Hawking Radiation

Intuitive

Concrete

$$ T= \frac{\hbar c^3}{8 \pi G M k_B} ,$$

where $k_B$ is the Boltzmann constant, $c$ the speed of light, $G$ the gravitational constant, $\hbar$ the reduced Planck constant and $M$ the mass of the black hole.

The temperature of a black hole is tiny. Putting in the numbers yields

$$ T= 6.169 \cdot 10^{-8} \text{ K } \ \frac{M_\odot }{M}, $$ where $M_\odot$ is the mass of the sun. In words this means that black hole with a mass equal to the mass of our sun would have a temperature of only $10^{-8}$ K. If the black hole is heavier, the temperature gets even tinier.

• For a nice explicit discussion of the question "Where does Hawking radiation originate?", see Hawking radiation, the Stefan-Boltzmann law, and unitarization by Steven B. Giddings

Abstract

Why is it interesting?

This formula for the Hawking radiation shows why black holes are so important and interesting. In this little formula everything comes together:

• Quantum mechanics, in the form of $\hbar$
• Gravity, in the form of $G$

It tells us that black holes are laboratories for quantum gravity.