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← Back to Part V: Advanced Topics

Chapter 25: Black Hole Thermodynamics

Black holes have entropy and temperature! The laws of black hole mechanics mirror thermodynamics perfectly, and Hawking radiation shows this analogy is exact.

Four Laws

0th Law: Constant Surface Gravity

κ is constant over the horizon (↔ T uniform in equilibrium)

1st Law: Energy Conservation

$ dM = \frac{\kappa}{8\pi G} dA + \Omega dJ + \Phi dQ $

2nd Law: Area Increase

Horizon area never decreases: δA ≥ 0 (↔ entropy increase)

3rd Law: κ = 0 Unattainable

Cannot achieve extremal (T=0) black hole in finite steps

Hawking Temperature & Entropy

Temperature

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

≈ 6×10^-8 K for M_☉

Bekenstein-Hawking Entropy

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

Entropy ∝ Area (not volume!)

Hawking Evaporation

$ t_{evap} \approx \frac{5120 \pi G^2 M^3}{\hbar c^4} $

For 1 M_☉: t ≈ 10⁶⁷ years (much longer than universe age!)

Small primordial black holes (M ~ 10¹² kg) could be evaporating today, producing gamma-ray bursts at the end of their life.

Interactive Simulation: BH Thermodynamics

Run this Python code to explore black hole temperatures, entropies, and evaporation times across the mass spectrum from primordial to supermassive black holes. See why all astrophysical black holes are colder than the CMB!

Black Hole Thermodynamics

Python

Compute Hawking temperature, entropy, and evaporation times

bh_thermodynamics.py192 lines

#!/usr/bin/env python3
"""
bh_thermodynamics.py - Black hole temperature, entropy, and evaporation
Explores the profound connection between gravity and thermodynamics
"""
import numpy as np
import matplotlib.pyplot as plt

# Physical constants
G = 6.674e-11      # N m²/kg²
c = 3e8            # m/s
hbar = 1.055e-34   # J s
kB = 1.381e-23     # J/K
M_sun = 1.989e30   # kg
year = 3.156e7     # seconds

def schwarzschild_radius(M):
    """Event horizon radius"""
    return 2 * G * M / c**2

def horizon_area(M):
    """Horizon area A = 4πr_s²"""
    r_s = schwarzschild_radius(M)
    return 4 * np.pi * r_s**2

def hawking_temperature(M):
    """Hawking temperature T_H = ℏc³/(8πGMk_B)"""
    return hbar * c**3 / (8 * np.pi * G * M * kB)

def bekenstein_hawking_entropy(M):
    """S_BH = k_B c³ A / (4Gℏ)"""
    A = horizon_area(M)
    return kB * c**3 * A / (4 * G * hbar)

def evaporation_time(M):
    """Time for complete evaporation: t ~ 5120πG²M³/(ℏc⁴)"""
    return 5120 * np.pi * G**2 * M**3 / (hbar * c**4)

def hawking_power(M):
    """Luminosity P = ℏc⁶/(15360πG²M²)"""
    return hbar * c**6 / (15360 * np.pi * G**2 * M**2)

print("=" * 70)
print("Black Hole Thermodynamics")
print("=" * 70)
print()
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  BEKENSTEIN-HAWKING FORMULAS")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  Hawking Temperature:")
print()
print("  \$ T_H = \\frac{\\hbar c^3}{8\\pi G M k_B} \$")
print()
print("  Bekenstein-Hawking Entropy:")
print()
print("  \$ S_{BH} = \\frac{k_B c^3 A}{4 G \\hbar} = \\frac{k_B A}{4 \\ell_P^2} \$")
print()
print("  where \$ \\ell_P = \\sqrt{G\\hbar/c^3} \\approx 1.6 \\times 10^{-35} \$ m (Planck length)")
print()

# Different black hole masses
masses_solar = [1, 10, 1e6, 4e6, 1e9]  # Solar masses
labels = ['Stellar BH', '10 M☉ BH', 'SMBH (10⁶)', 'Sgr A*', 'Quasar BH']

print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  BLACK HOLE PROPERTIES BY MASS")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()

for i, (m_sol, label) in enumerate(zip(masses_solar, labels)):
    M = m_sol * M_sun
    r_s = schwarzschild_radius(M)
    T = hawking_temperature(M)
    S = bekenstein_hawking_entropy(M)
    t_evap = evaporation_time(M)
    P = hawking_power(M)

print(f"  {label} (\$ M = {m_sol:.0e} \\, M_\\odot \$):")
    print(f"    Schwarzschild radius: \$ r_s = {r_s:.2e} \$ m")
    print(f"    Temperature: \$ T_H = {T:.2e} \$ K")
    print(f"    Entropy: \$ S = {S:.2e} \$ J/K")
    print(f"    Evaporation time: \$ t = {t_evap/year:.2e} \$ years")
    print()

# Compare to CMB temperature
T_cmb = 2.725  # K
M_cmb_equiv = hbar * c**3 / (8 * np.pi * G * kB * T_cmb)

print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  COSMIC PERSPECTIVE")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print(f"  CMB temperature: \$ T_{{CMB}} = 2.725 \$ K")
print(f"  BH mass with T = T_CMB: \$ M \\approx {M_cmb_equiv/M_sun:.2e} \\, M_\\odot \$")
print()
print("  → All astrophysical BHs are COLDER than the CMB!")
print("  → They absorb more than they emit ⟹ growing, not evaporating")
print()

# Create visualization
fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Mass range
M_range = np.logspace(-20, 10, 500) * M_sun  # From primordial to SMBH

# Plot 1: Temperature vs Mass
ax1 = axes[0, 0]
T_range = [hawking_temperature(M) for M in M_range]
ax1.loglog(M_range/M_sun, T_range, 'r-', linewidth=2)
ax1.axhline(y=T_cmb, color='blue', linestyle='--', label=f'CMB = {T_cmb} K')
ax1.axhline(y=1e12, color='orange', linestyle='--', alpha=0.5, label='Hadronic (10¹² K)')
ax1.set_xlabel(r'$M/M_{\odot}$', fontsize=12)
ax1.set_ylabel(r'$T_H$ (K)', fontsize=12)
ax1.set_title('Hawking Temperature vs Mass', fontsize=14, fontweight='bold')
ax1.legend(fontsize=9)
ax1.grid(True, alpha=0.3, which='both')
ax1.set_xlim(1e-20, 1e10)
ax1.fill_between(M_range/M_sun, T_range, T_cmb, where=np.array(T_range)>T_cmb,
                 alpha=0.2, color='red', label='Evaporating')

# Plot 2: Entropy vs Mass
ax2 = axes[0, 1]
S_range = [bekenstein_hawking_entropy(M) for M in M_range]
ax2.loglog(M_range/M_sun, S_range, 'b-', linewidth=2)
ax2.set_xlabel(r'$M/M_{\odot}$', fontsize=12)
ax2.set_ylabel(r'$S_{BH}$ (J/K)', fontsize=12)
ax2.set_title('Bekenstein-Hawking Entropy vs Mass', fontsize=14, fontweight='bold')
ax2.grid(True, alpha=0.3, which='both')
ax2.set_xlim(1e-20, 1e10)

# Add comparison line for S ∝ M²
M_ref = 1 * M_sun
S_ref = bekenstein_hawking_entropy(M_ref)
ax2.annotate(r'$S \propto M^2$', xy=(1e3, 1e80), fontsize=11, color='blue')

# Plot 3: Evaporation Time vs Mass
ax3 = axes[1, 0]
t_range = [evaporation_time(M)/year for M in M_range]
ax3.loglog(M_range/M_sun, t_range, 'g-', linewidth=2)
ax3.axhline(y=13.8e9, color='purple', linestyle='--', label='Age of Universe')
ax3.set_xlabel(r'$M/M_{\odot}$', fontsize=12)
ax3.set_ylabel(r'$t_{evap}$ (years)', fontsize=12)
ax3.set_title('Evaporation Time vs Mass', fontsize=14, fontweight='bold')
ax3.legend(fontsize=9)
ax3.grid(True, alpha=0.3, which='both')
ax3.set_xlim(1e-20, 1e10)
ax3.annotate(r'$t \propto M^3$', xy=(1e-10, 1e30), fontsize=11, color='green')

# Plot 4: Four Laws of BH Thermodynamics
ax4 = axes[1, 1]
ax4.axis('off')

laws_text = """
┌─────────────────────────────────────────────────────────┐
│         Four Laws of Black Hole Mechanics               │
├─────────────────────────────────────────────────────────┤
│                                                         │
│  0th Law:  κ = const on horizon  ↔  T uniform          │
│                                                         │
│  1st Law:  dM = (κ/8πG)dA + ΩdJ + ΦdQ                  │
│            ↔  dE = TdS + work terms                     │
│                                                         │
│  2nd Law:  δA ≥ 0  ↔  δS ≥ 0                           │
│            (Area theorem → entropy increase)            │
│                                                         │
│  3rd Law:  κ = 0 unattainable  ↔  T = 0 unattainable   │
│                                                         │
├─────────────────────────────────────────────────────────┤
│  κ = surface gravity = 1/(4M) for Schwarzschild        │
│  A = horizon area, J = angular momentum, Q = charge    │
└─────────────────────────────────────────────────────────┘
"""
ax4.text(0.5, 0.5, laws_text, transform=ax4.transAxes, fontsize=10,
         verticalalignment='center', horizontalalignment='center',
         family='monospace', bbox=dict(boxstyle='round', facecolor='black', alpha=0.8))

plt.tight_layout()
plt.savefig('bh_thermodynamics.png', dpi=150, bbox_inches='tight')

print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  INFORMATION PARADOX")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  If BHs evaporate completely, what happens to information?")
print()
print("  • Hawking radiation appears thermal (no information)")
print("  • But quantum mechanics requires unitarity (info preserved)")
print("  • Resolution likely involves quantum gravity")
print()
print("✓ Plot saved as 'bh_thermodynamics.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Thermodynamic Properties

This Fortran code calculates black hole thermodynamic properties and demonstrates the four laws of black hole mechanics with their thermodynamic correspondences.

BH Thermodynamics Visualization

Python

Hawking temperature, entropy, and evaporation time vs mass

bh_thermo_plot.py130 lines

import subprocess, tempfile, os, io, base64
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fortran_source = r"""
program bh_thermo_plot
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: pi = 4.0_dp * atan(1.0_dp)
    real(dp), parameter :: G = 6.674e-11_dp
    real(dp), parameter :: c = 3.0e8_dp
    real(dp), parameter :: hbar = 1.055e-34_dp
    real(dp), parameter :: kB = 1.381e-23_dp
    real(dp), parameter :: M_sun = 1.989e30_dp
    real(dp), parameter :: year = 3.156e7_dp

real(dp) :: M, r_s, A, T_H, S_BH, t_evap, log_M
    integer :: i, n_pts

n_pts = 200
    print '(A)', 'DATA_START'
    do i = 1, n_pts
        log_M = -1.0_dp + real(i-1, dp) * 11.0_dp / real(n_pts-1, dp)
        M = 10.0_dp**log_M * M_sun

r_s = 2.0_dp * G * M / c**2
        A = 4.0_dp * pi * r_s**2
        T_H = hbar * c**3 / (8.0_dp * pi * G * M * kB)
        S_BH = kB * c**3 * A / (4.0_dp * G * hbar)
        t_evap = 5120.0_dp * pi * G**2 * M**3 / (hbar * c**4)

write(*,'(ES16.8,A,ES16.8,A,ES16.8,A,ES16.8,A,ES16.8)') &
            M/M_sun, ',', T_H, ',', S_BH/kB, ',', t_evap/year, ',', r_s
    end do
    print '(A)', 'DATA_END'
end program bh_thermo_plot
"""

workdir = tempfile.mkdtemp()
src = os.path.join(workdir, 'bhthermo.f90')
exe = os.path.join(workdir, 'bhthermo')

with open(src, 'w') as f:
    f.write(fortran_source)

comp = subprocess.run(['gfortran', '-O2', '-o', exe, src],
                      capture_output=True, text=True, timeout=30)
if comp.returncode != 0:
    print("Compilation error:", comp.stderr)
else:
    result = subprocess.run([exe], capture_output=True, text=True, timeout=30, cwd=workdir)
    output = result.stdout

lines = []
    in_data = False
    for line in output.split('\n'):
        if 'DATA_START' in line:
            in_data = True
            continue
        if 'DATA_END' in line:
            break
        if in_data and line.strip():
            lines.append(line.strip())

m_arr, T_arr, S_arr, t_arr, rs_arr = [], [], [], [], []
    for line in lines:
        vals = [float(x) for x in line.split(',')]
        m_arr.append(vals[0])
        T_arr.append(vals[1])
        S_arr.append(vals[2])
        t_arr.append(vals[3])
        rs_arr.append(vals[4])

m_arr = np.array(m_arr)
    T_arr = np.array(T_arr)
    S_arr = np.array(S_arr)
    t_arr = np.array(t_arr)

fig, axes = plt.subplots(1, 3, figsize=(18, 6))
    fig.patch.set_facecolor('#0f172a')
    fig.suptitle('Black Hole Thermodynamics', color='white', fontsize=14, fontweight='bold')

ax = axes[0]
    ax.set_facecolor('#1e293b')
    ax.loglog(m_arr, T_arr, color='#ef4444', linewidth=2)
    ax.axhline(y=2.725, color='#94a3b8', linestyle='--', alpha=0.7, label='CMB temperature (2.725 K)')
    ax.set_xlabel('Mass ($M_\\odot$)', color='#cbd5e1')
    ax.set_ylabel('Hawking Temperature (K)', color='#cbd5e1')
    ax.set_title('$T_H = \\hbar c^3 / (8\\pi G M k_B)$', color='white')
    ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

ax2 = axes[1]
    ax2.set_facecolor('#1e293b')
    ax2.loglog(m_arr, S_arr, color='#4ade80', linewidth=2)
    ax2.set_xlabel('Mass ($M_\\odot$)', color='#cbd5e1')
    ax2.set_ylabel('Entropy ($S/k_B$)', color='#cbd5e1')
    ax2.set_title('$S_{BH} = A / (4 \\ell_P^2)$', color='white')
    ax2.tick_params(colors='#cbd5e1')
    for spine in ax2.spines.values():
        spine.set_color('#334155')
    ax2.grid(True, color='#334155', alpha=0.4)

ax3 = axes[2]
    ax3.set_facecolor('#1e293b')
    ax3.loglog(m_arr, t_arr, color='#a78bfa', linewidth=2)
    ax3.axhline(y=1.38e10, color='#fbbf24', linestyle='--', alpha=0.7, label='Age of Universe')
    ax3.set_xlabel('Mass ($M_\\odot$)', color='#cbd5e1')
    ax3.set_ylabel('Evaporation time (years)', color='#cbd5e1')
    ax3.set_title('$t_{evap} \\propto M^3$', color='white')
    ax3.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)
    ax3.tick_params(colors='#cbd5e1')
    for spine in ax3.spines.values():
        spine.set_color('#334155')
    ax3.grid(True, color='#334155', alpha=0.4)

plt.tight_layout()
    buf = io.BytesIO()
    plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
    buf.seek(0)
    img_b64 = base64.b64encode(buf.read()).decode()
    buf.close()
    plt.close()
    print(f'<img src="data:image/png;base64,{img_b64}" />')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Singularities Next: ADM Formalism →