Math 2.0

Relativity and gravitational theory Black holes Problem

Black hole information paradox

Builds on Hawking radiationBekenstein–Hawking entropy

Problem

Suppose a black hole forms from matter in a pure quantum state and then evaporates completely by Hawking radiation. In Hawking's semiclassical computation the radiation is exactly thermal, so the final state is mixed. A map from pure states to mixed states cannot be a unitary evolution, as quantum mechanics requires. The problem is to decide whether the evaporation of a black hole is unitary. If it is, the problem is to explain how the information about the initial state is carried out by the radiation.

Remarks

Hawking stated the paradox in 1976 and argued at first that unitarity fails in gravitational collapse. Page observed in 1993 that unitarity has a consequence for the entanglement entropy of the emitted radiation. It must rise and then fall back to zero, following what is now called the Page curve. The turning point comes roughly when the black hole has lost half of its Bekenstein–Hawking entropy. The AdS/CFT correspondence describes some black holes by an ordinary unitary quantum theory, and this has led many physicists to expect unitarity. Since 2019, semiclassical computations of entropy that include so-called islands have reproduced the Page curve in several models. There is no consensus on the mechanism, and the problem remains open as a question of physics.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

Show the path as a graph
Cauchy surface, concept, no evidence levelCauchy surfaceconceptCongruence of geodesics, concept, no evidence levelCongruence of geodesi…conceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptEnergy conditions, concept, no evidence levelEnergy conditionsconceptEvent horizon, concept, no evidence levelEvent horizonconceptGeodesic, concept, no evidence levelGeodesicconceptKilling horizon, concept, no evidence levelKilling horizonconceptKilling vector field, concept, no evidence levelKilling vector fieldconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptRicci curvature, concept, no evidence levelRicci curvatureconceptWave equation, concept, no evidence levelWave equationconceptPenrose–Hawking singularity theorems, statement, no evidence levelPenrose–Hawking singu…statementRaychaudhuri equation, statement, no evidence levelRaychaudhuri equationstatementHawking area theorem, statement, no evidence levelHawking area theoremstatementLaws of black hole mechanics, statement, no evidence levelLaws of black hole me…statementHawking radiation, statement, no evidence levelHawking radiationstatementBekenstein–Hawking entropy, statement, no evidence levelBekenstein–Hawking en…statementBlack hole information paradox, problem, no evidence levelBlack hole informatio…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cauchy surface concept
  2. Congruence of geodesics concept
  3. Einstein field equations concept
  4. Energy conditions concept
  5. Event horizon concept
  6. Geodesic concept
  7. Killing horizon concept
  8. Killing vector field concept
  9. Lorentzian manifold concept
  10. Ricci curvature concept
  11. Wave equation concept
  12. Penrose–Hawking singularity theorems statement
  13. Raychaudhuri equation statement
  14. Hawking area theorem statement
  15. Laws of black hole mechanics statement
  16. Hawking radiation statement
  17. Bekenstein–Hawking entropy statement
  18. Black hole information paradox problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Black_hole_information_paradox
About this page
Tag6ETFPK0E2G
URIm2://6ETFPK0E2G
Kindproblem
WritingW0 stub
MSC83C47, 83C57, 81P45
StatusThis copy of the site has no build data. Run m2 status 6ETFPK0E2G.
AboutBlack hole, Hawking radiation
UsesHawking radiation, Bekenstein–Hawking entropy
Changed3 November 2026

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