Where does information go in a black hole?

Hawking calculated that a black hole radiates with a thermal spectrum set by its mass, and argued that when it has fully evaporated the information about what formed it is gone.

That breaks a basic rule: quantum evolution is reversible, and a pure state cannot become a mixed one. Hawking’s paper said so explicitly, and proposed that the rule fails for gravity.

The objection came quickly, and Page set it out in 1980: if formation and evaporation can be described by an operator that respects CPT, then they can be described by one that takes pure states to pure states, and black holes are “in principle no more unpredictable than other quantum phenomena”. It is a paradox only on one assumption, that a black hole seen from outside is an ordinary quantum system evolving reversibly. Without that assumption, Hawking’s loss could be a feature of quantum gravity.

In 1993 Don Page showed how the entanglement between the radiation and the remaining hole should behave if information does come out. Treat the two as a random pure state, and the smaller subsystem carries less than half a unit of information on average, so the entanglement entropy rises while the hole is the larger system, then turns around and falls once the radiation carries as much entropy as the hole that is left.

That turning point, since named the Page time, converts a vague question into a quantitative one. Any account in which the information comes out has to reproduce it.

Since 2019 several groups have computed, in simplified models of gravity, entropies that follow the Page curve. Penington, and separately Almheiri, Engelhardt, Marolf and Maxfield, did it in May 2019, for the black hole rather than the radiation.

What changed was the region the radiation is taken to describe. Past the Page time the calculation counts a disconnected piece behind the horizon, an island, as part of that region, and it is that piece which bends the entropy back down. Almheiri, Mahajan, Maldacena and Zhao proposed that rule and computed the radiation’s curve with it, with the black hole interior becoming part of what the radiation describes.

Two groups then showed where that rule comes from: extra saddle points of the gravitational path integral, wormholes joining separate copies of the black hole, which they named replica wormholes.

It is a genuine advance and, by its own authors’ account, not a complete resolution: it gives the entropy of the radiation, not the quantum state that would carry the information out. The calculations work in particular model spacetimes, typically with the radiation drawn off into a region without gravity, and what the connecting geometries mean physically is not agreed.

Geng, Karch, Raju, Randall and colleagues argue that islands may not be consistent in ordinary gravity, where the graviton is massless. Raju argues that the paradox is resolved by a different route, with a complete copy of what fell in always available outside the hole.

Almheiri, Marolf, Polchinski and Sully proposed in 2012 that the horizon is not as featureless as relativity says, and that an infalling observer burns there. Maldacena and Susskind proposed the year after that entangled systems are joined by wormholes, which would connect the radiation to the black hole’s interior and leave the horizon smooth. The older alternatives were a remnant that keeps the information and Hawking’s original answer, that it is lost.

Each is a proposal and each abandons something that seemed safe. The versions differ on which.

No agreed account describes, physically, how the information in something that crossed a horizon gets out into radiation emitted from just outside it.

The problem is a collision between the two theories in a regime where both are supposed to apply, which is why it has been treated as an important clue to what a theory of quantum gravity has to look like.