Zeroth and First Laws
The Zeroth Law states that surface gravity \(\kappa\) is constant over the entire event horizon of a stationary black hole. This parallels the Zeroth Law of thermodynamics, which states that temperature is uniform in thermal equilibrium. The First Law relates changes in mass to changes in area and angular momentum, analogous to the conservation of energy.
Thermodynamics
- Zeroth: Temperature is uniform
- First: \(dE = T dS - P dV\)
Black Holes
- Zeroth: Surface gravity \(\kappa\) is constant
- First: \(dM = \frac{\kappa}{8\pi} dA + \Omega dJ\)
Second Law
As discussed, the Second Law of Black Hole Mechanics states that area never decreases. In thermodynamics, entropy never decreases. Identifying \(S \propto A\) makes this law identical in form.
Thermodynamics
- Second: \(dS \geq 0\)
Black Holes
- Second: \(dA \geq 0\)
Third Law
The Third Law of Thermodynamics states that absolute zero temperature cannot be reached. Similarly, the Third Law of Black Hole Mechanics states that surface gravity \(\kappa\) cannot be reduced to zero by any finite process. This implies that extremal black holes (where \(\kappa=0\)) cannot be formed from non-extremal ones.
Thermodynamics
- Third: \(T \neq 0\) achievable
Black Holes
- Third: \(\kappa \neq 0\) achievable