---
# System prepended metadata

title: black hole thermodynamics
tags: [mechanics, Thermodynamics & Fluid Mechanics]

---

# Black hole thermodynamics
## Black hole
![Black hole](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Black_Hole_Merger.jpg/525px-Black_Hole_Merger.jpg)
## The zeroth law
The horizon has constant surface gravity for a stationary black hole.
## The first law
$$E = mc^2$$

$$dE = \frac{\kappa}{8\pi}dA+\Omega dJ+\Phi dQ = T_HdS_{BH}+\Omega dJ$$
* energy : $E$
* surface gravity : $\kappa$
* horizon area : $A$
* black hole angular velocity : $\Omega$
* black hole angular momentum : $J$
* black hole electrostatic potential : $\Phi$
* electric charge : $Q$

$$dW = \Omega dJ$$
* work : $W$

$$S_{BH} = \frac{kA}{4l_p^2}$$
* Planck length : $l_p = \sqrt{G\bar h/c^3}$
*  Boltzmann constant : $k$

## The second law
$$\frac{dA}{dt}$$
* the horizon area : $A$

## The third law

$$\kappa \not= 0$$
* surface gravity : $\kappa$