Black-Hole Quantum Fields and Hawking Radiation

Quantum fields on a black-hole background combine horizon state selection, exponential redshift, exterior scattering, and renormalized stress–energy. The Hawking temperature is fixed near the horizon, the flux at infinity is filtered by greybody factors, and horizon regularity distinguishes the physical collapse state. Every result in this chapter holds on a prescribed or adiabatically approximated geometry; determining the metric evolution self-consistently requires the backreaction methods introduced at the end.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and stress renormalization. Black-Hole Geometry and Mechanics provides horizons, surface gravity, and Killing energy. Black-Hole Perturbation Theory supplies radial scattering and Wronskians. Causality and Global Structure develops collapse horizons and null infinity.

Enter this chapter

The recurring geometry is a stationary late exterior with horizon generator $\chi$ and surface gravity $\kappa$, normalized to the asymptotic time translation. The corresponding temperature is

$$T_H=\frac{\kappa}{2\pi}.$$

That formula does not by itself select a state, specify the spectrum at infinity, or evolve the mass. The chapter supplies those distinctions in this order:

1. Boulware, Hartle–Hawking, and Unruh States compares boundary populations and past/future horizon regularity.
2. Hawking Radiation from Gravitational Collapse derives the exponential affine-coordinate map and Planck factor.
3. Near-Horizon Universality states the Hadamard, adiabatic, and dispersion assumptions behind robustness.
4. Greybody Factors and Asymptotic Flux propagates horizon occupation through the exterior potential.
5. Renormalized Stress–Energy Near Horizons checks conservation, anomaly, state constants, and regular frames computationally.
6. Fixed-Background Limits and Backreaction identifies when cumulative mass loss or fluctuations require a dynamical metric.

For the next physical questions, continue to luminosity and mass-loss laws and criteria for self-consistent evaporation. Those pages distinguish an evolving mass approximation from a coupled state-and-geometry calculation. Questions about the radiation’s correlations continue to fine-grained versus coarse-grained information, with the quantum-information preparation it requires. A flux calculation, an evaporation history, and an information-recovery claim therefore have separate next steps.

State, redshift, and temperature

The Boulware state is empty in static Killing modes and singular at a horizon. The Hartle–Hawking state is regular on both horizons of a static bifurcate geometry and describes thermal equilibrium with equal incoming and outgoing populations. The Unruh state is empty in the incoming sector at past null infinity, regular on the future horizon, and carries outgoing late flux; it is the stationary idealization appropriate to collapse.

Collapse supplies the key map between an early affine coordinate $U$ and late retarded time $u$:

$$U=U_H-Ae^{-\kappa u}+\cdots.$$

Tracing $e^{-i\omega u}$ backward produces a logarithmic branch point at $U_H$. Fourier analysis gives the ratio $|\beta|^2/|\alpha|^2=e^{-2\pi\omega/\kappa}$ and hence the Bose factor. The derivation, wave-packet normalization, and correction are given in Hawking 1975, §2, Eqs. (2.16)–(2.29), pp. 208–211.

The proof architecture is not a local “pair appears at the horizon” picture. It combines a regular in-state, causal ray tracing, positive frequency with respect to affine coordinates, analytic continuation, and the Bogoliubov norm identity. A non-Hadamard in-state or an exactly extremal horizon is a counterexample to applying this standard chain without new analysis.

From the horizon to infinity

The near-horizon Planck factor is not the final spectrum. A separated radial mode encounters a curvature potential, and its conserved Wronskian defines a channel greybody factor. For a nonrotating hole,

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

Spin, angular momentum, and frequency shape $\Gamma$. For a stationary rotating or electrically charged hole, take the field mode as $e^{-i\omega t+im\phi}$ and write its covariant derivative as $D_a=\nabla_a-i e_{\mathrm f}A_a$. With the normalized horizon generator and horizon-to-infinity potential used in Horizon Area, Surface Gravity, Angular Velocity, and Potentials, the horizon-frame energy becomes $\omega-m\Omega_H-e_{\mathrm f}\Phi_H$. Here $e_{\mathrm f}$ is the field’s gauge-charge eigenvalue, not a geometric charge parameter of the hole. Hawking states the rotating/charged combination in Hawking 1975, §3, printed p. 213, Eqs. (3.1)–(3.3), PDF. A bosonic superradiant absorption coefficient can be negative; its sign must be retained so the final emitted flux remains positive. Page’s uncharged Schwarzschild calculation and tabulated channel luminosities are in Page 1976, §§II–IV and Tables I–VI, pp. 198–206.

Near-horizon universality is similarly qualified. The Hadamard scaling limit and exponential peeling suppress many collapse details, while suitable modified-dispersion models recover the same leading temperature under ground-state, stability, and adiabaticity conditions. Exterior greybody factors, transients, species, and boundary conditions remain nonuniversal.

Stress–energy and energy bookkeeping

Particle occupation is not enough to source geometry. The renormalized tensor must be conserved, have the correct anomaly, and be regular in a nonsingular frame. The chapter’s intrinsic $1+1$ benchmark solves

$$T_{uu}=H+t_u,\qquad T_{vv}=H+t_v,$$

and reproduces the Unruh flux $F=\kappa^2/(48\pi)$. At infinity $T_{uu}\to F$; at the future horizon $T_{vv}\to-F$. The conserved outward Killing current links positive radiation energy with negative energy crossing the horizon. Candelas’s four-dimensional state and stress analysis is in Candelas 1980, §§II–IV, pp. 2185–2202.

The numerical benchmark is deliberately limited. It validates state constants, anomaly sign, conservation, coordinate regularity, and convergence, not four-dimensional luminosity. A four-dimensional calculation needs angular modes, a declared Hadamard subtraction, finite counterterms, and a regular tetrad.

The fixed-background boundary

For a large black hole, mass changes slowly per crossing time, so a stationary or adiabatic approximation can be accurate locally. Yet a tiny flux integrated over the evaporation time causes order-one mass loss. Fixed geometry fails when local quantum stress is large, cumulative mass change is appreciable, or fluctuations make the mean stress unrepresentative.

The next theory must solve a self-consistent semiclassical metric equation with matching renormalization conventions. Fine-grained radiation entropy and microscopic endpoints require still different observables. This chapter supplies the field-theory inputs but does not claim those downstream results.

Review the chapter

A calculation reports a Planck occupation at the horizon and calls it the luminosity and evaporation law. List the missing steps and identify which remain within this chapter.

Solution

First select a state and verify future-horizon regularity; the collapse problem calls for the Unruh choice. Next solve the radial scattering problem and multiply each channel by its signed greybody factor to obtain the asymptotic particle and energy flux. Then compute or check a conserved renormalized stress tensor in a regular frame. Those steps belong here. Turning the flux into a changing mass and horizon requires an adiabatic or self-consistent semiclassical metric, while an endpoint or information-recovery claim requires additional downstream theory.

References

• Candelas, Philip. “Vacuum Polarization in Schwarzschild Spacetime.” Physical Review D 21 (1980): 2185–2202. DOI.
• Fredenhagen, Klaus, and Rudolf Haag. “On the Derivation of Hawking Radiation Associated with the Formation of a Black Hole.” Communications in Mathematical Physics 127 (1990): 273–284. DOI.
• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43 (1975): 199–220. DOI.
• 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.