Black Hole Entropy

Black Hole Entropy

One of the most profound results in theoretical physics is that black holes carry entropy proportional to their surface area — not their volume. This is the Bekenstein–Hawking entropy:

$S = \frac{k_B A}{4 \ell_P^2}$

where $A$ is the event horizon area and $\ell_P$ is the Planck length:

$\ell_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.616 \times 10^{-35} \text{ m}$

Substituting $\ell_P^2 = \hbar G / c^3$, this becomes:

$S = \frac{k_B c^3 A}{4 G \hbar}$

Event Horizon Area

For a Schwarzschild black hole, the event horizon (Schwarzschild radius) is:

$r_s = \frac{2GM}{c^2}$

The surface area of a sphere of radius $r_s$ is:

$A = 4\pi r_s^2 = \frac{16\pi G^2 M^2}{c^4}$

Entropy Formula

Substituting $A$ into the entropy formula:

$S = \frac{k_B c^3}{4G\hbar} \cdot \frac{16\pi G^2 M^2}{c^4} = \frac{4\pi G M^2 k_B}{\hbar c}$

A solar-mass black hole has entropy $\approx 10^{54}$ J/K — astronomically larger than a normal star of the same mass ($\sim 10^{34}$ J/K). This entropy-area relation hints at holography: all information inside may be encoded on the 2D surface.

Your Task

Implement three functions. All physical constants must be defined inside each function body.

• planck_length() — returns $\ell_P = \sqrt{\hbar G / c^3}$ in meters
• event_horizon_area(M) — returns $A = 4\pi (2GM/c^2)^2$ in m²
• black_hole_entropy(M) — returns $S = 4\pi G M^2 k_B / (\hbar c)$ in J/K