Bekenstein Entropy
In 1972, Jacob Bekenstein proposed that black holes must have entropy proportional to their horizon area to preserve the second law of thermodynamics. This revolutionary idea suggested that black holes carry information, laying the groundwork for holography and quantum gravity.
Bekenstein's Argument
Consider dropping a hot object with entropy $S$ into a black hole. The object disappears behind the horizon, and from the outside perspective, its entropy is lost. This would violate the second law of thermodynamics unless the black hole itself gains entropy.
The Generalized Second Law
$$S_{total} = S_{matter} + S_{BH} \quad \text{must never decrease}$$
Bekenstein proposed that black hole entropy must be proportional to the horizon area, the only extensive quantity that increases when matter falls in.
The Bekenstein-Hawking Formula
$$S_{BH} = \frac{k_B c^3 A}{4G\hbar} = \frac{k_B A}{4\ell_P^2}$$
where:
- $A$ is the horizon area
- $\ell_P = \sqrt{G\hbar/c^3} \approx 1.6 \times 10^{-35}$ m is the Planck length
- $k_B$ is Boltzmann's constant
The factor of 1/4 was determined by Hawking in 1974 when he computed the black hole temperature.
Numerical Examples
Solar Mass Black Hole
For $M = M_{\odot} \approx 2 \times 10^{30}$ kg:
- Schwarzschild radius: $r_s \approx 3$ km
- Horizon area: $A = 4\pi r_s^2 \approx 10^{14}$ m²
- Entropy: $S_{BH} \approx 10^{54} k_B$
This is enormous—about $10^{77}$ times larger than the entropy of the Sun!
Supermassive Black Hole (M87*)
For $M \approx 6.5 \times 10^9 M_{\odot}$:
- Entropy: $S_{BH} \approx 10^{73} k_B$
Supermassive black holes are the most entropic objects in the universe.
The Bekenstein Bound
Bekenstein also derived a universal bound on entropy:
$$S \leq \frac{2\pi k_B RE}{\hbar c}$$
where $R$ is the radius of a sphere containing the system and $E$ is its total energy. This bound is saturated by black holes, which are therefore the maximum entropy objects for a given size.
Holographic Principle
The fact that black hole entropy scales with area (2D) rather than volume (3D) suggests that the information content of a region of space is encoded on its boundary—a profound hint toward the holographic principle.
Microscopic Origin
The Bekenstein-Hawking entropy can be understood as counting microstates:
$$S = k_B \ln \Omega$$
where $\Omega$ is the number of microstates. For a black hole:
$$\Omega \sim \exp\left(\frac{A}{4\ell_P^2}\right)$$
- String Theory: Strominger and Vafa (1996) counted microstates for certain extremal black holes in string theory, exactly reproducing the Bekenstein-Hawking formula.
- Loop Quantum Gravity: Horizon states are quantized, leading to a discrete area spectrum and recovering the entropy formula.
- AdS/CFT: Black hole microstates correspond to states in the dual CFT with energy$E \sim 1/R_{AdS}$.