Hawking Radiation
In 1974, Stephen Hawking discovered that black holes emit thermal radiation due to quantum effects near the event horizon. This revolutionary finding united general relativity, quantum mechanics, and thermodynamics, proving that black holes have a temperature and slowly evaporate.
Hawking Temperature
$$T_H = \frac{\hbar c^3}{8\pi G k_B M} = \frac{\hbar \kappa}{2\pi k_B c}$$
This relates the black hole temperature to its surface gravity. Key features:
- Temperature is inversely proportional to mass: $T_H \propto M^{-1}$
- Smaller black holes are hotter
- For a solar mass black hole: $T_H \approx 60$ nanokelvin (far below the CMB at 2.7 K)
- For a 1 kg black hole: $T_H \approx 10^{23}$ K (incredibly hot!)
Physical Mechanism
Hawking radiation arises from quantum vacuum fluctuations near the event horizon:
- Virtual particle-antiparticle pairs constantly form in the vacuum
- Near the horizon, one particle can fall into the black hole while the other escapes
- The escaping particle becomes real radiation
- The infalling particle has negative energy relative to infinity, reducing the black hole mass
This is a heuristic picture. The rigorous derivation uses quantum field theory in curved spacetime, computing the Bogoliubov transformation between ingoing and outgoing vacuum states.
Black Hole Evaporation
The luminosity (power radiated) by a black hole is:
$$L = \frac{\hbar c^6}{15360 \pi G^2 M^2}$$
Since $L = -dE/dt = -c^2 dM/dt$, we can integrate to find the evaporation time:
$$t_{evap} = \frac{5120 \pi G^2 M^3}{\hbar c^4} \propto M^3$$
Evaporation Times
- Solar mass: $t_{evap} \approx 10^{67}$ years (age of universe $\approx 10^{10}$ years)
- Moon mass: $t_{evap} \approx 10^{50}$ years
- Mountain mass ($10^{12}$ kg): $t_{evap} \approx 10^{11}$ years
- 1 kg: $t_{evap} \approx 10^{-25}$ seconds (explosive!)
As the black hole loses mass, it heats up and radiates faster, leading to a runaway process ending in an explosive evaporation in the final moments.
Spectrum and Greybody Factors
The Hawking radiation has a thermal spectrum:
$$\frac{dN}{dt d\omega} = \frac{\Gamma_s(\omega)}{2\pi} \frac{1}{e^{\omega/T_H} - 1}$$
where $\Gamma_s(\omega)$ is the greybody factor accounting for scattering off the gravitational potential. For low frequencies, $\Gamma_s \to 1$ and the spectrum is perfectly thermal (Planck distribution).
The radiation includes all particle species: photons, neutrinos, gravitons, and (if they exist) other particles.