The Superradiant Frequency Condition

Superradiance occurs when a future-ingoing bosonic wave carries negative asymptotic-time energy into a rotating or charged horizon. The reflected flux then exceeds the incident flux. With the shared phase $e^{-i\omega t+im\phi}$, the sign is controlled by the frequency measured by the horizon generator, $\omega-m\Omega_H-q\Phi_H$. This page derives the product condition, fixes the flux orientation, and checks its neutral and Schwarzschild limits.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr–Newman horizons; PDEs and Green Functions for Gravitation supplies conserved currents and scattering; Formulations of General Relativity supplies charged field equations; Relativistic Matter and Compact Objects supplies matter and charge conventions; Black-Hole Perturbation Theory supplies the mode normalization.

Horizon generator and electrostatic potential

Set $c=1$ while retaining $G$. The dimensionless black-hole spin is

$$\chi_{\mathrm{BH}}=\frac{J}{GM^2}.$$

For Kerr, $\lvert\chi_{\mathrm{BH}}\rvert\leq1$. Let the future-horizon generator be

$$\chi_H^a=(\partial_t)^a+\Omega_H(\partial_\phi)^a.$$

Choose the electromagnetic gauge with $A_t\to0$ at infinity and define the horizon potential difference by

$$\Phi_H=-\left.\chi_H^aA_a\right|_{\mathcal H^+}.$$

For a bosonic field of charge $q$, use $D_a=\nabla_a-iqA_a$ and the mode

$$\Psi=e^{-i\omega t+im\phi}S(\theta)R(r).$$

Acting along the generator gives

$$\chi_H^aD_a\Psi =-i\widetilde\omega_H\Psi, \qquad \widetilde\omega_H =\omega-m\Omega_H-q\Phi_H.$$

This is the gauge-invariant horizon-frame frequency once the potential difference to infinity is fixed. Future regularity selects the ingoing radial factor $e^{-i\widetilde\omega_H r_*}$ in horizon-regular coordinates. A negative $\widetilde\omega_H$ changes the $\partial_t$ Killing-energy sign, not the causal direction of the wave.

Outward-positive horizon flux

Take positive radial flux to mean flux toward increasing $r$ at both endpoints. Thus an ordinary absorbed wave has negative horizon flux. For a canonically normalized bosonic mode, the conserved number-current factor at the horizon is proportional to $-\widetilde\omega_H\lvert T\rvert^2$. Multiplication by the positive asymptotic energy per quantum $\omega$ gives

$$\mathcal F_H^{(+r)} =-\mathcal N_H\, \omega\bigl(\omega-m\Omega_H-q\Phi_H\bigr) \lvert T\rvert^2, \qquad \mathcal N_H>0.$$

The field- and normalization-dependent factor $\mathcal N_H$ is positive. Consequently, energy flows outward in the asymptotic-time sense at the horizon exactly when

$$\boxed{ \omega\bigl(\omega-m\Omega_H-q\Phi_H\bigr)<0 }.$$

For positive-frequency incidence, this becomes $m\Omega_H+q\Phi_H>\omega>0$ when the upper endpoint is positive. Brito, Cardoso, and Pani derive the classical flux condition for rotating and charged bosonic fields in Brito, Cardoso, and Pani 2020, § 4.7, pp. 57–68.

The same sign follows from horizon mechanics. A monochromatic mode transports angular momentum and charge in the ratios

$$\frac{\delta J}{\delta E}=\frac{m}{\omega}, \qquad \frac{\delta Q}{\delta E}=\frac{q}{\omega}.$$

The horizon-generator energy is

$$\delta E_\chi =\delta E-\Omega_H\delta J-\Phi_H\delta Q =\frac{\widetilde\omega_H}{\omega}\,\delta E.$$

Regular classical matter has nonnegative $\delta E_\chi$ through the future horizon. If $\omega>0$ but $\widetilde\omega_H<0$, this requires $\delta E<0$: negative asymptotic-time energy enters the hole, reducing its mass, angular momentum, or charge while the horizon-generator flux remains nonnegative.

This formulation also exposes the gauge check. A stationary gauge transformation that shifts $A_t$ by a constant changes the canonical frequency label and the potential assigned to either endpoint, but it cannot change the potential difference or $\widetilde\omega_H$. Fixing $A_t=0$ at infinity makes the displayed $\omega$ the energy measured by asymptotic stationary observers. Using a horizon potential from another gauge while leaving $\omega$ unchanged would move the apparent band artificially.

Scattering balance and amplification

At infinity, let $I$ and $R$ denote incident and reflected amplitudes in a flux-compatible normalization. The outward-positive flux there is

$$\mathcal F_\infty^{(+r)} =\mathcal N_\infty\omega^2 \left(\lvert R\rvert^2-\lvert I\rvert^2\right), \qquad \mathcal N_\infty>0.$$

Stationarity makes the radial Wronskian constant, so $\mathcal F_\infty^{(+r)}=\mathcal F_H^{(+r)}$ after the endpoint normalizations are translated. Unit incident flux gives

$$\mathcal R+\mathcal T=1, \qquad \mathcal R=\frac{F_{\mathrm{ref}}}{F_{\mathrm{inc}}}, \qquad \mathcal T\propto \frac{\widetilde\omega_H}{\omega}\lvert T\rvert^2.$$

In the superradiant band, $\mathcal T<0$ and therefore $\mathcal R>1$. Teukolsky and Press give the spin-dependent Kerr flux normalizations and amplification calculation in Teukolsky and Press 1974, pp. 443–461. For electromagnetic and gravitational Teukolsky variables, raw radial amplitudes require the appropriate Starobinsky normalization before they can be inserted into this energy balance.

Number flux, charge flux, and energy flux should not be conflated. The radial Wronskian directly controls a signed wave-action current. Multiplying by $\omega$ converts it to asymptotic-time energy for a monochromatic mode, while multiplication by $m$ or $q$ gives angular-momentum or charge transport. Because these factors differ, a negative energy flux does not mean a negative local norm or a future-outgoing group velocity. It means that the Killing field $\partial_t$ is not the positive horizon generator in the rotating or charged problem.

At the threshold $\widetilde\omega_H=0$, the signed horizon $\partial_t$-energy flux vanishes and $\mathcal R=1$ for a regular scattering solution. This is an important numerical benchmark: inserting $\lvert\widetilde\omega_H\rvert$ would produce absorption on both sides of the threshold and erase the sign change.

Superradiance is a single-pass scattering statement. It becomes an exponential instability only if a separate mechanism returns the amplified wave for repeated encounters faster than leakage and damping remove it.

Limits and field scope

For a neutral field, $q=0$, and the condition reduces to

$$\omega(\omega-m\Omega_H)<0.$$

With $\omega>0$ and a corotating mode, this is $m\Omega_H>\omega>0$. In the Schwarzschild neutral limit, $\Omega_H=\Phi_H=0$, the product becomes $\omega^2$, which cannot be negative: there is no rotational or charged superradiance. In nonrotating charged Reissner–Nordström, the remaining band is $q\Phi_H>\omega>0$ for like-signed $q\Phi_H>0$.

For a massive scattering field, propagation from infinity additionally requires $\omega^2>\mu_b^2$, where $\mu_b$ is the boson mass. A quasi-bound state instead has $\omega^2<\mu_b^2$ and decays at infinity; its instability condition is obtained from a complex eigenfrequency problem rather than a unit incident scattering experiment. The flux-sign result is classical and bosonic. Classical Dirac fields obey a different conserved-current relation: their future-causal number current forbids reflected-number amplification, without invoking quantum occupation numbers. The stationary energy comparison and its scope are developed in Dirac Fields and Horizon Flux.

Nonlinear saturation and hair formation belong to Black-Hole Dynamics; astrophysical environments belong to Black-Hole Astrophysics.

The gain is computed next in Scattering Amplification Factors.

Exercises

A neutral Kerr mode has $\omega>0$, $m>0$, and horizon-frame frequency $\widetilde\omega_H=-\omega/5$. If its signed transmission ratio is $\mathcal T=-0.03$, determine the reflection ratio and explain the horizon energy sign.

Solution

Flux balance gives

$$\mathcal R=1-\mathcal T=1.03.$$

The reflected energy flux is three percent larger than the incident flux. Because $\omega\widetilde\omega_H=-\omega^2/5<0$, the outward-positive horizon Killing-energy flux is positive even though the field is future-ingoing. Equivalently, negative $\partial_t$ energy crosses into the hole. The horizon-generator energy remains nonnegative, so causal regularity and the area law are not violated.

References

• Brito, R., Cardoso, V., and Pani, P. (2020). Superradiance: New Frontiers in Black Hole Physics. Lecture Notes in Physics 971. Springer. DOI.
• Teukolsky, S. A., and Press, W. H. (1974). “Perturbations of a rotating black hole. III. Interaction of the hole with gravitational and electromagnetic radiation.” Astrophysical Journal 193, 443–461. ADS.