Chapter 15: Black Hole Information Paradox

Part V: Information & Physics

Bekenstein-Hawking Entropy

$ S_{BH} = \frac{k_B c^3 A}{4 G \hbar} = \frac{A}{4 \ell_P^2} $

One bit of information per four Planck areas of event horizon surface

Jacob Bekenstein (1972) proposed that black holes carry entropy proportional to their event horizon area $A$, not their volume. Stephen Hawking (1974) confirmed this by showing black holes radiate thermally — Hawking radiation — at a temperature inversely proportional to mass:

$ T_H = \frac{\hbar c^3}{8\pi G M k_B} $

For a solar-mass black hole, $T_H \approx 62$ nanokelvin — colder than the CMB. Only primordial black holes less than about $10^{11}$ kg would be evaporating today.

Penrose Diagram: Black Hole Evaporation

Penrose diagram showing Hawking radiation escaping to future null infinity, infalling matter, and the "island" region contributing to the Page curve.

The Information Paradox

Hawking's 1976 calculation showed that the radiation emitted is exactly thermal— it carries no information about what fell in. When the black hole completely evaporates, all information appears destroyed. This conflicts with quantum mechanics:

Unitarity: Quantum evolution is unitary — information cannot be destroyed.
No-cloning: Information cannot be duplicated — it cannot be both inside and in the radiation.
Monogamy of entanglement: Late-time Hawking photons would need to be entangled with too many partners simultaneously (firewall paradox).

Holographic Principle & Page Curve

The holographic principle (t'Hooft, Susskind, 1993) states that the maximum entropy in any volume is proportional to its surface area, not its volume:

$ S_{\max} \leq \frac{A}{4\ell_P^2} $

Don Page (1993) showed that if unitarity holds, the entanglement entropy of radiation must follow the Page curve: rising until half the black hole has evaporated (the "Page time"), then decreasing back to zero.

Recent breakthroughs (2019–2022) using the island formula and replica wormholes in Jackiw-Teitelboim gravity derived the Page curve from first principles, suggesting that quantum gravity effects resolve the paradox by allowing information to escape through topological contributions to the Euclidean path integral.

Python: Entropy, Page Curve & Hawking Temperature

Plot Bekenstein-Hawking entropy vs black hole mass, the Page curve vs evaporation time, and Hawking temperature vs mass for a range of black hole sizes.

Python

script.py128 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Physical constants (SI)
G  = 6.674e-11    # m^3 kg^-1 s^-2
c  = 3e8          # m/s
hbar = 1.055e-34  # J s
kB = 1.381e-23    # J/K
M_sun = 1.989e30  # kg

# Planck length: l_P = sqrt(hbar G / c^3)
l_P = np.sqrt(hbar * G / c**3)
print(f"Planck length: l_P = {l_P:.3e} m")

# Schwarzschild radius: r_s = 2GM/c^2
# Surface area: A = 4 pi r_s^2
# Bekenstein-Hawking entropy: S_BH = A/(4 l_P^2) = 4 pi G M^2 / (hbar c)

def BH_entropy(M):
    """Bekenstein-Hawking entropy in bits (S / kB ln2)"""
    r_s = 2 * G * M / c**2
    A   = 4 * np.pi * r_s**2
    S   = A / (4 * l_P**2)   # in nats (natural units)
    return S / np.log(2)      # convert to bits

def hawking_temperature(M):
    """Hawking temperature in Kelvin"""
    return hbar * c**3 / (8 * np.pi * G * M * kB)

# Mass range: 1 solar mass to 10^9 solar masses
M_range = np.logspace(0, 9, 300) * M_sun
S_range = BH_entropy(M_range)
T_range = hawking_temperature(M_range)

# Print some key values
for factor, label in [(1, '1 M_sun'), (10, '10 M_sun'), (1e6, '10^6 M_sun (Sgr A*)'),
                      (1e9, '10^9 M_sun (quasar)')]:
    M = factor * M_sun
    S = BH_entropy(M)
    T = hawking_temperature(M)
    print(f"{label}: S_BH = {S:.3e} bits, T_Hawking = {T:.3e} K")

print()
print(f"Sun if collapsed: S_BH = {BH_entropy(M_sun):.3e} bits")
print(f"Sgr A* black hole: S_BH = {BH_entropy(4e6 * M_sun):.3e} bits")
print()

# Page curve: entanglement entropy vs evaporation time
# Before Page time: S_ent ~ linear growth (radiation is thermal-like)
# After Page time:  S_ent decreases as BH shrinks
# At t = 0: S=0; Page time ~ t_evap/2; at t_evap: S=0

N_points = 300
t = np.linspace(0, 1, N_points)  # normalized time (0=formation, 1=evaporation)
t_page = 0.5  # Page time at half evaporation

# Thermodynamic (Hawking) curve: S_rad grows monotonically
S_hawking = t * 1.0  # linearly grows (radiation piling up)

# Page curve (unitarity): entanglement entropy follows Hawking then decreases
S_page = np.where(t < t_page, t / t_page, (1 - t) / (1 - t_page))
S_page = S_page * 0.5  # scale for comparison

# BH remaining entropy during evaporation (shrinks)
S_BH_evap = 1 - t  # normalized: starts at max, ends at 0

print("Page time = halfway point of evaporation")
print("After Page time, entanglement entropy DECREASES (information escapes)")
print("This requires quantum gravity corrections — islands/replica wormholes")

# ── Plotting ──────────────────────────────────────────────────────────────────
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
fig.patch.set_facecolor('#0a0a1a')

# Plot 1: S_BH vs mass
ax1 = axes[0]
ax1.loglog(M_range / M_sun, S_range, color='#818cf8', linewidth=2.5)
ax1.axvline(1,    color='#fbbf24', linestyle='--', alpha=0.7, linewidth=1.5, label='1 M_sun')
ax1.axvline(4e6,  color='#f87171', linestyle='--', alpha=0.7, linewidth=1.5, label='Sgr A* (4×10⁶ M_sun)')
ax1.axvline(1e9,  color='#34d399', linestyle='--', alpha=0.7, linewidth=1.5, label='Quasar (10⁹ M_sun)')
ax1.set_xlabel('Mass (solar masses)', color='white', fontsize=11)
ax1.set_ylabel('Bekenstein-Hawking Entropy (bits)', color='white', fontsize=11)
ax1.set_title("Black Hole Entropy vs Mass\n$S_{BH} = A / 4\ell_P^2$", color='white', fontsize=12, fontweight='bold')
ax1.legend(fontsize=8, facecolor='#1a1a2e', edgecolor='#818cf8', labelcolor='white')
ax1.set_facecolor('#0a0a1a')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.2, color='#818cf8')
for sp in ax1.spines.values(): sp.set_color('#818cf8')

# Plot 2: Page curve
ax2 = axes[1]
ax2.plot(t, S_hawking, color='#f87171', linewidth=2.5, linestyle='--', label='Hawking (thermal, information lost)')
ax2.plot(t, S_page,    color='#818cf8', linewidth=2.5, label='Page curve (unitary, information preserved)')
ax2.plot(t, S_BH_evap, color='#fbbf24', linewidth=2, linestyle=':', label='Remaining BH entropy')
ax2.axvline(t_page, color='#34d399', linestyle='--', linewidth=1.5, alpha=0.8, label=f'Page time (t = {t_page})')
ax2.set_xlabel('Normalized evaporation time', color='white', fontsize=11)
ax2.set_ylabel('Entropy (normalized)', color='white', fontsize=11)
ax2.set_title("Page Curve\nBlack Hole Evaporation", color='white', fontsize=12, fontweight='bold')
ax2.legend(fontsize=8, facecolor='#1a1a2e', edgecolor='#818cf8', labelcolor='white')
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.2, color='#818cf8')
for sp in ax2.spines.values(): sp.set_color('#818cf8')

# Plot 3: Hawking temperature vs mass
ax3 = axes[2]
ax3.loglog(M_range / M_sun, T_range, color='#a5b4fc', linewidth=2.5, label='Hawking temperature')
ax3.axhline(2.725, color='#fbbf24', linestyle='--', alpha=0.8, linewidth=1.5, label='CMB temperature (2.725 K)')
ax3.fill_between(M_range / M_sun, 2.725, T_range,
                 where=(T_range > 2.725), alpha=0.12, color='#f87171',
                 label='Evaporating (T > T_CMB)')
ax3.set_xlabel('Mass (solar masses)', color='white', fontsize=11)
ax3.set_ylabel('Hawking Temperature (K)', color='white', fontsize=11)
ax3.set_title("Hawking Temperature vs Mass\n$T_H = \\hbar c^3 / (8\\pi G M k_B)$", color='white', fontsize=12, fontweight='bold')
ax3.legend(fontsize=8, facecolor='#1a1a2e', edgecolor='#818cf8', labelcolor='white')
ax3.set_facecolor('#0a0a1a')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.2, color='#818cf8')
for sp in ax3.spines.values(): sp.set_color('#818cf8')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.show()
print("\nHolographic principle: all information in a volume encoded on its boundary surface.")
print("Recent resolution: quantum extremal surfaces / replica wormholes give Page curve.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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