In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature $\hbar\kappa/2\pi k \approx 10^{-6} (M_\odot/M)$ °K where $\kappa$ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about $10^{15}$ g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law: $S + \frac{1}{4}A$ never decreases where $S$ is the entropy of matter outside black holes and $A$ is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.
Lacking a satisfactory quantum theory of gravity, Hawking adopts a semi-classical approximation: matter fields (scalar, electromagnetic, neutrino) are treated quantum mechanically on a classical space-time metric obeying the Einstein equations. He shows that this approximation remains good as long as the radius of curvature stays large compared to the Planck length ($10^{-33}$ cm), i.e. everywhere except near singularities and during the first $10^{-43}$ s of the universe. In curved space-time the splitting of a field into positive and negative frequencies has no invariant meaning: a field initially in the vacuum state may no longer be in it after crossing a curved region, which is interpreted as particle creation by the gravitational field.
Applied to black holes, this mechanism predicts an emission of particles to infinity at exactly the rate one would expect from an ordinary body with temperature $\kappa/2\pi$ (in geometric units), about $10^{-6} (M_\odot/M)$ K. For a solar-mass black hole this temperature is far below the 3 K of the cosmic microwave background: such black holes absorb more than they emit and grow. Much smaller primordial black holes, formed by density fluctuations in the early universe, would however be hotter: as they radiate they lose mass, get hotter still and radiate faster. Around $10^{12}$ K (a mass of about $10^{14}$ g), the emission of all particle species could dissipate the remaining mass on a strong-interaction timescale ($\sim 10^{-23}$ s), producing an explosion of about $10^{35}$ ergs. The lifetime is otherwise of the order of $10^{-28} M^3$ s.
Hawking offers a heuristic picture of the mechanism: virtual particle pairs near the horizon, whose negative-energy member tunnels through the horizon (inside which it can exist as a real particle) while the positive-energy member escapes to infinity, making up the thermal emission. This negative energy flux decreases the area of the horizon, in violation of the classical area-increase law. The prediction confirms the Generalized Second Law proposed by Bekenstein (entropy plus some multiple of the area never decreases), fixing the temperature at $\kappa/2\pi$ and the black hole entropy at $\frac{1}{4}A$; without emission this law would be violated by a black hole immersed in radiation colder than itself.
The calculation is carried out for a non-rotating, uncharged black hole (Schwarzschild solution). Since the equilibrium solutions are stationary, no particle creation would be expected from them: it is essential to consider the time-dependent collapse phase. Following the modes of a massless scalar field from past null infinity to future null infinity through the collapsing body, Hawking computes the mixing of positive and negative frequencies (Bogoliubov coefficients) and shows that the number of particles emitted to infinity in each mode of frequency $\omega$ is that of a body of temperature $\kappa/2\pi$: $\Gamma_\omega/(e^{2\pi\omega/\kappa} - 1)$, where $\Gamma_\omega$ is the fraction of the mode that would be absorbed by the black hole. In the spirit of the no-hair theorems, the emission rate depends on the details of the collapse only through the mass, angular momentum and charge of the final black hole.
For a rotating and/or charged black hole (Kerr-Newman solution), the classical first law of black holes implies that bosonic modes of frequency $\omega < m\Omega + e\Phi$ (where $\Omega$ and $\Phi$ are the angular frequency and the electrostatic potential of the black hole) are scattered with increased amplitude: this is classical "superradiance", which corresponds to stimulated emission. Hawking shows that a spontaneous emission adds to it: the thermal factor becomes $\{e^{2\pi(\omega - m\Omega - e\Phi)/\kappa} \mp 1\}^{-1}$ (sign according to spin), so that the black hole preferentially emits in the modes that carry away its angular momentum and charge. For half-integer spin fields there is no classical superradiance, but the spontaneous emission remains.
Particle creation is a global process, not localised in the collapsing body: an observer falling through the horizon would not see an infinite number of particles coming out of it. The mass decrease of the black hole is interpreted as a negative energy flux across the horizon, tied to the quantum indeterminacy of the local energy density (or to the indeterminacy of the very position of the horizon). As long as the mass remains large compared to the Planck mass ($10^{-5}$ g), the evolution is slow and the black hole can be described by a sequence of stationary solutions. Below that, the approximation breaks down, but the remaining energy being tiny, the black hole can hardly do anything but disappear completely. The baryons and leptons of the original body cannot reappear, their rest-mass energy having been carried away by the thermal radiation: Hawking speculates that the evaporation of small primordial black holes might explain why the universe contains so few baryons compared to photons.