Black Hole Information Paradox
When a black hole evaporates via Hawking radiation, what happens to the information that fell into it? This questionâthe black hole information paradoxâhas 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 radiationâa mixed state
The problem: Thermal radiation carries no information about the initial state. Information appears to be lost, violating unitarityâa 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 increaseâin 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.