Hawking radiation

Remarks

Hawking found the effect in 1974 by tracing late outgoing modes back in time. They pass through the collapsing body just before the event horizon forms and are blueshifted enormously. Comparing them with the vacuum modes of the past gives Bogoliubov coefficients whose squared norms follow a Planck distribution at temperature $T_H$. Rigorous versions of the result for linear fields have been proved in mathematical physics.

Example: for a Schwarzschild black hole of mass $M$, $\kappa = c^4/(4GM)$, so $T_H = \hbar c^3/(8\pi G M k_B)$. For one solar mass this is about $6 \times 10^{-8}$ kelvin, far below the temperature of the cosmic microwave background. A smaller black hole is hotter, so it radiates faster and evaporates.

Going deeper

For a student

Hawking radiation turned the laws of black hole mechanics into real thermodynamics, fixing the Bekenstein–Hawking entropy at a quarter of the area. It is closely related to the Unruh effect, in which an accelerated observer in flat space sees the vacuum as a thermal bath. Because the radiation is exactly thermal in Hawking's computation, it seems to carry no information about what fell in, which leads to the black hole information paradox.