The Adiabatic Evaporation Approximation

An evaporating black hole may be approximated by a sequence of stationary geometries only when its parameters change little during the time required waves and geometry to relax. For a large Schwarzschild hole this condition is parametrically strong: the light-crossing and quasinormal times scale as $GM$, whereas the fractional mass-loss time scales as $G^2M^3$. Near extremality, at large species number, or near the effective cutoff, a longer relaxation scale can invalidate the approximation well before a naive lifetime formula does.

Helpful background. Quantum Fields in Curved Spacetime supplies instantaneous Hawking spectra; Einstein Equations supplies slowly varying metric ansätze; Black-Hole Geometry and Mechanics supplies surface gravity and trapping horizons; Semiclassical Gravity supplies the backreaction expansion; and Black-Hole Dynamics supplies relaxation and flux balance.

Three competing timescales

For Schwarzschild, choose the asymptotic time associated with the unit- normalized stationary generator. The characteristic scales are

$$t_{\mathrm{cross}}\sim r_s=2GM, \qquad t_{\mathrm{relax}}\sim\kappa^{-1}=4GM, \qquad t_{\mathrm{loss}}=\frac{M}{\lvert\dot M\rvert}.$$

With

$$\dot M=-\frac{\alpha}{G^2M^2},$$

the loss time is $t_{\mathrm{loss}}=G^2M^3/\alpha$. A useful local adiabatic parameter is

$$\epsilon_{\mathrm{ad}} =\frac{\lvert\dot\kappa\rvert}{\kappa^2} =\frac{4\alpha}{GM^2}.$$

Thus $\epsilon_{\mathrm{ad}}\ll1$ for $M\gg\sqrt{\alpha/G}$. Numerical coefficients depend on which relaxation time is chosen, but the invariant content is scale separation. Auffinger’s quasistatic emission and mass-evolution construction is summarized in Auffinger 2022, §§3.1–3.4, PDF.

Building a slowly evolving geometry

At leading adiabatic order, calculate greybody factors and $\langle T_{\mu\nu}\rangle_{\mathrm{ren}}$ on a stationary geometry with parameters $\lambda^A=(M,J,Q)$. Evolve those parameters by their fluxes, then reevaluate the stationary scattering problem:

$$\dot\lambda^A =\mathcal F^A[\lambda(t)] +O(\epsilon_{\mathrm{ad}}\mathcal F^A).$$

Near a shrinking spherical horizon, an advanced-coordinate ansatz can encode ingoing negative energy,

$$ds^2 =-\left(1-\frac{2GM(v)}{r}\right)dv^2 +2\,dv\,dr+r^2d\Omega^2, \qquad \frac{dM}{dv}\lt0.$$

An outgoing retarded-coordinate description is natural near future null infinity. These patches organize different flux regions; writing one Vaidya mass function is not a substitute for solving the global semiclassical equation with a conserved stress tensor.

The construction is derivative expansion, not a fixed background over the whole lifetime. Over one relaxation time the fractional parameter change is $O(\epsilon_{\mathrm{ad}})$, but over $t_{\mathrm{loss}}$ it is order unity. One must resum that secular change by updating $M(t)$ continuously. Freezing $M=M_0$ and integrating its stationary flux for a full lifetime violates the very approximation used to obtain the flux.

State and control conditions

The intended state is Unruh-like: regular on the future horizon, no incoming thermal bath, and outgoing flux at infinity. A Hartle–Hawking state has a stationary bath and tests equilibrium rather than evaporation. At each step, the renormalized stress tensor must remain conserved on the updated geometry, and its finite counterterm convention must be held fixed.

Adiabaticity is only one control test. One also requires

• horizon curvature and emitted frequencies below the effective cutoff;
• stress-tensor fluctuations small enough for mean-field gravity;
• greybody tables resolved as parameters and particle thresholds change;
• distance from a branch point or extremal bound large compared with one evolution step;
• constraint and Bondi-flux residuals smaller than the claimed correction.

Hu and Verdaguer emphasize that mean backreaction and stress fluctuations are distinct validity questions in their black-hole discussion Hu and Verdaguer 2008, §8. Passing a timescale ratio cannot establish the other conditions.

Near-extremal and large-N failure

For Kerr or Reissner–Nordström, $\kappa^{-1}$ diverges toward extremality, and the throat or least-damped mode can relax much more slowly than $GM$. Simultaneously, superradiant, discharge, or massive-particle thresholds can change the flux rapidly. The relevant parameter is then $t_{\mathrm{longest\ relaxation}}/t_{\mathrm{loss}}$, not the Schwarzschild estimate. More precisely, near an extremal bound the small energy above extremality, $E_{\mathrm{exc}}=M-M_{\mathrm{ext}}(J,Q)$, can vary rapidly even while $M/\lvert\dot M\rvert$ is large. One must compare relaxation with $E_{\mathrm{exc}}/\lvert\dot E_{\mathrm{exc}}\rvert$ and with the variation times of $\kappa$, $\Omega_{\mathrm H}$, and $\Phi_{\mathrm H}$. A small fractional loss of total mass alone is not an adiabaticity criterion there.

If $N$ comparable light species contribute independently, $\alpha\propto N$, so the adiabatic parameter grows with $N$. The condition $GM^2\gg\alpha$ resembles the species-lowered validity scale. It must be combined with the renormalization of $G$ and the gravitational cutoff rather than treated as a species-independent mass bound.

Near $M\sim G^{-1/2}$, curvatures become Planckian and the derivative, semiclassical, and loop expansions can fail together. The formal cubic solution may still be integrated algebraically, but it no longer predicts a physical endpoint.

Independent scaling check

For a constant-$\alpha$ Schwarzschild history, the ratio of formal remaining lifetime to light-crossing time is

$$\frac{t_{\mathrm{evap}}}{t_{\mathrm{cross}}} \sim\frac{GM^2}{6\alpha}.$$

It is large in precisely the same regime where $\epsilon_{\mathrm{ad}}$ is small. This independent lifetime comparison recovers the local derivative criterion up to order-one constants. Both fail parametrically when $GM^2$ approaches $\alpha$.

Exercise

Using $\epsilon_{\mathrm{ad}}=4\alpha/(GM^2)$, find the mass $M_{\mathrm a}$ at which this parameter reaches one. For an initial $M_0\gg M_{\mathrm a}$ and constant $\alpha$, what fraction of the formal cubic lifetime remains at that point?

Solution

Setting $\epsilon_{\mathrm{ad}}=1$ gives

$$M_{\mathrm a}=2\sqrt{\frac{\alpha}{G}}.$$

The formal lifetime remaining from mass $M$ is $G^2M^3/(3\alpha)$. Relative to the initial formal lifetime,

$$\frac{t_{\mathrm{remaining}}(M_{\mathrm a})} {t_{\mathrm{evap}}(M_0)} =\left(\frac{M_{\mathrm a}}{M_0}\right)^3 =\frac{8\alpha^{3/2}}{G^{3/2}M_0^3}.$$

This fraction is tiny for a very massive initial hole, but that does not license extrapolation through it. Once $\epsilon_{\mathrm{ad}}$ is order one, the instantaneous stationary spectrum and cubic law have lost their controlled error estimate. If $\alpha$ is small, $M_{\mathrm a}$ can even lie below the Planck mass, in which case curvature or quantum-gravity control fails before this formal adiabatic threshold is reached.

References

• Auffinger, J. (2022), “Primordial Black Hole Constraints with Hawking Radiation—A Review,” arXiv:2206.02672v1. Open PDF.
• Hu, B. L., and E. Verdaguer (2008), “Stochastic Gravity: Theory and Applications,” Living Reviews in Relativity 11, 3. DOI.