Laws of Black Hole Mechanics
In 1973, Bardeen, Carter, and Hawking discovered that black holes obey four laws that closely parallel the laws of thermodynamics. This deep analogy suggested a fundamental connection between gravity, thermodynamics, and quantum mechanics—a connection later confirmed by Hawking's discovery of black hole radiation.
1. The Zeroth Law
Statement
The surface gravity $\kappa$ is constant over the event horizon of a stationary black hole.
Surface Gravity: The surface gravity is defined by:
$$\kappa^2 = -\frac{1}{2}(\nabla_\mu \chi_\nu)(\nabla^\mu \chi^\nu)\bigg|_{r=r_+}$$
where $\chi^\mu$ is the Killing vector that generates time translations.
For Specific Black Holes
- Schwarzschild: $\kappa = \frac{c^4}{4GM} = \frac{1}{4r_s}$
- Kerr: $\kappa = \frac{c(r_+ - r_-)}{2(r_+^2 + a^2)}$
- Reissner-Nordström: $\kappa = \frac{c^2(r_+ - r_-)}{2r_+^2}$
Thermodynamic Analogy
The zeroth law of thermodynamics states that temperature is constant throughout a system in thermal equilibrium. Here, $\kappa$ plays the role of temperature, suggesting $T \propto \kappa$.
2. The First Law
Statement
For perturbations between nearby stationary black hole solutions:
$$dM = \frac{\kappa c^2}{8\pi G}dA + \Omega_H dJ + \Phi_H dQ$$
where:
- $M$ is the black hole mass
- $A$ is the horizon area
- $\kappa$ is the surface gravity
- $\Omega_H$ is the angular velocity of the horizon
- $J$ is the angular momentum
- $\Phi_H$ is the electrostatic potential at the horizon
- $Q$ is the electric charge
Thermodynamic Analogy
Compare to the first law of thermodynamics:
$$dE = TdS - PdV + \mu dN$$
The area $A$ plays the role of entropy, mass $M$ plays the role of energy, and surface gravity $\kappa$ plays the role of temperature.
Schwarzschild Case
For a non-rotating, uncharged black hole ($J = 0, Q = 0$):
$$dM = \frac{\kappa c^2}{8\pi G}dA = \frac{c^4}{32\pi GM^2}dA$$
Since $A = 16\pi G^2M^2/c^4$, we have $dA = 32\pi G^2M\,dM/c^4$, confirming the consistency.
3. The Second Law (Hawking's Area Theorem)
Statement
In any classical process, the total area of all event horizons never decreases:
$$\frac{dA}{dt} \geq 0$$
This holds for any process: matter falling in, black hole collisions, mergers, etc.
Examples
- Particle falling in: When a particle of mass m falls into a black hole of mass M, the final area is always greater than the initial area.
- Black hole merger: When two black holes of masses $M_1$ and $M_2$ merge to form a black hole of mass $M_3$, the area theorem requires $A_3 \geq A_1 + A_2$, which implies$M_3^2 \geq M_1^2 + M_2^2$.
- Penrose process: Even when extracting energy from a rotating black hole, the area increases.
Thermodynamic Analogy
The second law of thermodynamics states that entropy never decreases: $dS \geq 0$. This strongly suggests that horizon area is proportional to entropy.
Generalized Second Law
The generalized second law (GSL) extends this to include ordinary matter entropy:
$$\frac{d}{dt}(S_{matter} + S_{BH}) \geq 0$$
where $S_{BH} \propto A$. This allows matter entropy to decrease as long as the black hole area increases sufficiently.
4. The Third Law
Statement
It is impossible to reduce the surface gravity $\kappa$ to zero by a finite sequence of operations.
In other words, one cannot reach an extremal black hole (where the inner and outer horizons coincide) through a finite number of physical processes.
Physical Interpretation
- For a Kerr black hole, $\kappa = 0$ when $a = M$ (extremal rotation)
- For a Reissner-Nordström black hole, $\kappa = 0$ when $Q = M$ (extremal charge)
- To reach extremality would require adding infinite angular momentum or charge
- This is analogous to the third law: it's impossible to cool a system to absolute zero in a finite number of steps
Thermodynamic Analogy
The third law of thermodynamics (Nernst's theorem) states that it's impossible to reach absolute zero temperature ($T = 0$) in a finite number of steps. Since $T \propto \kappa$, the black hole third law is the exact analog.
Summary: The Laws and Their Thermodynamic Analogs
| Law | Black Hole Mechanics | Thermodynamics |
|---|---|---|
| Zeroth | $\kappa$ constant on horizon | $T$ constant in equilibrium |
| First | $dM = \frac{\kappa}{8\pi G}dA + \Omega_H dJ + \Phi_H dQ$ | $dE = TdS - PdV + \mu dN$ |
| Second | $dA/dt \geq 0$ | $dS \geq 0$ |
| Third | Cannot reach $\kappa = 0$ | Cannot reach $T = 0$ |
The Profound Implication
The precise mathematical analogy between black hole mechanics and thermodynamics strongly suggested that black holes actually are thermodynamic objects. But this raised a paradox:
- If black holes have entropy, they must have temperature
- If they have temperature, they must radiate
- But nothing can escape from a black hole!
This paradox was resolved in 1974 when Stephen Hawking discovered that quantum effects cause black holes to radiate thermally, with exactly the temperature predicted by the analogy.
Historical Context
- 1970: Penrose proposes process to extract energy from rotating black holes
- 1971: Hawking proves the area theorem (second law)
- 1972: Bekenstein proposes black holes have entropy proportional to area
- 1973: Bardeen, Carter, and Hawking formalize the four laws of black hole mechanics
- 1974: Hawking discovers black hole radiation, confirming $T_H = \hbar\kappa/(2\pi k_Bc)$
- 1974: Bekenstein-Hawking entropy formula established: $S_{BH} = \frac{k_Bc^3A}{4G\hbar}$