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Eventide Prophet

Black Hole Information Paradox Physics

\[T_H = \frac{\hbar c^{3}}{8\pi G M k_B}\]
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When a black hole evaporates, where does everything that fell in go? Hawking showed in 1974 that black holes radiate, and in 1976 argued that the information inside is lost, which quantum mechanics forbids. Page sharpened the puzzle in 1993. Recent work points toward information escaping, but the question is still open. Eventide Prophet on a pixel-art unisex tee; equation, symbols and sources in the panel below.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Ter vergelijking: maat M is 20 in (50.80 cm) breed, ongeveer 3.143 × 1034 Plancklengtes (lP = 1.616255 × 10−35 m). Een meetlint is sneller.

Measurements in inches, ±1.5 in tolerance.

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The physics

Hawking showed that black holes glow faintly, with a temperature that goes down as their mass goes up. Bekenstein's entropy is proportional to the horizon area. If evaporation preserves information, the entropy of the radiation has to stay below the black hole's: that is the open information puzzle.

Eventide Prophet stands at the horizon holding a question nobody has answered: when a black hole evaporates, where does what fell in go?

Start with what is established on paper. In 1972 Jacob Bekenstein argued that black holes carry entropy. In 1974 Stephen Hawking found that they radiate, with the temperature in the first formula, and that fixed the entropy, the second formula, proportional to the horizon's area. A black hole left alone shrinks as it radiates, and in principle can vanish entirely.

The trouble is what the radiation looks like. In Hawking's calculation it is exactly thermal: its details depend on the black hole's mass, charge and spin and nothing else. Throw in an encyclopaedia or an equal mass of sand and the glow coming out is the same. If the black hole disappears and only that glow is left, the information about the encyclopaedia is gone. In 1976 Hawking argued just that. Quantum mechanics, as normally understood, forbids it: the exact state now determines the exact state before.

In 1993 Don Page sharpened the problem. If information does get out, the entanglement entropy of the radiation, the third line on this page, must rise at first and then fall back to zero, and it can never exceed the black hole's own entropy. Hawking's original calculation never turns over. Recent theoretical work has found ways to reproduce Page's turnover, and many physicists now expect information is preserved. How it gets out, through what physics, and whether those calculations capture real black holes, are not settled. Hawking radiation itself has never been observed.

This is the black hole information paradox. It is open.

Eventide Prophet waits at the edge. The question has not been answered, only asked better.

Equations

\[T_H = \frac{\hbar c^{3}}{8\pi G M k_B}\]
\[S_{BH} = \frac{k_B c^{3} A}{4 G \hbar}\]
\[\begin{gathered} S_{\text{rad}}(t) \le S_{BH}(t) \\ \text{unitary evaporation} \\ (\text{Page 1993}) \end{gathered}\]
SymbolMeaningUnit
\(T_H\) Hawking temperature \(\mathrm{K}\)
\(M\) black-hole mass \(\mathrm{kg}\)
\(k_B\) Boltzmann constant \(\mathrm{J/K}\)
\(S_{BH}\) Bekenstein–Hawking entropy \(\mathrm{J/K}\)
\(A\) horizon area \(\mathrm{m²}\)
\(\hbar, c, G\) as defined above \(\mathrm{J·s, m/s, m³·kg⁻¹·s⁻²}\)
\(S_{\text{rad}}\) entanglement entropy of the emitted radiation \(\mathrm{J/K}\)

Sources

  1. Bekenstein (1973) Black Holes and Entropy, Phys. Rev. D 7, 2333
  2. Hawking (1974) Black hole explosions?, Nature 248, 30
  3. Hawking (1975) Particle creation by black holes, Commun. Math. Phys. 43, 199
  4. Hawking (1976) Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460
  5. Encyclopaedia Britannica — Hawking radiation
  6. Bekenstein (1972) Black holes and the second law, Lett. Nuovo Cimento 4, 737
  7. Page (1993) Information in black hole radiation, PRL 71, 3743

Source Hawking, S. W. (1975). Particle creation by black holes. Communications In Mathematical Physics 43, 199–220.doi.org/10.1007/BF02345020

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