Hawking radiation from horizon redshift

Retain signature (−,+,+,+) and geometric G=c=1, with the late stationary horizon generator normalized by the clock at infinity. Let u be retarded time at future null infinity and U an affine incoming coordinate at past null infinity. Rays escaping at increasingly late u passed increasingly close to the last ray that could escape. To leading late-time order their relation is exponential, with a positive constant A that depends on the collapse history but not the universal temperature factor.

Tracing an outgoing positive-frequency wave backward gives an oscillatory power of UH−U. Its Fourier decomposition in the incoming affine coordinate has positive- and negative-frequency pieces with a universal magnitude ratio. Wavepackets make the mode normalization and late-time localization well behaved. For a bosonic channel, canonical normalization converts the ratio into a Planck occupation. The asymptotic relation concerns late outgoing rays; it is not a claim that a single geometric-optics ray already contains a quantum particle.

For a nonrotating Schwarzschild hole, the curvature potential partly reflects outgoing modes. The occupation reaching infinity in one angular channel is multiplied by a greybody transmission probability Γ between zero and one. Γ is an intensity or flux probability, not a transmission amplitude, and it depends on frequency and angular momentum. Summing channels and including their energy weights gives a luminosity; one mean occupation alone does not determine the total power. Fermionic fields instead have a plus sign in the thermal denominator.

The collapse state is modeled by an Unruh-type state: no incoming thermal bath from past infinity, regular behavior across the future horizon, and late outgoing flux. The Hartle–Hawking state on an eternal stationary background describes thermal equilibrium with both incoming and outgoing radiation. The Boulware state is a static vacuum at infinity but is singular on the Schwarzschild horizon. These choices are physically different. A statement about a static equilibrium thermometer must not silently be transferred to every freely falling detector in a collapse state.

The standard derivation uses a free or sufficiently weakly interacting quantum field with the appropriate short-distance, Hadamard behavior on a regular background. In an adiabatic evaporation treatment, changes of the surface gravity must be slow, for example $|\partial_u\kappa|/\kappa^2\ll1$ with u normalized at infinity. The outgoing spectrum is approximately stationary over the measurement interval, and backreaction is treated in a controlled expansion. Small curvature for a large hole helps justify that regime; it does not establish the detailed microscopic fate of a Planck-scale endpoint.

Backward propagation of a late outgoing mode reaches extremely high incoming frequencies. The calculation therefore has a short-distance assumption, often discussed as the trans-Planckian issue. This identifies where robustness or ultraviolet completion must be investigated, rather than by itself disproving the low-energy thermal result. Greybody filtering, finite formation transients, and slowly changing parameters also limit an exactly blackbody interpretation. A detector response, a renormalized stress tensor, and an asymptotic occupation are related calculations with different specifications; none should be substituted for another without checking those specifications.