Evaporating Backgrounds and Adiabatic Backreaction

A slowly evaporating black hole is neither stationary nor automatically outside the Hawking approximation. A wavepacket can see an approximately constant peeling rate when its oscillation time is short compared with the evolution time, yet long enough to resolve a spectrum. That double inequality licenses an instantaneous Hawking form with derivative corrections; it does not solve the semiclassical Einstein equation.

Required background. Radiation from gravitational collapse supplies the affine ray map, and surface gravity and redshift fixes the stationary normalization to which the evolving result is compared.

Helpful background. Stress-tensor flux in two-dimensional reductions gives a tractable flux diagnostic, and adiabaticity and Stokes phenomena supplies the derivative expansion.

Peeling rate on an evolving geometry

Let $v=p(u)$ map an outgoing ray at future retarded time $u$ to an affine coordinate on the initial null surface. Define the time-dependent peeling function

$$\kappa(u)=-\frac{p''(u)}{p'(u)}.$$

This definition is kinematic and exact wherever $p'(u)>0$. It integrates to

$$p'(u)=C\exp\!\left[-\int^u\kappa(s)\,ds\right].$$

Constant $\kappa$ reproduces the exponential Hawking map. For a slowly changing function, the local expansion about $u_0$ is

$$\log p'(u)=\log p'(u_0)-\kappa_0\Delta u -\frac12\dot\kappa_0(\Delta u)^2+\cdots,$$

where $\Delta u=u-u_0$. The natural first control parameter is

$$\epsilon_\kappa(u_0)= \frac{|\dot\kappa_0|}{\kappa_0^2}\ll1.$$

Higher accuracy also requires dimensionless higher derivatives such as $|\ddot\kappa|/\kappa^3$ to be small in the chosen interval. Kinoshita and Tanahashi derive a near-equilibrium temperature from a saddle-point Bogoliubov analysis under slow-evolution hypotheses (2012, §§ II–IV); Barceló, Liberati, Sonego, and Visser formulate the corresponding approximately exponential ray-map criterion (2011, §§ 2–5).

An outgoing Vaidya-like metric illustrates the geometry without claiming self-consistency:

$$ds^2=\left(1-\frac{2M(u)}{r}\right)du^2 +2\,du\,dr-r^2d\Omega_2^2,$$

with $\dot M<0$. In a quasistatic Schwarzschild comparison,

$$\kappa(u)\simeq\frac{1}{4M(u)}.$$

This $\kappa$ is an approximate local parameter tied to the selected ray map and normalization. In a genuinely dynamical spacetime, event, trapping, and apparent horizons need not coincide, so one must not attach the same formula to every horizon definition.

The structure map now has time-dependent entries. A state transported from collapse and an evolving ray map precede the packet-level thermal approximation; time-dependent scattering must be evaluated over the same window.

Adiabatic bridge from fixed-background Hawking radiation to slow evolution. The diagram is schematic and not to scale; each packet has its own interval of approximate stationarity, and flux feedback is not yet a self-consistent geometry.

The failure map asks whether a spectral resolution time fits inside the evolution time. If it does not, an instantaneous temperature is not an observable approximation for that packet.

Timescale test for an evolving horizon. This schematic, not-to-scale map requires state transport, slow peeling, matching scattering data, and a packet window before it licenses an instantaneous spectral claim.

The packet window

A packet centered at $u_0$ with duration $\Delta u$ resolves the local Hawking scale only if it contains several oscillation times,

$$\kappa_0\Delta u\gg1.$$

During the same packet, the fractional change of $\kappa$ must be small,

$$\frac{|\dot\kappa_0|\Delta u}{\kappa_0}\ll1.$$

Together,

$$\frac{1}{\kappa_0}\ll\Delta u \ll\frac{\kappa_0}{|\dot\kappa_0|}.$$

Such a window exists parametrically exactly when $\epsilon_\kappa\ll1$. Write $\Delta u=c/\kappa_0$ with $1\ll c\ll1/\epsilon_\kappa$. Then the fractional drift during the packet is bounded by

$$\frac{|\delta\kappa|}{\kappa_0} \simeq c\epsilon_\kappa.$$

This is the useful uncertainty estimate: derivative corrections to a packet spectrum are at most parametrically small in $c\epsilon_\kappa$, with coefficients depending on the packet, state, scattering potential, and definition of the extracted spectrum. There is no frequency-independent “instantaneous temperature” at arbitrary time resolution.

Application: estimate the first adiabatic correction

Choose a smooth packet $f(u-u_0)$, record its $c=\kappa_0\Delta u$, and compare two calculations:

1. freeze the geometry and radial potential at $u_0$, using $T_0=\kappa_0/(2\pi)$;
2. use the quadratic ray-map expansion containing $\dot\kappa_0$ and evolve the scattering data across the same interval.

For the frozen spectrum

$$n_0(\omega)=\frac{1}{e^{2\pi\omega/\kappa_0}-1},$$

a small shift $\delta\kappa$ changes the occupation by

$$\frac{\delta n}{n_0} =\frac{2\pi\omega}{\kappa_0} \frac{e^{2\pi\omega/\kappa_0}} {e^{2\pi\omega/\kappa_0}-1} \frac{\delta\kappa}{\kappa_0} +\text{nonthermal derivative terms}.$$

Using $|\delta\kappa|/\kappa_0\lesssim c\epsilon_\kappa$ gives a reproducible scale for the thermal-drift part. The nonthermal part is obtained from the full packet transform and must be reported separately; it cannot be absorbed into $T_0$ at all frequencies. Multiply by a time-dependent greybody factor only after checking that its variation is also small over $\Delta u$.

For the quasistatic Schwarzschild estimate $L=-\dot M=a/M^2$ in Planck units,

$$\epsilon_\kappa =\frac{|\dot\kappa|}{\kappa^2} =4|\dot M| =\frac{4a}{M^2}.$$

Thus a macroscopic mass supplies a broad adiabatic window when the fixed-background luminosity coefficient $a$ is small compared with $M^2$. This is an estimate within the assumed mass-loss law, not a derivation of that law.

The backreaction boundary

Computing $\langle T_{uu}\rangle_{\rm ren}$ on a prescribed $M(u)$ is a fixed-background calculation. Updating the parameter through

$$\dot M(u)=-L[M(u)]$$

is a quasistatic balance model. A self-consistent solution instead requires a state $\omega[g]$ evolved on the same geometry and the renormalized equation

$$G_{\mu\nu}[g]+\Lambda g_{\mu\nu} +\text{renormalized curvature terms} =8\pi G\,\langle T_{\mu\nu}[g]\rangle_{\omega,\rm ren},$$

together with constraints, conservation, initial data, and controlled higher-derivative terms. The conceptual and computational gap between these levels remains material in current semiclassical-gravity assessments (del Río 2025, §§ 2–5). A slow mass update is useful input to that problem, not its solution.

Adversarial test: collapse the timescale hierarchy

Set $\Delta u\sim1/\kappa_0$ to gain time localization and simultaneously take $\epsilon_\kappa=O(1)$. The packet then barely resolves a thermal frequency while $\kappa$ changes by order unity. Fitting a Planck curve can return a number, but the quadratic and higher terms in $\log p'$ are as important as the linear term. Different packet shapes and centers need not agree on that number.

The strongest surviving claim is a nonstationary particle-production calculation for the chosen state and ray map. It is no longer an instantaneous Hawking spectrum. If the geometry itself came only from inserting a guessed $M(u)$, even that result remains fixed-background evidence.

Domain and failure conditions

See the chapter domain and failure-conditions table. This page licenses packet-level adiabatic spectra only when the double timescale window exists, the state remains regular, the peeling definition is stated, and scattering varies slowly on the same interval. It excludes the Planckian endpoint, rapid accretion or merger, uncontrolled horizon interchange, and self-consistent backreaction. Evidence was checked through 10 August 2026; newer proposed global evaporation solutions require their own equation, state, renormalization, and stability checks.

Exercise

Show that the quasistatic Schwarzschild evaporation time is much longer than the emission time for $M^2\gg a$.

Solution

The emission time is $\tau_H\sim\kappa^{-1}=4M$. From $|\dot M|=a/M^2$, the mass-evolution time is $\tau_M=M/|\dot M|=M^3/a$. Therefore

$$\frac{\tau_H}{\tau_M}=\frac{4a}{M^2}=\epsilon_\kappa.$$

The hierarchy $\tau_H\ll\tau_M$ is exactly the adiabatic condition $M^2\gg4a$.

Handoff

Extremal and late-time limits examines a different failure of uniformity: $\kappa\to0$ removes the very scale used to organize the packet window.

References

• Barceló, Carlos, Stefano Liberati, Sebastiano Sonego, and Matt Visser. “Hawking-like Radiation from Evolving Black Holes and Compact Horizonless Objects.” Journal of High Energy Physics 2011, no. 2 (2011): 003. doi:10.1007/JHEP02(2011)003.
• del Río, Adrián. “The Backreaction Problem for Black Holes in Semiclassical Gravity.” General Relativity and Gravitation 57 (2025): 30. doi:10.1007/s10714-025-03352-x.
• Kinoshita, Shunichiro, and Norihiro Tanahashi. “Hawking Temperature for Near-Equilibrium Black Holes.” Physical Review D 85 (2012): 024050. doi:10.1103/PhysRevD.85.024050.