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The black hole information paradox and the Page curve

If a black hole evaporates into thermal radiation, what happens to everything that fell in? Since 1976 this question has pitted general relativity against quantum mechanics. The 2019 calculations changed the picture.

8 min readUpdated 8 October 2026

The problem

Quantum mechanics rests on a strict principle: information is never lost. Burn a book, and in principle, by measuring the smoke, light and ashes perfectly, you could rebuild every page. The laws are reversible.

Yet Hawking radiation is thermal: it depends only on the black hole's mass, charge and spin, not on what fell in. If the black hole evaporates entirely, the information seems erased from the Universe. In 1976 Hawking concluded that gravity violates quantum mechanics. Many physicists rejected that conclusion: one of the two pillars of physics has to give.

The Page curve

In 1993, Don Page proposed a test: follow the entropy of the radiation as the black hole evaporates. If information is lost, it keeps rising to the end. If information is preserved, it must rise, peak around the middle of the black hole's life, then fall back to zero. Move the time slider:

The Page curve, step by step
time →entropyPage time
  • Black hole entropy
  • Radiation according to Hawking (1974)
  • Radiation if information escapes (Page curve)

Before the Page time, both scenarios look the same: the radiation looks thermal and its entropy rises.

Simplified model: the entropy carried by the radiation equals the entropy lost by the black hole, whose entropy scales as M², with M ∝ (1 - t)^(1/3). In this model the Page time falls at about 65% of the lifetime; more realistic calculations put it a little earlier.

Go deeperWhy the entropy has to come down

Here entropy measures the entanglement between the radiation and the black hole. At first, each emitted particle is entangled with the black hole, and that entanglement grows. But a black hole cannot be entangled beyond its own entropy, which shrinks as it evaporates. After the "Page time", new particles must therefore be entangled with earlier ones, and the total radiation entropy falls to zero: a pure state carrying all the information.

2012: the firewall

Almheiri, Marolf, Polchinski and Sully pushed the argument to the limit: if information comes out, late particles cannot be entangled both with the early radiation and with their partner behind the horizon. That entanglement would have to break, creating a "firewall" of energy at the horizon, contradicting the idea that a freely falling observer notices nothing there.

2013: ER = EPR

Juan Maldacena and Leonard Susskind proposed a way out: two entangled systems would be linked by a microscopic wormhole. The black hole interior and the distant radiation would then not be independent regions, and the conflict would vanish. This conjecture echoes the central idea of this site: spacetime geometry would be made of entanglement.

2019: islands

Two teams, Geoffrey Penington's and that of Almheiri, Engelhardt, Marolf and Maxfield, showed that when the radiation entropy is computed taking spacetime geometry into account, a region inside the black hole, since called an "island", must be counted together with the radiation. The result reproduces the Page curve exactly. For the first time, a gravity-based calculation indicated that information comes out.

These calculations were done in simplified models, often in Anti-de Sitter space. How the information is actually encoded in the radiation, and what happens in our own Universe, remains one of the big questions in physics.

Frequently asked questions

Has the information paradox been solved?

Not entirely. Since 2019, calculations show that the radiation entropy does follow the Page curve, which indicates that information comes out. But we cannot yet describe in detail how it is encoded in the radiation, nor how this applies to our Universe.

Did Hawking change his mind?

Yes. In 2004 he publicly accepted that information was probably not destroyed, and lost a famous bet to physicist John Preskill.

What does "information" mean here?

Everything that tells two starting states apart: a book and a star may produce black holes that look the same, but quantum mechanics requires the difference to remain written somewhere.

Sources

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