Flux Through Black-Hole Horizons

A black-hole horizon is an inner radiative boundary. Perturbations regular on the future horizon can transport energy and angular momentum into it; for a rotating hole, a superradiant mode instead carries negative asymptotic Killing energy into the horizon while extracting spin. The invariant positive quantity is not the horizon’s $t$-energy alone but the flux associated with its future generator. This page develops that sign structure for linear perturbations of a stationary, nonextremal, asymptotically flat black hole and for quasi-stationary changes between nearby stationary states.

Helpful background. Einstein Equations supplies the Raychaudhuri and constraint equations; Waves, Green Functions, and Geometric Optics supplies characteristic flux and regularity; Gravitational Radiation and Waveforms supplies the $e^{-i\omega t}$ convention; Causality and Global Structure supplies future event horizons; and Black-Hole Geometry and Mechanics supplies Killing generators, surface gravity, area, and horizon charges.

Future-horizon orientation and regularity

Let $t^a$ and $\phi^a$ be the stationary and axial Killing fields, with $\phi^a$ oriented toward increasing azimuth. The future Kerr-horizon generator is

$$\chi^a=t^a+\Omega_H\phi^a, \qquad \chi^b\nabla_b\chi^a=\kappa\chi^a,$$

where $\Omega_H$ is the horizon angular velocity and $\kappa\gt0$ the surface gravity in the chosen asymptotic time normalization. Orient the axial coordinate so that $\Omega_H\ge0$. Flux “through the horizon” will mean directed into the future event horizon. It is an inner boundary of the exterior region, so a positive inward horizon flux appears with a minus sign in an exterior-source balance.

With time and azimuthal dependence $e^{-i\omega t+im\phi}$, regularity requires the ingoing solution in a future-horizon-penetrating coordinate system. For a nonrotating hole this dependence is $e^{-i\omega v}$ near the horizon; the outgoing solution is not smooth there. The regular ingoing boundary condition and its relation to retarded black-hole response are set out in Berti, Cardoso, and Starinets 2009, §3.1, pp. 17–18, PDF. For a rotating horizon, the phase measured along $\chi^a$ has frequency

$$\widetilde\omega_H=\omega-m\Omega_H.$$

This corotating frequency, together with $\omega$, controls the sign of the asymptotic Killing-energy flux. The future-generator flux itself is nonnegative for the classical perturbations considered here.

Shear, expansion, and area growth

Let $v$ parametrize the generators with the normalization above, and let $\theta$ and $\sigma_{ab}$ be their expansion and shear. The nonaffine null Raychaudhuri equation is

$$\frac{d\theta}{dv} =\kappa\theta-\frac12\theta^2 -\sigma_{ab}\sigma^{ab} -8\pi G\,T_{ab}\chi^a\chi^b.$$

For a weak gravitational perturbation, shear is first order and expansion begins at second order; include matter whose $T_{ab}\chi^a\chi^b$ is of that same retained order. Then $\theta^2$ is higher order, and $\kappa$ can be held at its background value. Imposing the event-horizon final condition $\theta\to0$ after the perturbation gives

$$\theta(v) =\int_v^\infty e^{-\kappa(v'-v)} \left[ \sigma_{ab}\sigma^{ab} +8\pi G\,T_{ab}\chi^a\chi^b \right]_{v'}dv'.$$

Thus shear and positive matter generator energy make $\theta\ge0$ and $dA/dv=\int\theta\,dA\ge0$. The future boundary condition displays the familiar teleology of an event horizon: its expansion is not determined by data on one cut alone. A dynamical or trapping horizon uses different local flux laws and must not be substituted without translating the horizon definition and orientation.

Because shear is first order while the absorbed gravitational energy is quadratic, horizon heating is a second-order effect in a linear waveform amplitude. The perturbation ordering, future boundary condition, and required averaging are derived in Poisson 2004, §IV A–B, eqs. (4.1)–(4.13), pp. 10–11, PDF. For Schwarzschild, $\Omega_H=0$, so positive generator energy is positive asymptotic energy absorption.

The instantaneous area derivative must still be distinguished from a local flux. Define the positive generator-power density integrated over a horizon cut by

$$\mathcal P_\chi(v) =\int\left[\frac{\sigma_{ab}\sigma^{ab}}{8\pi G} +T_{ab}\chi^a\chi^b\right]dA.$$

Integrating Raychaudhuri over the cut, at this perturbative order, gives

$$\mathcal P_\chi =\frac{1}{8\pi G}\left( \kappa\frac{dA}{dv}-\frac{d^2A}{dv^2}\right).$$

An event horizon can begin expanding before a pulse arrives, so $\kappa\dot A/(8\pi G)$ is not the instantaneous absorbed power. Integrating between stationary endpoint limits eliminates the $\ddot A$ term and gives $\int\mathcal P_\chi dv=\kappa\Delta A/(8\pi G)$. Averaging away bounded rapid variations, or assuming area growth changes on a timescale $\tau\gg\kappa^{-1}$, likewise permits the familiar rate form.

Energy, angular momentum, and superradiance

For the rest of this section, use averaged rates for a quasi-stationary transition between nearby Kerr states:

$$\mathcal F_E^H=\left\langle\frac{dM_H}{dv}\right\rangle, \qquad \mathcal F_J^H=\left\langle\frac{dJ_H}{dv}\right\rangle.$$

Equivalently these are endpoint changes divided by the averaging interval; no unique instantaneous mass of a dynamical event-horizon cut is assumed. The first law between the stationary endpoints gives $\Delta M_H-\Omega_H\Delta J_H=\kappa\Delta A/(8\pi G)$, hence

$$\mathcal F_E^H-\Omega_H\mathcal F_J^H =\frac{\kappa}{8\pi G}\left\langle\frac{dA}{dv}\right\rangle\ge0.$$

For a real-frequency separated mode with $\omega\ne0$, the averaged energy and angular-momentum fluxes obey

$$\mathcal F_J^H=\frac{m}{\omega}\mathcal F_E^H.$$

These relations, with the same $e^{-i\omega t+im\phi}$ phase, appear in Poisson 2004, §IV C, eqs. (4.14)–(4.18), pp. 11–12, PDF. For $\omega\gt0$ and nonzero transmitted amplitude away from threshold, the horizon energy flux has the sign of $\omega-m\Omega_H$:

$$\mathcal F_E^H \begin{cases} >0, & \omega>m\Omega_H,\\ =0, & \omega=m\Omega_H,\\ <0, & 0<\omega<m\Omega_H\quad(m>0). \end{cases}$$

The last line is superradiant extraction. Both $M_H$ and $J_H$ decrease, yet $\mathcal F_E^H-\Omega_H\mathcal F_J^H$ remains nonnegative and the horizon area does not decrease. At threshold, a regular finite-amplitude separated perturbation is stationary along the generator, so its shear and dissipative flux vanish; this zero does not follow by dividing the first-law relation by $\omega-m\Omega_H$. This sign statement assumes a stationary Killing frequency and a scattering or sufficiently narrow-band mode; a rapidly evolving merger horizon has no exact global $\omega$ or $\Omega_H$ with which to make the same mode-by-mode claim.

The threshold is also sensitive to normalization in exactly one controlled way. Rescaling the asymptotic time changes $\omega$, $\Omega_H$, $\kappa$, and the Killing energy together, but leaves the dimensionless comparison $\omega-m\Omega_H$ relative to that normalization consistent. By contrast, changing the sign of $m$, reversing the axial Killing field, or switching from $e^{-i\omega t}$ to $e^{+i\omega t}$ without translating the whole convention reverses the apparent criterion. A trustworthy calculation states all three choices.

For wave packets, the horizon flux is obtained spectrally or directly from a regular horizon field. Individual frequency components can have different signs; the integrated result need not be inferred from a single central frequency. Near extremality, where $\kappa$ is small, the quasi-stationary relation between generator flux and area requires extra care, and the nonextremal derivation above should not be extrapolated by simply setting $\kappa=0$.

Joining the boundaries

With $\mathcal F_E^\infty$ defined positive outward at $\mathscr I^+$ and $\mathcal F_E^H$ defined positive into the future horizon, an exterior-source energy balance reads

$$\frac{dE_{\rm source}}{dt} =-\mathcal F_E^\infty-\mathcal F_E^H \quad\text{up to stored-field and nonadiabatic terms}.$$

The angular-momentum balance has the same orientation. Here the source budget excludes the black holes’ evolving mass and spin; absorption is an internal transfer if those are included in the total Bondi system. A negative superradiant horizon term returns Killing energy to the exterior and can partly offset radiation escaping to infinity. Fluxes computed from Teukolsky variables, metric shear, or curvature scalars agree only after their tetrad normalization, horizon time, field normalization, and mode amplitudes are translated.

Exercise

For a Kerr mode with $m>0$ and $0<\omega<m\Omega_H$, suppose the horizon generator flux

$$\mathcal P_\chi =\mathcal F_E^H-\Omega_H\mathcal F_J^H$$

is positive, with the fluxes averaged as above. Use $\mathcal F_J^H=(m/\omega)\mathcal F_E^H$ to determine the signs of $\mathcal F_E^H$, $\mathcal F_J^H$, and $\langle dA/dv\rangle$.

Solution

Substitution gives

$$\mathcal P_\chi =\mathcal F_E^H\left(1-\frac{m\Omega_H}{\omega}\right).$$

The parenthesis is negative in the stated frequency range. Since $\mathcal P_\chi>0$, one must have $\mathcal F_E^H<0$. Because $m/\omega>0$, also $\mathcal F_J^H<0$: the hole loses mass and angular momentum. Nevertheless

$$\left\langle\frac{dA}{dv}\right\rangle =\frac{8\pi G}{\kappa}\mathcal P_\chi>0.$$

Superradiance extracts rotational Killing energy without violating the area increase implied by positive generator flux.

References

• Berti, Emanuele, Vitor Cardoso, and Andrei O. Starinets. “Quasinormal Modes of Black Holes and Black Branes.” Classical and Quantum Gravity 26 (2009): 163001. DOI. Open PDF.
• Poisson, Eric. “Absorption of Mass and Angular Momentum by a Black Hole: Time-Domain Formalisms for Gravitational Perturbations, and the Small-Hole/Slow-Motion Approximation.” Physical Review D 70 (2004): 084044. DOI. Open PDF.