Black Hole Information Paradox

When a black hole evaporates via Hawking radiation, what happens to the information that fell into it? This question—the black hole information paradox—has been one of the deepest puzzles in theoretical physics for nearly 50 years, touching the foundations of quantum mechanics and spacetime.

The Paradox

Setup

Start with a pure quantum state (say, an encyclopedia)
Throw it into a black hole
The black hole radiates thermally via Hawking radiation
Eventually the black hole completely evaporates
We're left with only thermal radiation—a mixed state

The problem: Thermal radiation carries no information about the initial state. Information appears to be lost, violating unitarity—a fundamental principle of quantum mechanics.

$$|\psi\rangle_{\text{pure}} \xrightarrow{\text{BH evaporation}} \rho_{\text{thermal}}$$

This violates the principle that quantum evolution is unitary and reversible.

The Page Curve

Don Page (1993) computed what the entropy of Hawking radiation should look like if information is conserved:

Early time: Radiation is thermal, entropy increases linearly as the black hole evaporates
Page time ($t \sim t_{evap}/2$): Entropy reaches maximum, then begins to decrease
Late time: Entropy approaches zero as black hole completely evaporates

Hawking's calculation predicts monotonic increase—in stark conflict with unitarity. Recent progress using quantum extremal surfaces and the "island formula" has reproduced the Page curve.

Proposed Resolutions

1. Information in Correlations

Information is gradually encoded in subtle correlations between early and late Hawking quanta. Requires breakdown of effective field theory near the horizon. (Mainstream view since ~2020)

2. Remnants

Evaporation halts at Planck scale, leaving a stable remnant containing the information. Problem: infinite species of remnants violates effective field theory.

3. Information Loss

Quantum mechanics is modified for gravity; unitarity is violated. Black holes truly destroy information. (Hawking's original position; now disfavored)

4. Black Hole Complementarity

Information is both inside and outside the black hole, but no observer can see both copies. Leads to the firewall paradox.

5. Soft Hair and Supertranslations

Hawking, Perry, and Strominger (2016) proposed black holes have "soft hair" from asymptotic symmetries storing information. Debate continues on whether this resolves the paradox.

Recent Progress: Islands and the Page Curve

In 2019-2020, Penington, Almheiri, Engelhardt, Marolf, and Maxfield computed the entropy using quantum extremal surfaces and "islands":

$$S(R) = \min\left[\text{extremal}\left(\frac{A(\partial I)}{4G} + S_{QFT}(R \cup I)\right)\right]$$

where $I$ is an "island" inside the black hole that contributes to the radiation entropy. This successfully reproduces the Page curve, suggesting information is preserved.

Status: Active area of research. Island formula works in toy models (2D gravity, AdS/CFT) but extending to realistic 4D black holes remains challenging.

Interactive Simulations

Page Curve and Entanglement Entropy During Evaporation

Python

script.py71 lines

import numpy as np

# Constants
G = 6.674e-11
c = 2.998e8
hbar = 1.055e-34
k_B = 1.381e-23
M_sun = 1.989e30

print("=" * 65)
print("THE PAGE CURVE: ENTROPY OF HAWKING RADIATION")
print("=" * 65)
print("\nThe Page curve shows how entanglement entropy of radiation")
print("evolves during black hole evaporation.\n")

M0 = 10 * M_sun
r_s0 = 2 * G * M0 / c**2
l_P = np.sqrt(hbar * G / c**3)
S_BH_initial = 4 * np.pi * r_s0**2 / (4 * l_P**2)

print(f"Initial BH: M = 10 M_sun")
print(f"Initial Bekenstein-Hawking entropy: S_BH = {S_BH_initial:.4e} k_B")

# Page time: roughly when half the entropy has been radiated
# This is when S_rad = S_BH (remaining)
print(f"\nPage time: when S_radiation = S_remaining_BH")
print(f"This occurs when M(t) = M0 / sqrt(2)")
M_page = M0 / np.sqrt(2)
S_page = 4 * np.pi * (2*G*M_page/c**2)**2 / (4*l_P**2)
print(f"M at Page time: {M_page/M_sun:.4f} M_sun")
print(f"S_BH at Page time: {S_page:.4e} k_B = S_initial/2")

# Simulate the Page curve
N_steps = 20
fractions = np.linspace(0, 0.999, N_steps)

print(f"\n{'f_evap':>8} {'M/M0':>8} {'S_BH':>14} {'S_rad(Hawking)':>16} {'S_rad(Page)':>14}")
print("-" * 65)

for f in fractions:
    M = M0 * (1 - f)**(1.0/3.0)  # M^3 decreases linearly with time
    r_s = 2 * G * M / c**2
    S_BH = 4 * np.pi * r_s**2 / (4 * l_P**2)

# Hawking (incorrect): entropy of radiation grows monotonically
    S_rad_hawking = S_BH_initial - S_BH

# Page (correct): entropy follows the Page curve
    # Before Page time: S_rad = S_BH_initial - S_BH (grows)
    # After Page time: S_rad = S_BH (decreases with BH)
    if S_rad_hawking <= S_BH:
        S_rad_page = S_rad_hawking
    else:
        S_rad_page = S_BH

print(f"{f:>8.3f} {M/M0:>8.4f} {S_BH:>14.4e} {S_rad_hawking:>16.4e} {S_rad_page:>14.4e}")

print("\nKey insight: Hawking&apos;s calculation gives monotonically")
print("increasing S_rad (information lost!). The Page curve shows")
print("S_rad must eventually decrease, returning to zero (unitarity).")

# Scrambling time
print("\n--- Scrambling Time ---")
print("Time for infalling information to be mixed into Hawking radiation:")
for m in [10, 1e6, 1e9]:
    M = m * M_sun
    r_s = 2 * G * M / c**2
    S = 4 * np.pi * r_s**2 / (4 * l_P**2)
    t_scr = r_s / c * np.log(S)
    print(f"  M = {m:.0e} M_sun: t_scramble = {t_scr:.4e} s ({t_scr*1000:.2f} ms)" if t_scr < 1 else f"  M = {m:.0e} M_sun: t_scramble = {t_scr:.4e} s")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Scrambling Time and Information Recovery Timescales

Fortran

program.f9059 lines

program information_paradox
    implicit none
    double precision :: G, c, hbar, k_B, M_sun, pi, l_P, yr
    double precision :: M, r_s, S_BH, T_H, t_evap, t_scramble
    double precision :: masses(7)
    character(len=20) :: names(7)
    integer :: i

G = 6.674d-11
    c = 2.998d8
    hbar = 1.055d-34
    k_B = 1.381d-23
    M_sun = 1.989d30
    pi = 3.14159265358979d0
    l_P = sqrt(hbar * G / c**3)
    yr = 3.156d7

print *, '============================================================'
    print *, 'INFORMATION PARADOX: KEY TIMESCALES'
    print *, '============================================================'
    print *, ''
    print *, 'Scrambling time: t_scr ~ (r_s/c) * ln(S_BH)'
    print *, 'Page time:       t_Page ~ t_evap / 2'
    print *, 'Evaporation:     t_evap = 5120*pi*G^2*M^3/(hbar*c^4)'
    print *, ''

masses(1) = 1.0d0;    names(1) = '1 M_sun'
    masses(2) = 10.0d0;   names(2) = '10 M_sun'
    masses(3) = 100.0d0;  names(3) = '100 M_sun'
    masses(4) = 1.0d4;    names(4) = '1e4 M_sun'
    masses(5) = 1.0d6;    names(5) = '1e6 M_sun (Sgr A*)'
    masses(6) = 1.0d9;    names(6) = '1e9 M_sun'
    masses(7) = 1.0d10;   names(7) = '1e10 M_sun'

print '(A20, A14, A14, A14, A14)', &
          'Object', 'S_BH/k_B', 't_scramble', 't_Page (yr)', 't_evap (yr)'
    print *, '----------------------------------------------------------------------'

do i = 1, 7
        M = masses(i) * M_sun
        r_s = 2.0d0 * G * M / c**2
        S_BH = 4.0d0 * pi * r_s**2 / (4.0d0 * l_P**2)
        T_H = hbar * c**3 / (8.0d0 * pi * G * M * k_B)
        t_evap = 5120.0d0 * pi * G**2 * M**3 / (hbar * c**4)
        t_scramble = (r_s / c) * log(S_BH)

print '(A20, ES14.3, ES14.3, ES14.3, ES14.3)', &
              names(i), S_BH, t_scramble, t_evap/(2.0d0*yr), t_evap/yr
    end do

print *, ''
    print *, 'Key observations:'
    print *, '  t_scramble << t_Page << t_evap'
    print *, '  Information is scrambled fast but takes ~t_Page to start'
    print *, '  appearing in the radiation (after the Page time).'
    print *, ''
    print *, '  The paradox: Hawking radiation appears thermal (no info)'
    print *, '  but unitarity demands information is preserved.'
end program information_paradox

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Research Seminars: Information Paradox & Black Hole Evaporation

Advanced research talks on the information paradox, black hole disappearance, and modern resolutions.

Black Hole Disappearance - Ahmed Almheiri

Near-Extremal Black Hole Evaporation - Mykhaylo Usatyuk

A Black Hole Airy Tail: Part 1 - Pratik Rath

A Black Hole Airy Tail: Part 2 - Stefano Antonini

ER Toward Typical EPR - Martin Sasieta

Emergent Mixed States for Baby Universes and Black Holes - Jonah Kudler-Flam

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