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black hole information paradox

Curiosities: Black Hole Information Paradox

1. Classical Black Holes

In General Relativity, a black hole is a spacetime region with an event horizon.
The simplest solution is the Schwarzschild metric:

ds2=−(1−2GMr)dt2+(1−2GMr)−1dr2+r2dΩ2.ds^2 = -\left(1-\frac{2GM}{r}\right) dt^2 + \left(1-\frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2.

Classically, nothing (not even light) escapes once it crosses r=2GMr=2GM.
Thus, information about the matter that fell in seems lost.

2. Hawking Radiation

Hawking (1974) combined quantum field theory with curved spacetime.
He found that black holes emit thermal radiation:

⟨Nω⟩=1eℏω/kBTH−1,\langle N_\omega \rangle = \frac{1}{e^{\hbar \omega / k_B T_H} - 1},

with Hawking temperature:

TH=ℏc38πGMkB.T_H = \frac{\hbar c^3}{8\pi GM k_B}.

And the Bekenstein–Hawking entropy:

SBH=kBc34GℏA,S_{\text{BH}} = \frac{k_B c^3}{4 G \hbar} A,

where AA is the horizon area.

3. The Information Paradox

If Hawking radiation is purely thermal, then:

  • Initial state: pure (quantum field + collapsing matter)
  • Final state: mixed thermal state

This implies non-unitary evolution:

ρpure    ⟶    ρthermal.\rho_{\text{pure}} \;\;\longrightarrow\;\; \rho_{\text{thermal}}.

But quantum mechanics requires unitarity.
Thus, either:

  1. Quantum mechanics fails, or
  2. General relativity’s description of black holes fails.

This conflict is the Black Hole Information Paradox.

4. The Page Curve

Page argued (1993) that if unitarity holds, the entanglement entropy of radiation Srad(t)S_{\text{rad}}(t) must follow the Page curve:

  • Increases initially (Hawking’s result)
  • Peaks at Page time tPage∼(1/2) tevapt_{\text{Page}} \sim (1/2)\, t_{\text{evap}}
  • Then decreases to zero as radiation becomes pure

Hawking’s calculation predicts a monotonically increasing entropy, in contradiction with unitarity.

5. Proposed Resolutions

(a) Information loss (Hawking’s original view)

Quantum mechanics is modified; evolution not unitary.
But this undermines the foundation of QM.

(b) Remnants

Black holes leave Planck-mass remnants storing information.
Criticism: requires infinite species, problematic thermodynamics.

(c) Information escapes with radiation

Quantum gravity corrections allow subtle correlations in Hawking radiation.
Supported by AdS/CFT correspondence, which is manifestly unitary.

(d) Firewalls

At the horizon, effective field theory breaks down, creating high-energy “firewalls” that destroy infalling observers.
Violates the equivalence principle.

(e) Islands and Quantum Extremal Surfaces

Recent progress (2019–2020):
Using the replica trick and quantum extremal surfaces, the island formula reproduces the Page curve:

S(R)=min⁡I  ext{Area(∂I)4Gℏ+Sbulk(R∪I)}.S(R) = \min_{\mathcal{I}} \; \text{ext} \left\{ \frac{\text{Area}(\partial \mathcal{I})}{4G\hbar} + S_{\text{bulk}}(R \cup \mathcal{I}) \right\}.

This suggests unitarity is preserved.

6. Status

  • Consensus today: information is not lost.
  • AdS/CFT and the island formula strongly indicate unitary evaporation.
  • The precise microscopic mechanism in real (non-AdS) black holes remains under investigation.

The paradox continues to be a window into quantum gravity.