Lectures on Hawking Radiation

Overview

The effect follows from quantum field theory on a fixed background and, in the end, from one geometric fact: outgoing rays near a forming horizon are redshifted exponentially. The lectures derive the temperature from that fact, follow its consequences for evaporation and for information, and close with what experiment can say.

Key results

A black hole radiates as a black body at $T_{\mathrm H} = \hbar\kappa/2\pi c k_{\mathrm B}$ , which is $6\times10^{-8}$ K for one solar mass. The first law then fixes the entropy at a quarter of the horizon area in Planck units.
Every derivation of the temperature, by Bogoliubov coefficients, the Euclidean section, the trace anomaly or tunnelling, uses the same two inputs: exponential peeling at the rate $\kappa$ , and a state that looks like the vacuum across the horizon.
Evaporation runs away. A solar-mass black hole would last about $10^{67}$ years; one of $5\times10^{14}$ g, if formed in the early universe, is ending its life now.
If evaporation is unitary, the entropy of the radiation must turn over at $t_{\mathrm{Page}} \approx 0.54\,\tau$ . Replica wormholes reproduce this Page curve in two-dimensional and AdS models; the four-dimensional case and the mechanism remain open.
Condensate experiments report spontaneous, correlated emission at the predicted temperature. Gamma-ray limits exclude primordial black holes as most of the dark matter between $10^{15}$ and $10^{17}$ g; the asteroid-mass window above is still open.

[Image: The Page curve: Hawking's calculation (dashed) against unitary evaporation (solid), which turns over at the Page time (Lecture 7).]

The Page curve: Hawking's calculation (dashed) against unitary evaporation (solid), which turns over at the Page time (Lecture 7).

Abstract

Hawking radiation is sometimes described as an effect of quantum gravity. It is better understood as a consequence of quantum field theory on a fixed background, and ultimately of a single geometrical fact: outgoing rays near a forming horizon are redshifted exponentially. These lectures are organised around that observation. After the classical laws of black hole mechanics and a short treatment of the Unruh effect, we derive the Hawking temperature step by step, then compare the Euclidean, anomaly and tunnelling derivations to see which ingredients each of them actually requires. The numbers matter as much as the formalism. A black hole of one solar mass radiates at about $6\times10^{-8}$ K and would need some $10^{67}$ years to evaporate, whereas a primordial black hole of $5\times10^{14}$ g would be completing its evaporation today. The longest lecture concerns the information problem, from Hawking's 1976 argument to the Page curve and the island calculations that reproduce it. The course closes with what can be tested: analogue horizons in Bose–Einstein condensates, and the searches for evaporating black holes in astrophysical data and at colliders. Each lecture ends with annotated suggestions for further reading.