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Page Curve, Islands and Replica Wormholes

If quantum mechanics holds, the radiation's entropy must follow a precise curve over time.

Family status
Frontier
Proposed
1993 and 2019
Sources
10
Reader votes
0
Basis: theoretical
Calculations since 2019 in simplified models show information can escape, but the mechanism in real black holes is open.

1The claim

If no information is lost, the entropy of Hawking radiation must rise and then fall. Calculations since 2019 reproduce that curve from gravity in simplified model universes.

2Why it might be true

Suppose a black hole forms from matter in a pure quantum state, one with no missing information. As the black hole evaporates, each particle of Hawking radiation is entangled with a partner left inside. So the entropy of the radiation grows steadily during the early part of evaporation. If no information is lost, it can never exceed the entropy of the black hole that remains. The moment the two are equal is the Page time, and after it the radiation's entropy must fall. Page estimated the full rise-and-fall shape in 1993 by treating black hole and radiation as one random quantum state.

Hawking's own method gives a different answer, in which the radiation's entropy keeps rising until the black hole is gone. The clash appears at the Page time, while the black hole is still large and the calculation should hold. Proposals in this family, such as complementarity, firewalls and ER=EPR, changed the physics to avoid the clash or live with it. None of them derived the Page curve from a calculation in gravity.

In 2019 Penington, and separately Almheiri, Engelhardt, Marolf and Maxfield, found the fall in gravity calculations. Past the Page time, a new surface just inside the horizon takes over the entropy calculation. The region behind that surface, called an island, then counts as part of the radiation. Almheiri, Mahajan, Maldacena and Zhao stated this as a general rule, the island formula. Replica wormholes, extra geometries in the gravitational path integral, showed why the rule holds. All of these calculations use simplified settings, such as two-dimensional gravity or anti-de Sitter space.

3Evidence so far

  • Page's 1993 estimate uses only quantum mechanics, and shows that the smaller part of a random pure state is almost maximally mixed
  • Penington and the Almheiri-Engelhardt-Marolf-Maxfield group found the same turn at the Page time independently, both in May 2019
  • Replica wormholes in the gravitational path integral produce the island rule, worked out explicitly in two-dimensional Jackiw-Teitelboim gravity
  • Penington, Shenker, Stanford and Yang studied the transition by summing geometries of different topology, and argued for interior reconstruction from gravity

A scientific documentary visualization of the Page curve and replica wormholes. In the upper half of the frame a glowing cyan line is drawn as a graph that rises from the left, crests at a warm-gold peak, and descends to the right, the rise-then-fall Page curve, traced out by a small luminous cursor over the eight seconds. In the lower foreground a replica-wormhole saddle, two rounded translucent cyan lobes joined by a short pinched neck, brightens as the cursor passes the peak and descends, marking it as the gravitational geometry responsible for the entropy turning back down. The visualization conveys the variant's central claim that the entropy of Hawking radiation must rise and then fall for unitarity to hold, and that the 2019 replica-wormhole calculations reproduced that curve from gravity in simplified model universes.

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4What would test it

4 predictions

  • The fine-grained entropy of the radiation follows the Page curve, rising until the Page time and then falling to zero
  • At the Page time, the surface that sets the entropy jumps inside the horizon, and the island joins the radiation
  • After the Page time, the collected radiation encodes part of the black hole interior, which can in principle be reconstructed from it
  • Information thrown into an old black hole can be decoded from the radiation after a scrambling time, as Hayden and Preskill argued

5Where it breaks

4 objections, plus 4 shared

  • The calculations work in two-dimensional gravity and anti-de Sitter settings, and nobody has extended them to realistic four-dimensional black holes
  • They give the entropy of the radiation but not its exact quantum state, which a 2021 review calls an important unresolved problem
  • The method relies on a gravitational path integral whose saddles and integration contour are not precisely defined, as its own authors note
  • Geng, Karch, Randall and collaborators argue that islands may not be consistent in ordinary gravity, where gravitons have no mass

Go deeper

Technical register

Stated precisely

If no information is lost, the entropy of a black hole's Hawking radiation must rise and then fall. Don Page estimated the curve in 1993, treating black hole and radiation as one random quantum state. For 26 years, no calculation based on gravity could reproduce the fall. Calculations in 2019 did, by finding regions inside the black hole, called islands, that count as part of the radiation.

Page time: the time at which the fine-grained entropy of the radiation equals the Bekenstein-Hawking entropy of the remaining black hole. Page modelled black hole plus radiation as a random pure state in a Hilbert space of fixed dimension mn. For m at most n, the average entropy of the smaller subsystem is close to ln m minus m/2n, nearly maximal. The exact shape of the curve depends on the black hole and on the fields it radiates.

Generalized entropy is the area of a surface divided by 4G, plus the entropy of quantum fields outside it. Engelhardt and Wall (2015) proposed extremizing it to find the quantum extremal surface. The island formula also allows disconnected pieces, the islands, and takes the smallest extremal value. Before the Page time the winning solution has no island. After it, the quantum extremal surface sits slightly inside the event horizon, about one scrambling time in the past, and the island lies behind it.

Replica wormholes: the n-th Rényi entropy is computed on n copies of the geometry glued along the radiation region. Past the Page time the dominant saddle connects the copies through the interior, a replica wormhole. In the limit n to 1 these saddles give the island formula (Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini). Penington, Shenker, Stanford and Yang interpret the wormholes as part of an ensemble average implicit in the gravity description.

Settings: Penington used AdS/CFT with absorbing boundary conditions. Almheiri, Engelhardt, Marolf and Maxfield used two-dimensional Jackiw-Teitelboim gravity coupled to a heat sink. Almheiri, Mahajan, Maldacena and Zhao gave the matter a higher-dimensional holographic dual and named the island rule. They describe the link between radiation and interior as 'in the spirit of ER=EPR', the subject of the ER=EPR variant in this family.

anti-de Sitter space
A spacetime of constant negative curvature, the natural setting for the AdS/CFT correspondence. Distinct from our universe, which has positive cosmological constant (de Sitter), but the framework provides the cleanest testing ground for holographic ideas.
entropy
A measure of disorder, equivalent to the number of microscopic arrangements consistent with what we see at large scales.
Hawking radiation
A faint thermal glow predicted to leak from black holes, slowly draining their mass over time.

6Who built it, and when

  1. 1

    Page (1993). Average entropy of a subsystem

    Phys. Rev. Lett. 71, 1291

    established

    1,735 citations

  2. 2

    Page (1993). Information in black hole radiation

    Phys. Rev. Lett. 71, 3743

    established

    1,316 citations

  3. 3

    established

    965 citations

  4. 4

    established

    1,130 citations

  5. 5

    established

    1,205 citations

  6. 6

    established

    no count yet

  7. 7

    Almheiri et al. (2020). Replica Wormholes and the Entropy of Hawking Radiation

    JHEP 05, 013

    established

    1,039 citations

  8. 8

    Almheiri et al. (2021). The entropy of Hawking radiation

    Rev. Mod. Phys. 93, 035002

    established

    no count yet

  9. 9

    established

    no count yet

  10. 10

    Penington et al. (2022). Replica wormholes and the black hole interior

    JHEP 03, 205

    established

    1,141 citations

Black Hole Information Paradox, all 7 variants

Lead variant first, then by year

  1. Black Hole Complementarity1993
  2. Page Curve, Islands and Replica Wormholesthis page1993 and 2019
  3. Fuzzballs2005
  4. Firewall Paradox2013
  5. ER = EPR2013
  6. Soft Hair2016
  7. Hawking's Information Loss1976

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