition from the wave equation itself and turn these numbers into seconds and electronvolts in [conserved flux, physical units and a first growth estimate](https://sandboxphysics.com/stories/black-hole-superradiance/flux-units-and-a-first-growth-estimate/).

## Try it in the Lab

Predict the threshold frequency for spin 0.7, then drag the wavelength marker until the gain changes sign. [Open the black hole superradiance Lab](https://sandboxphysics.com/labs/black-hole-superradiance/)

## Sources

1. [Brito, Cardoso and Pani, Superradiance, §4.7](https://arxiv.org/abs/1501.06570v8)
2. [Dolan, Instability of the massive Klein-Gordon field on the Kerr spacetime, §II](https://arxiv.org/abs/0705.2880v2)

## Keep reading

Spinning black holes and boson clouds · Reading path

### From an amplified wave to a growing cloud

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. [You are here **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/)