====== Hawking Radiation ======
//see also [[advanced_notions:black_hole]] //
<tabbox Intuitive> 

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox 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 [[https://arxiv.org/abs/1511.08221|Hawking radiation, the Stefan-Boltzmann law, and unitarization]] by Steven B. Giddings
 
<tabbox Abstract> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<tabbox 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 [[theories:speculative_theories:quantum_gravity|quantum gravity]]. 
</tabbox>