{ "id": "eperhtw56d", "slug": "black-hole-information-paradox", "title": "Black hole information paradox", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics" }, { "id": "physical.physics.relativity-and-gravity", "label": "Relativity and gravitation", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.relativity-and-gravity" }, { "id": "physical.physics.relativity-and-gravity.quantum-gravity-unification", "label": "Quantum gravity and unification", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.relativity-and-gravity.quantum-gravity-unification" }, { "id": "physical.physics.relativity-and-gravity.quantum-gravity-unification.qgu-string-holography", "label": "String-theoretic gravity and holography", "api_url": 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"api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.physics.relativity-and-gravity.quantum-gravity-unification", "path": [ { "id": "geo.weu", "label": "Western Europe", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu" }, { "id": "geo.weu.t1946", "label": "Western Europe · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946" }, { "id": "geo.weu.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical" }, { "id": "geo.weu.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.physics" }, { "id": "geo.weu.t1946.physical.physics.relativity-and-gravity", "label": "Relativity and gravitation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.physics.relativity-and-gravity" }, { "id": "geo.weu.t1946.physical.physics.relativity-and-gravity.quantum-gravity-unification", "label": "Quantum gravity and unification", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1946.physical.physics.relativity-and-gravity.quantum-gravity-unification" } ] } ], "excerpt": "The black hole information paradox is a conflict between quantum mechanics and general relativity: Hawking's 1976 calculation suggests evaporating black holes destroy information, which unitary quantum evolution forbids.", "snippet": "The black hole information paradox is a conflict between quantum mechanics and general relativity: Hawking's 1976 calculation suggests evaporating black holes destroy information, which unitary quantum evolution forbids.", "node": "physical.physics.relativity-and-gravity.quantum-gravity-unification.qgu-string-holography.string-black-hole-physics", "markdown": "# Black hole information paradox\n\nThe black hole information paradox is a conflict between quantum mechanics and general relativity that arises when a black hole evaporates: Hawking's calculation predicts that the radiation is thermal and carries no information about what formed the hole, so a pure quantum state would evolve into a mixed state, which unitary quantum evolution forbids. [Stephen Hawking](https://www.edgepedia.org/stephen-hawking) of the [University of Cambridge](https://www.edgepedia.org/university-of-cambridge) argued in 1976 that this information loss is a fundamental limitation on predictability, analogous to but additional to the ordinary quantum-mechanical uncertainty principle, and that there is no S matrix for black-hole formation and evaporation.[1](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.2460)\n\n| Key fact | Detail |\n|---|---|\n| Origin | Hawking (1976): pair creation at the horizon means part of the information falls into the hole; the final state is a density matrix, not a pure state.[1](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.2460) |\n| The tension | Hawking radiation depends only on the black hole's mass, charge, and angular momentum, so different initial states that form identical holes would produce identical radiation.[2](https://www.sciencedirect.com/science/article/pii/S0370157321003720) |\n| Unitarity expectation | If evaporation is described by an S matrix, information must come out in the radiation, though possibly so slowly or so spread out that it never appears in a perturbative analysis.[3](https://arxiv.org/abs/hep-th/9306083) |\n| Small-corrections theorem | Mathur showed that small corrections to the leading-order Hawking computation cannot remove the entanglement between radiation and hole; corrections must be order one.[4](https://axi.lims.ac.uk/paper/0909.1038) |\n| Firewall argument | Almheiri, Marolf, Polchinski, and Sully (2012) showed that pure radiation, horizon emission of information, and a smooth horizon cannot all hold simultaneously.[5](https://arxiv.org/abs/1207.3123) |\n| Holographic evidence | In AdS/CFT, the unitarity of the boundary CFT strongly suggests all information about a state that forms a black hole is returned in the Hawking radiation.[6](https://arxiv.org/abs/hep-th/9903237) |\n| Recent progress | Replica wormholes in the gravitational path integral lead to the island rule, which yields a Page curve consistent with unitary evaporation.[7](https://arxiv.org/abs/1911.12333) |\n\n## Hawking's argument\n\nHawking showed in 1974 that black holes radiate heat and can eventually evaporate, apparently erasing all information about what fell into them.[8](https://www.livescience.com/space/black-holes/science-history-stephen-hawking-writes-a-tiny-paper-and-turns-our-understanding-of-black-holes-inside-out-march-1-1974) In a 1976 paper he contended that black holes produce particles in pairs, one of which drops into the hole while the other may get away to infinity; since some information about the system's state disappears into the hole, the end result is described by a density matrix instead of a pure quantum state.[1](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.2460) The radiation depends only on the black hole's mass, charge, and angular momentum, so the detailed state of the infalling matter is absent from the output.[2](https://www.sciencedirect.com/science/article/pii/S0370157321003720)\n\nA useful diagnostic is the fine-grained (von Neumann) entropy of the [Hawking radiation](https://www.edgepedia.org/hawking-radiation), \\( S_{R} = -\\mathrm{Tr}\\, \\rho_{R} \\log \\rho_{R} \\), where \\( \\rho_{R} \\) is the density matrix of the radiation. This entropy initially increases because the radiation is entangled with its partners in the black hole interior. If evaporation is unitary, the entropy must eventually fall back to zero following the [Page curve](https://www.edgepedia.org/page-curve); Hawking's calculation instead predicts an entropy that rises monotonically.[7](https://arxiv.org/abs/1911.12333)\n\n## Why small corrections do not help\n\nA natural hope was that subleading quantum-gravity corrections, small for each radiated quantum but accumulating over the large number of emitted quanta, would restore unitarity while keeping the semiclassical horizon. Samir D. Mathur, a physicist at The Ohio State University, proved that this fails: small corrections to the leading-order Hawking computation cannot remove the entanglement between the radiation and the hole, using the strong subadditivity property of quantum entanglement entropy.[9](https://www.sciencedirect.com/topics/physics-and-astronomy/hawking-radiation) • [4](https://axi.lims.ac.uk/paper/0909.1038) The conclusion is that recovering information requires order-one corrections, which means the horizon itself cannot remain featureless.[4](https://axi.lims.ac.uk/paper/0909.1038)\n\n## Complementarity and firewalls\n\n[Black hole complementarity](https://www.edgepedia.org/black-hole-complementarity), associated with [Leonard Susskind](https://www.edgepedia.org/leonard-susskind) and collaborators, framed the paradox as a contradiction between unitarity and the principle of equivalence.[10](https://onlinelibrary.wiley.com/doi/10.1155/2018/4130417) In 2012, Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully (AMPS) contended that three claims are mutually inconsistent: that Hawking radiation forms a pure state, that the radiation's information escapes from near the horizon while low-energy effective field theory remains valid outside some microscopic distance, and that an observer falling in finds nothing special at the horizon. Their most conservative resolution was that the infalling observer burns up at the horizon, the firewall.[5](https://arxiv.org/abs/1207.3123) Daniel Harlow of Princeton University countered that a version of complementarity evades firewalls, because AMPS use low-energy physics to translate between the quantum mechanics of different observers, a translation polluted by short-distance physics of the stretched horizon.[11](https://arxiv.org/abs/1207.6243) Follow-up work found that large-N rescaling and the AMPS argument show black hole complementarity is inconsistent under its usual assumptions.[12](https://arxiv.org/abs/1210.6733)\n\n## Holography and string theory\n\nThe [AdS/CFT correspondence](https://www.edgepedia.org/ads-cft-correspondence), proposed by [Juan Maldacena](https://www.edgepedia.org/juan-maldacena) of the [Institute for Advanced Study](https://www.edgepedia.org/institute-for-advanced-study) in 1997, gives a definition of quantum gravity in anti-de Sitter space in terms of a conformal field theory on the boundary without gravity.[13](https://arxiv.org/abs/hep-th/9711200) Since the boundary theory's evolution is unitary, any bulk process resembling the formation and evaporation of a black hole has to be captured by unitary dynamics on the boundary, and the CFT's unitary character gives strong indication that the Hawking radiation carries back every detail of an initial state that collapses into a black hole.[14](https://adscft.org/advanced/black-hole-information-and-quantum-gravity/information-loss-in-ads-cft/) • [6](https://arxiv.org/abs/hep-th/9903237) String theory also accounts for black hole entropy microscopically: Andrew Strominger and Cumrun Vafa derived the Bekenstein-Hawking area-entropy relation \\( S_{BH} = A/4 \\) for a class of five-dimensional extremal black holes by counting the degeneracy of BPS soliton bound states.[15](https://arxiv.org/abs/hep-th/9601029)\n\nThe fuzzball proposal of Mathur and collaborators takes a stronger step: black hole microstates are horizon-sized quantum gravity objects called fuzzballs, with neither horizons nor singularities, instead of vacuum regions with a central singularity. The core principle is that horizons arise only if one describes gravity with too few degrees of freedom; string theory has enough degrees of freedom to resolve the physics.[16](https://arxiv.org/abs/2204.13113) • [17](https://arxiv.org/abs/1812.11641)\n\n## The Page curve and the island rule\n\nIn 2019, Geoffrey Penington showed that in AdS/CFT with absorbing boundary conditions there is a phase transition in the location of the quantum Ryu-Takayanagi surface at precisely the Page time, with the new surface lying slightly inside the event horizon. This yields the Page curve and the Hayden-Preskill decoding criterion through entanglement wedge reconstruction: part of the interior is encoded in the early Hawking radiation, so the decreasing entanglement entropy of the black hole is consistent with the semiclassical entanglement of late-time Hawking modes without a firewall.[18](https://arxiv.org/abs/1905.08255)\n\nA direct gravitational derivation followed. Almheiri, Hartman, Maldacena, Shaghoulian, and Tajdini showed that the large discrepancy between the von Neumann entropy calculated by Hawking and the requirements of unitarity is fixed by including new saddles in the gravitational path integral, the [replica wormholes](https://www.edgepedia.org/replica-wormhole), which connect different copies of the black hole in the replica method. As the replica number \\( n \\to 1 \\), these wormholes lead to the island rule for computing fine-grained gravitational entropy, in which a region behind the horizon, the island, is included together with the radiation in the entropy computation.[7](https://arxiv.org/abs/1911.12333) At late times the replica wormhole saddle dominates despite topological suppression, and entanglement across the horizon no longer contributes to the entropy of the outgoing radiation while the entanglement needed for semiclassical physics at the horizon is maintained.[7](https://arxiv.org/abs/1911.12333) Islands also appear in higher dimensions: a five-dimensional asymptotically AdS construction exhibits the eternal-black-hole version of the paradox, and it is avoided by the presence of islands.[19](https://arxiv.org/abs/1911.09666)\n\n## Status\n\nThe recent island-formula results show that the von Neumann entropy of Hawking radiation can follow the Page curve, consistent with unitary, information-preserving time evolution, and reveal large corrections needed for that consistency.[20](https://arxiv.org/abs/2006.06872) Reviews of the subject describe the paradox as encapsulating a deep conundrum between gravitation and quantum physics, and assess its resolution as a continuing goal of quantum gravity research.[21](https://www.nature.com/articles/s42254-026-00942-9) Exponentially small correlations in the emitted radiation are sufficient to resolve the original paradox as Hawking put it forward.[2](https://www.sciencedirect.com/science/article/pii/S0370157321003720)\n\n## References\n\n1. [S. W. Hawking (1976). Breakdown of predictability in gravitational collapse. Physical Review D 14, 2460.](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.2460)\n2. [Rajagopal et al. (2021). Lessons from the information paradox. Physics Reports.](https://www.sciencedirect.com/science/article/pii/S0370157321003720)\n3. [D. N. Page (1993). Information in Black Hole Radiation. arXiv:hep-th/9306083.](https://arxiv.org/abs/hep-th/9306083)\n4. [S. D. Mathur (2009). The information paradox: A pedagogical introduction.](https://axi.lims.ac.uk/paper/0909.1038)\n5. [A. Almheiri, D. Marolf, J. Polchinski, J. Sully (2012). Black Holes: Complementarity or Firewalls? arXiv:1207.3123.](https://arxiv.org/abs/1207.3123)\n6. [Maldacena (1999). AdS/CFT and the Information Paradox. arXiv:hep-th/9903237.](https://arxiv.org/abs/hep-th/9903237)\n7. [A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, A. Tajdini (2019). Replica Wormholes and the Entropy of Hawking Radiation. arXiv:1911.12333.](https://arxiv.org/abs/1911.12333)\n8. [Science history: Stephen Hawking writes a tiny paper, Live Science.](https://www.livescience.com/space/black-holes/science-history-stephen-hawking-writes-a-tiny-paper-and-turns-our-understanding-of-black-holes-inside-out-march-1-1974)\n9. [Hawking Radiation, ScienceDirect Topics overview.](https://www.sciencedirect.com/topics/physics-and-astronomy/hawking-radiation)\n10. [O. C. Stoica (2018). Revisiting the Black Hole Entropy and the Information Paradox. Advances in High Energy Physics.](https://onlinelibrary.wiley.com/doi/10.1155/2018/4130417)\n11. [D. Harlow (2012). Complementarity, not Firewalls. arXiv:1207.6243.](https://arxiv.org/abs/1207.6243)\n12. [D.-i. Hwang, B.-H. Lee, D.-h. Yeom (2012). Is the firewall consistent? arXiv:1210.6733.](https://arxiv.org/abs/1210.6733)\n13. [J. Maldacena (1997). The Large N Limit of Superconformal Field Theories and Supergravity. arXiv:hep-th/9711200.](https://arxiv.org/abs/hep-th/9711200)\n14. [Information Loss in AdS/CFT, AdS/CFT Duality.](https://adscft.org/advanced/black-hole-information-and-quantum-gravity/information-loss-in-ads-cft/)\n15. [A. Strominger, C. Vafa (1996). Microscopic Origin of the Bekenstein-Hawking Entropy. arXiv:hep-th/9601029.](https://arxiv.org/abs/hep-th/9601029)\n16. [I. Bena, E. J. Martinec, S. D. Mathur, N. P. Warner (2022). Fuzzballs and Microstate Geometries. arXiv:2204.13113.](https://arxiv.org/abs/2204.13113)\n17. [S. D. Mathur (2018). What does the information paradox say about the universe? arXiv:1812.11641.](https://arxiv.org/abs/1812.11641)\n18. [G. Penington (2019). Entanglement Wedge Reconstruction and the Information Paradox. arXiv:1905.08255.](https://arxiv.org/abs/1905.08255)\n19. [A. Almheiri, R. Mahajan, J. E. Santos (2019). Entanglement islands in higher dimensions. arXiv:1911.09666.](https://arxiv.org/abs/1911.09666)\n20. [A. Almheiri et al. (2020). The entropy of Hawking radiation. arXiv:2006.06872; Rev. Mod. Phys. 93, 035002 (2021).](https://arxiv.org/abs/2006.06872)\n21. [R. B. Mann (2026). Black-hole thermodynamics. Nature Reviews Physics 8, 425-436.](https://www.nature.com/articles/s42254-026-00942-9)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › Quantum gravity and unification › String-theoretic gravity and holography › String-theoretic black hole physics*\n\n*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/black-hole-information-paradox", "markdown_url": "https://www.edgepedia.org/black-hole-information-paradox.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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Edgepedia Community License 1.0.", "speakable": "The black hole information paradox is a conflict between quantum mechanics and general relativity: Hawking's 1976 calculation suggests evaporating black holes destroy information, which unitary quantum evolution forbids." }