Luminosity and Mass-Loss Laws

Hawking temperature sets the characteristic emission energy, but the luminosity is not that of a perfect blackbody. Curvature outside the horizon filters every spin and partial wave through a greybody transmission factor. Summing the transmitted Unruh-state spectrum gives the outward energy flux; only after that fixed-background calculation is known may it be fed into slowly evolving mass, angular-momentum, and charge balance laws. For a neutral Schwarzschild hole the resulting lifetime scales as $G^2M_0^3$.

Helpful background. Quantum Fields in Curved Spacetime supplies the Unruh state and Hawking spectrum; Einstein Equations supplies the stationary metric family; Black-Hole Geometry and Mechanics supplies horizon potentials; Semiclassical Gravity supplies the renormalized stress tensor; and Black-Hole Dynamics supplies flux and horizon-balance laws.

Greybody-filtered particle spectra

Let $\chi=\partial_t+\Omega_{\mathrm H}\partial_\phi$ be the horizon generator, with $\partial_t$ normalized to unit time translation at infinity. Choose the gauge-invariant horizon potential

$$\Phi_{\mathrm H} =-\left.\chi^\mu A_\mu\right|_{\mathrm H} +\left.\chi^\mu A_\mu\right|_{\infty}.$$

For a species $i$ of spin $s_i$, charge $q_i$, and mass $\mu_i$, the instantaneous stationary emission rate in one partial wave is

$$\mathcal N_{i\ell m}(\omega) \equiv\frac{d^2N_{i\ell m}}{dt\,d\omega} =\frac{g_i}{2\pi} \frac{\Gamma_{i\ell m}(\omega;M,J,Q)} {\exp[(\omega-m\Omega_{\mathrm H}-q_i\Phi_{\mathrm H})/T_{\mathrm H}] -(-1)^{2s_i}} \Theta(\omega-\mu_i).$$

The species rate is

$$\frac{d^2N_i}{dt\,d\omega}=\sum_{\ell,m}\mathcal N_{i\ell m}.$$

The factor $g_i$ counts only internal or polarization degeneracies not already included in the mode labels. Here $\Gamma_{i\ell m}$ is the signed absorption probability, equivalently the greybody factor. Its scattering problem has a purely ingoing solution at the future horizon and an incident plus reflected wave at infinity; it is the ratio of absorbed to incident conserved flux. Purely outgoing conditions at infinity instead define a resonance problem and must not be used to normalize absorption. The greybody factor contains the potential-barrier physics that a Stefan–Boltzmann estimate omits. Auffinger records the emission formula, chemical potentials, and greybody definition in Auffinger 2022, §§3.1 and 3.3, Eqs. (15)–(20), PDF.

For an uncharged nonrotating hole the chemical potentials vanish and $\Gamma\geq0$. In rotating bosonic superradiant channels, the signed absorption probability and thermal denominator must be treated together; one must not replace them independently by positive blackbody factors.

Luminosity and balance laws

Weighting the spectrum by conserved quantum numbers gives asymptotic fluxes

$$\begin{aligned} L_\infty&=\sum_{i,\ell,m}\int_0^\infty d\omega\, \omega\,\mathcal N_{i\ell m}(\omega),\\ \mathcal J_\infty&=\sum_{i,\ell,m}\int_0^\infty d\omega\, m\,\mathcal N_{i\ell m}(\omega),\\ \mathcal Q_\infty&=\sum_{i,\ell,m}\int_0^\infty d\omega\, q_i\,\mathcal N_{i\ell m}(\omega). \end{aligned}$$

In an asymptotically flat Unruh-state problem with no incoming bath, conservation gives

$$\dot M=-L_\infty, \qquad \dot J=-\mathcal J_\infty, \qquad \dot Q=-\mathcal Q_\infty,$$

after including every radiative and discharge channel in the definitions. The factor $m$ belongs inside the mode sum: different azimuthal modes carry different angular momentum even at the same $\omega$. For massless radiation these can be formulated as Bondi-time balances at future null infinity. Massive quanta instead reach timelike infinity; their contribution is an asymptotic energy/charge flux across a large timelike extraction surface in the quasistationary loss model, not a flux through future null infinity. Neither form is a local equation at the horizon. Exterior energy stored between the horizon and the extraction surface must be included beyond the stationary-flux approximation.

Page’s primary Kerr calculation evaluates the spin-dependent partial-wave transmission problem and mass/angular-momentum rates Page 1976, §§II–V. Numerical coefficients depend on particle content, masses, and thresholds; they are not universal constants of geometry.

Schwarzschild scaling and lifetime

For Schwarzschild,

$$T_{\mathrm H}=\frac{1}{8\pi GM}, \qquad A=16\pi G^2M^2.$$

A dimensional blackbody estimate gives $AT_{\mathrm H}^4\propto 1/(G^2M^2)$. The exact greybody sum preserves this mass scaling for fixed effectively massless species but changes its coefficient:

$$\frac{dM}{dt} =-\frac{\alpha_{\mathrm{eff}}(M)}{G^2M^2}.$$

The dimensionless $\alpha_{\mathrm{eff}}$ grows when $T_{\mathrm H}$ crosses a particle-mass threshold and can vary with running couplings or new degrees of freedom. If it is approximately constant over the interval of interest,

$$M(t)^3=M_0^3-\frac{3\alpha_{\mathrm{eff}}}{G^2}t, \qquad t_{\mathrm{evap}} =\frac{G^2M_0^3}{3\alpha_{\mathrm{eff}}}.$$

The second expression is the formal extrapolation to $M=0$. The controlled prediction stops at a chosen mass $M_{\mathrm{stop}}$ while the approximations still hold: $t_{\mathrm{stop}}=G^2(M_0^3-M_{\mathrm{stop}}^3)/(3\alpha_{\mathrm{eff}})$.

Auffinger derives the evolution integrals and cubic lifetime scaling in Auffinger 2022, §§3.3–3.4, Eqs. (24)–(26), PDF. The result explains how area, temperature, species, and greybody factors enter one timescale: $A$ and $T$ fix the $M^{-2}$ scaling, while the transmitted species sum fixes $\alpha_{\mathrm{eff}}$.

State, approximation, and checks

The state is Unruh-like: no incoming thermal flux from past null infinity, regular future-horizon behavior, and outgoing flux at future null infinity. A Hartle–Hawking equilibrium state instead has compensating incoming radiation and no net mass loss. This state check prevents a thermal spectrum from being mistaken for evaporation.

The greybody spectrum is initially a fixed-background QFT result. Promoting $M,J,Q$ to slowly varying parameters assumes that the light-crossing and relaxation times are short compared with their loss times. Curvatures and typical frequencies must remain below the effective cutoff, and changes during one emission timescale must be small. The first omitted effects include nonadiabatic mode mixing, the self-consistent metric response, correlations between emissions, and higher-loop corrections.

Dimensional analysis checks the mass law: $[G^2M^2]=\text{length}^2$, so its inverse has the units of energy per time. The sign check is equally direct: positive $L_\infty$ decreases Bondi mass. Near the Planck regime, or near an extremal throat with a new long timescale, the cubic extrapolation is not controlled and makes no endpoint prediction. It is a leading history only while every instantaneous spectrum remains well approximated by the corresponding stationary scattering problem.

Exercise

Assume one effectively massless species set gives constant $\alpha_{\mathrm{eff}}=\alpha$. Find the time to evolve from $M_0$ to $M_0/2$. What fraction of the formal total lifetime has elapsed, and how does doubling all independent species change the answer if every greybody spectrum is identical?

Solution

Integrating the mass law gives

$$t(M) =\frac{G^2}{3\alpha}\left(M_0^3-M^3\right).$$

At $M=M_0/2$,

$$t_{1/2} =\frac{G^2M_0^3}{3\alpha}\left(1-\frac18\right) =\frac78\,t_{\mathrm{evap}}.$$

Thus losing half the mass uses seven eighths of the formal constant-species lifetime; the final half is lost much faster because luminosity grows as $M^{-2}$. Doubling identical independent species doubles $\alpha$, so both $t_{1/2}$ and $t_{\mathrm{evap}}$ are halved. Thresholds or unequal greybody factors would invalidate that simple linear species scaling.

References

• Auffinger, J. (2022), “Primordial Black Hole Constraints with Hawking Radiation—A Review,” arXiv:2206.02672v1. Open PDF.
• Page, D. N. (1976), “Particle Emission Rates from a Black Hole. II. Massless Particles from a Rotating Hole,” Physical Review D 14, 3260–3273. DOI.