Greybody Factors and Asymptotic Flux

The Hawking occupation is thermal near the horizon, but the spectrum measured at infinity is filtered by the exterior curvature potential. Each angular and spin channel has a greybody factor determined by a classical scattering problem. The occupation number and transmission factor are distinct: the former comes from horizon analyticity and the state, while the latter comes from propagation through the exterior geometry.

Helpful background. Flat-Spacetime Quantum Field Theory provides flux-normalized modes. Partial Differential Equations and Green Functions supplies one-dimensional scattering. Causality and Global Structure develops horizon boundary conditions. Quantum Fields in Curved Spacetime provides the Hawking state. Black-Hole Geometry and Mechanics supplies horizon potentials and angular velocity. Black-Hole Perturbation Theory provides separated radial equations and Wronskians.

The exterior scattering problem

For a Schwarzschild field channel labeled by spin $s$ and angular momentum $\ell,m$, a suitable radial variable obeys

$$\frac{d^2\psi_{s\ell}}{dr_*^2} +\left[\omega^2-V_{s\ell}(r)\right]\psi_{s\ell}=0.$$

For the minimally coupled massless scalar, $\Phi=r^{-1}\psi_{\ell\omega}(r)Y_{\ell m}e^{-i\omega t}$ gives

$$V_{0\ell}(r)=f(r)\left[\frac{\ell(\ell+1)}{r^2} +\frac{2GM}{r^3}\right], \qquad f(r)=1-\frac{2GM}{r}.$$

This real potential vanishes at both asymptotic ends. Its large-$\ell$ peak approaches the photon sphere, but the $\ell=0$ peak is at $r=8GM/3$; the photon-sphere description is an eikonal limit. Other spins require their own master variables and potentials. A scalar mode incident from infinity is normalized as

$$\psi\sim \begin{cases} e^{-i\omega r_*}+\mathcal R\,e^{+i\omega r_*}, & r_*\to+\infty,\\ \mathcal T\,e^{-i\omega r_*}, & r_*\to-\infty. \end{cases}$$

The conserved radial Wronskian gives

$$1-|\mathcal R|^2=|\mathcal T|^2$$

for this Schwarzschild flux normalization. The greybody factor is $\Gamma_{s\ell}=|\mathcal T|^2$. It lies between zero and one and depends on frequency, spin, angular momentum, dimension, and black-hole parameters.

This scattering equation is classical. For a real Schwarzschild potential, reciprocity equates the flux transmission of this incident-from-infinity problem to that of a wave emerging from the horizon. That is why its absorption factor can filter the outgoing Hawking occupation. The equality concerns flux-normalized channels, not arbitrary amplitudes.

Spectrum and flux at infinity

For a stationary rotating or electrically charged black hole, take the field mode to be proportional to $e^{-i\omega t+im\phi}$ and use $D_a=\nabla_a-i e_{\mathrm f}A_a$. Here $e_{\mathrm f}$ is the field’s gauge-charge eigenvalue, distinct from the hole’s geometric charge parameters. Use the normalized horizon generator and the rationalized Maxwell convention, gauge reference, and horizon-to-infinity potential defined in Horizon Area, Surface Gravity, Angular Velocity, and Potentials. The horizon-frame frequency is then

$$\widetilde\omega=\omega-m\Omega_H-e_{\mathrm f}\Phi_H.$$

Only the consistently normalized product $e_{\mathrm f}\Phi_H$ may be carried between formulas. Hawking distinguishes the field charge from the black-hole charge in Hawking 1975, §3, printed p. 213, Eqs. (3.1)–(3.3), PDF.

The asymptotic number flux has the schematic form

$$\frac{d^2N}{dt\,d\omega} =\frac{1}{2\pi} \sum_{\ell,m} \frac{\Gamma_{s\ell m}(\omega)} {\exp(\widetilde\omega/T_H)-(-1)^{2s}}.$$

Multiplying by $\omega$ gives the energy flux measured relative to asymptotic time; multiplying by $m$ or $e_{\mathrm f}$ gives angular-momentum or charge flux. The formula requires channel-dependent degeneracies and normalization conventions to be stated explicitly.

For Schwarzschild, $\widetilde\omega=\omega$ and $\Gamma\ge0$. The denominator is Bose–Einstein for integer spin and Fermi–Dirac for half-integer spin. The result is not an exact blackbody because $\Gamma_{s\ell}(\omega)$ suppresses low-frequency high-$\ell$ modes and varies across the spectrum.

For example, one real massless scalar has one polarization and $2\ell+1$ azimuthal modes. Its Schwarzschild energy spectrum is therefore

$$\frac{d^2E}{dt\,d\omega} =\frac{\omega}{2\pi} \sum_{\ell=0}^{\infty}(2\ell+1) \frac{\Gamma_{0\ell}(\omega)}{e^{8\pi GM\omega}-1}.$$

There is no extra factor of two for this real scalar. Integrating over positive $\omega$ gives its power; adding another independent field or polarization requires adding that field’s channels.

Low-frequency and Wronskian checks

For a minimally coupled massless scalar in four-dimensional Schwarzschild, write $\mu=GM$ for the gravitational length. The low-frequency absorption cross section approaches the horizon area,

$$\sigma_{\mathrm{abs}}(\omega\to0)=A_H=16\pi\mu^2.$$

Using $\sigma_{\mathrm{abs}}\simeq\pi\Gamma_0/\omega^2$ for the dominant $\ell=0$ channel gives

$$\Gamma_0\simeq16(\mu\omega)^2.$$

The area limit follows from matching a regular, ingoing near-horizon scalar solution to the low-frequency exterior solution Das, Gibbons, and Mathur 1997, Eq. (16), arXiv version p. 2, PDF. It assumes a minimally coupled massless scalar and a regular spherically symmetric horizon; it is not a universal low-frequency coefficient for every spin.

This provides an analytic boundary check for a numerical transmission code. At high frequency, summing many partial waves approaches geometric-optics capture rather than setting every individual channel to one.

Page’s numerical calculation fixes separated radial normalizations and converts channel transmission into particle and energy luminosities; see Page 1976, §§II–IV and Tables I–VI, pp. 198–206. The later article at Physical Review D 14, 3260 treats rotating holes; it is not an erratum to the nonrotating paper.

Rotation and the superradiant sign

For Kerr, the horizon flux is proportional to $\widetilde\omega$, while the incident flux at infinity is proportional to $\omega$. The Wronskian relation becomes schematically

$$1-|\mathcal R|^2 =\frac{\widetilde\omega}{\omega}|\mathcal T|^2.$$

For bosonic superradiant channels with $\widetilde\omega<0$, reflection is amplified and the absorption probability defined by net horizon flux is negative. The Hawking denominator is negative in the same regime, so their ratio gives a positive emitted number flux. Calling $\Gamma$ a probability between zero and one in this regime would lose the sign and produce an incorrect spectrum.

Fermionic channels do not exhibit the same classical superradiant amplification. Spin, statistics, angular harmonics, and chemical potentials must therefore be kept together rather than inserted into a scalar Schwarzschild formula afterward.

Scope and failure modes

A greybody calculation must verify ingoing horizon and outgoing infinity boundary conditions, the Wronskian residual, low-frequency limits, angular convergence, and positivity of the final physical flux. Numerical contamination by the exponentially dominant unwanted solution can imitate transmission. Near a superradiant threshold, dividing by a small horizon-frame frequency can also amplify roundoff.

Greybody factors do not alter the local Hawking temperature and do not evolve the black-hole mass. They provide the fixed-background map from horizon occupation to asymptotic flux. Adiabatic evaporation uses this flux in a separate, slowly varying geometry.

Exercise

For a Schwarzschild channel with $|\mathcal R|^2=0.91$, use the Wronskian to find the greybody factor. What changes in Kerr when the same calculation lies in a superradiant channel?

Solution

In Schwarzschild flux normalization, $\Gamma=1-|\mathcal R|^2=0.09$. In a Kerr superradiant channel the Wronskian includes the factor $\widetilde\omega/\omega<0$, so $|\mathcal R|^2$ exceeds one and the net absorption coefficient is negative. One must retain that sign and combine it with the negative bosonic thermal denominator; replacing the result by an absolute-value “probability” would give the wrong emitted flux.

The comparison shows why amplitude normalization alone is insufficient. Flux includes the group velocity or horizon-frame frequency, and conservation checks the signed current rather than the squared transmission amplitude by itself.

References

• Das, Sumit R., G. W. Gibbons, and Samir D. Mathur. “Universality of Low Energy Absorption Cross Sections for Black Holes.” Physical Review Letters 78 (1997): 417–419. DOI. Open PDF.
• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43 (1975): 199–220. DOI. Open PDF.
• Page, Don N. “Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole.” Physical Review D 13 (1976): 198–206. DOI.
• Page, Don N. “Particle Emission Rates from a Black Hole. II. Massless Particles from a Rotating Hole.” Physical Review D 14 (1976): 3260–3273. DOI.