Horizons and Hawking Radiation

Hawking radiation connects horizon geometry, quantum-state regularity, and wave propagation to observable radiation. A near-horizon thermal relation is only one part of the calculation; exterior scattering and state choice determine the measured flux.

Enter this chapter

Begin with Causal, Killing, Trapping, and Apparent Horizons: The QFT Interface to distinguish causal, Killing, and trapping horizons before assigning quantum observables.

Helpful background. Unruh Effect and Uniformly Accelerated Detectors and Ground, KMS, and Symmetry-Selected States develop the definitions used by the chapter’s opening pages.

Topics

1. Causal, Killing, Trapping, and Apparent Horizons: The QFT Interface — Which causal, event, Killing, trapping, apparent, and cosmological horizon notions supply the geometric data actually used in a quantum-field calculation?
2. Stationary Horizons, Surface Gravity, and Redshift — How do a stationary horizon’s generator, surface gravity, redshift, affine normalization, and near-horizon coordinates enter field modes and thermality claims?
3. Hawking Radiation from Gravitational Collapse — How does gravitational collapse map regular early-time field data into late-time outgoing radiation, and which state, asymptotic, geometric, and approximation assumptions are essential?
4. Ray Tracing and the Hawking Bogoliubov Map — How does the null ray-tracing map determine the Hawking Bogoliubov kernel, wavepacket spectrum, transient corrections, and departures from exact thermality?
5. Stress-Tensor Flux and Two-Dimensional Reductions — How is outgoing Hawking energy flux computed from renormalized stress components, and when do two-dimensional reductions faithfully represent the higher-dimensional field problem?
6. Euclidean Periodicity, KMS Structure, and Horizon Thermality — When do Euclidean regularity, imaginary-time periodicity, KMS analyticity, and Lorentzian horizon regularity describe the same stationary thermal state?
7. Boulware, Hartle–Hawking, and Unruh States — How are Boulware, Hartle–Hawking, and Unruh states defined by asymptotic occupation, KMS behavior, and past or future horizon regularity?
8. Greybody Scattering and Flux Accounting — How do radial potentials, boundary conditions, transmission and reflection coefficients, partial waves, and flux normalization convert horizon emission into asymptotic spectra?
9. Rotating and Charged Horizons, Superradiance, and Chemical Potentials — How do rotation, charge, horizon chemical potentials, superradiant mode signs, and state-existence obstructions modify horizon scattering and thermal formulas?
10. Cosmological and Multiple-Horizon State Obstructions — When can a single state be regular or thermal across cosmological or multiple horizons with different generators, temperatures, and accessible static regions?
11. Trans-Planckian Sensitivity of the Hawking Derivation — Which steps of the Hawking derivation involve arbitrarily blueshifted precursors, and under what state, dispersion, adiabaticity, and EFT assumptions is the low-energy result robust?
12. Modified Dispersion, Analogue Horizons, and Universality — What do modified-dispersion models and analogue horizons reveal about the robustness of horizon mode conversion, and what can they not establish about quantum gravity?
13. Evaporating Backgrounds and Adiabatic Backreaction — When may a slowly evolving evaporating geometry be treated through instantaneous horizon data, and which timescale, state, flux, and backreaction corrections control that approximation?
14. Extremal, Near-Extremal, and Late-Time Limits — How do extremality, near-extremal scaling, zero surface gravity, long throats, order of limits, and late-time tails alter standard horizon-state and radiation arguments?

How the ideas fit together

In a collapse calculation, rapidly varying ray propagation relates early-time vacuum correlations to late-time detector observables. The resulting near-horizon behavior must still be propagated through the exterior potential to obtain an asymptotic spectrum Fredenhagen and Haag 1990, §§ 2–4.

Greybody factors express that scattering, while rotation and charge change the relevant horizon energy and can allow superradiance. State regularity, thermality, and net flux therefore cannot be inferred from one another without their extra assumptions. Late-time evaporation further requires backreaction beyond a prescribed stationary geometry.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Horizon or geometry	State or derivation input	Near-horizon result	Scattering input	Operational observable	Licensed domain	Failure and required downgrade
Collapse to a nonextremal asymptotically stationary black hole	Regular Hadamard in-state on past null infinity; exponential late-time ray map	Planck ratio at $T_H=\kappa/(2\pi)$ for late wavepackets	Partial-wave transmission still required	Outgoing number packets or renormalized flux at future null infinity	Fixed background, late compared with collapse transients, before appreciable evaporation	An eternal metric alone supplies no collapse preparation; a singular in-state removes the standard production claim.
Static bifurcate Killing horizon	Invariant horizon-regular Hartle–Hawking–Israel state, when it exists	KMS analyticity with Killing temperature $\kappa/(2\pi)$	Equilibrium has both incoming and outgoing sectors	Detector detailed balance or local correlators; net flux can vanish	Static wedge and declared generator normalization	Euclidean periodicity without a Lorentzian state licenses only a geometric period, not a detector or flux claim.
Eternal static exterior in the Boulware state	Vacuum with respect to static time at both infinities	No thermal population at infinity; stress is singular at the horizons	Radial scattering defines vacuum polarization modes	Static-observer stress or detector response away from the horizon	Exterior region bounded away from the horizon	Calling it a regular black-hole vacuum fails at the horizon.
Eternal static exterior in the Unruh state	Vacuum incoming from past infinity; outgoing sector regular on the future horizon	Outgoing Hawking occupation, no matching incoming bath	Greybody factors transmit the outgoing sector	Positive late-time outward flux at future infinity	Collapse-like future exterior; not regular on the past horizon	The state label alone does not determine luminosity without the radial problem.
Rotating or charged Killing horizon	Mode energy $\tilde\omega=\omega-m\Omega_H-q\Phi_H$ and a specified state	Thermal factor uses $\tilde\omega$	Wronskian permits amplification when $\omega\tilde\omega<0$	Signed energy and charge/angular-momentum fluxes	Region and boundary conditions without an uncontrolled superradiant instability	An unshifted Planck factor violates horizon energy accounting; a global bosonic Hartle–Hawking state may not exist.
Static region between two horizons	Candidate state tested at both horizons	Separate periods $2\pi/\lvert\kappa_i\rvert$	Each boundary has its own ingoing/outgoing conditions	Local detector responses and inter-horizon flux	Global equilibrium only when periods and generator normalization are compatible	Averaging unequal temperatures does not remove either conical or Lorentzian singularity; use a nonequilibrium state.
Dispersive or analogue horizon	Preferred frame, dispersion law, prepared initial state, adiabatic mode conversion	Low-frequency near-Planckian spectrum under stated hierarchy	Extra roots and mode-conversion channels included	Analogue quasiparticle correlations or flux	The specified medium/model, not quantum gravity in general	Matching a spectrum alone cannot establish gravitational ultraviolet dynamics or backreaction.
Slowly evolving horizon	Transported Hadamard state and $\lvert\dot\kappa\rvert/\kappa^2\ll1$	Instantaneous thermal form plus controlled derivative corrections	Time-dependent transmission evaluated on matching timescales	Wavepacket flux over windows short relative to evaporation	Semiclassical adiabatic interval	If emission and evolution timescales are comparable, use the full nonstationary problem; no instantaneous temperature claim.
Near-extremal or extremal geometry	Order of state construction, $\kappa\to0$, and $u\to\infty$ stated explicitly	Near-extremal thermal sector may have a singular or nonuniform extremal limit	Long throat and low-frequency sectors treated separately	Finite-time correlators, flux, or late-time tails	Only the declared order of limits	Setting $\kappa=0$ in a nonextremal formula can lose transients and modes; no conclusion transfers without a direct extremal analysis.

Where to go next

Continue to Local Observables, Stress Tensors, and Anomalies to construct composite fields from Hadamard-subtracted products.

References

• 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:10.1007/BF02096757.