Physics:Quantum black hole information paradox

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Short description: Paradox concerning information loss in black-hole evaporation

The quantum black hole information paradox is a conflict between quantum mechanics, quantum field theory, and the gravitational description of black holes. In quantum mechanics, the complete state of a closed system is expected to evolve unitarily, so information about an earlier state is in principle preserved. In semiclassical black-hole physics, however, Hawking radiation appears thermal and seems to depend only on a few macroscopic properties of the black hole.

The paradox arises when a black hole forms from matter in a pure quantum state and then evaporates completely through Hawking radiation. If the final radiation contains only thermal information about the black hole's mass, charge, and angular momentum, then many different initial states could lead to the same final state. This would mean that information has been permanently lost, contradicting ordinary quantum-mechanical unitarity.[1]

The paradox is one of the central problems in quantum gravity. It links black-hole thermodynamics, Hawking radiation, entropy, holography, the Page curve, and the microscopic structure of spacetime. Many physicists now believe that information is preserved during black-hole evaporation, but there is still debate about the precise mechanism by which Hawking's original semiclassical calculation must be corrected.[2][3]

The information paradox asks whether information that falls into a black hole is lost, hidden, or encoded in subtle correlations in Hawking radiation.

Basic idea

A black hole is classically described by a small number of external parameters, such as mass, electric charge, and angular momentum. The no-hair idea suggests that many details of the matter that formed the black hole are not visible from outside the horizon.

Hawking's calculation showed that black holes should emit radiation with a thermal spectrum.[4] If the radiation is exactly thermal, then it carries no detailed memory of the matter that collapsed. When the black hole has fully evaporated, only radiation remains. The question is whether that radiation contains the original information.

In quantum mechanics, the wave function of a closed system evolves by a unitary operator. This means that the state at one time determines the state at any other time:

|Ψ(t1)⟩=U(t1,t2)|Ψ(t2)⟩.

If black-hole evaporation turns a pure state into a mixed thermal state, then the process is not unitary. That is the core of the paradox.

Hawking radiation

Hawking radiation is produced by quantum fields in the curved spacetime around a black hole. In Hawking's semiclassical calculation, gravity is treated classically while quantum fields are allowed to fluctuate on the black-hole background.

For a Schwarzschild black hole, the Hawking temperature is

T=ℏc38πkGM.

This formula shows that smaller black holes are hotter. As a black hole radiates, it loses mass and becomes hotter, so the evaporation accelerates. The lifetime of a Schwarzschild black hole is proportional to the cube of its initial mass.

The problem is not merely that a black hole radiates. The problem is that the radiation appears to be thermal and therefore insensitive to the detailed quantum state of the matter that formed the black hole.

Pure states and mixed states

In quantum mechanics, a pure state contains maximal information about a system's quantum state. A mixed state represents statistical uncertainty or incomplete information. Unitary quantum evolution preserves the purity of a closed system.

Hawking's original argument suggested that a black hole could form from matter in a pure state and evaporate into a mixed state of thermal radiation. Such a pure-to-mixed transition would violate unitarity.

This can also be described using von Neumann entropy. A pure state has zero von Neumann entropy, while a mixed state has nonzero entropy. If black-hole evaporation is unitary, then the fine-grained entropy of the total radiation should eventually return to zero. If Hawking's original calculation is taken literally, the entropy instead continues to increase.

Page curve

Don Page argued that if black-hole evaporation is unitary, the entropy of the Hawking radiation should follow a characteristic pattern. It should initially increase as the black hole becomes entangled with its radiation, then reach a maximum, and finally decrease back to zero when the black hole has evaporated completely.[5]

This expected entropy pattern is called the Page curve. Deriving the Page curve from a theory of gravity is often treated as a major sign that the information paradox has been resolved in that framework.

Recent work using replica wormholes and quantum extremal surfaces has reproduced Page-curve behavior in specific models of black-hole evaporation.[6][7][8]

Holography

The holographic principle suggests that the information inside a gravitational region may be encoded on a lower-dimensional boundary. In the context of black holes, holography offers a way for the physics outside the black hole to contain information that appears classically hidden behind the horizon.

The AdS/CFT correspondence provides a concrete example of holography. It relates certain gravitational theories in anti-de Sitter spacetime to ordinary quantum field theories without gravity on the boundary. Since the boundary theory is unitary, many physicists take AdS/CFT as evidence that black-hole evaporation should also be unitary in the gravitational description.

This view has strongly influenced modern thinking about the information paradox. It suggests that Hawking radiation is not exactly featureless thermal radiation, but contains subtle correlations that encode the black hole's initial state.[3]

Replica wormholes and islands

Recent developments have shown how the Page curve can emerge from gravitational path integrals in certain simplified models. The key ingredients include replica wormholes and the island formula.

In these calculations, the entropy of Hawking radiation receives contributions from additional saddle points in the gravitational path integral. These contributions can make part of the black-hole interior, called an island, count as part of the radiation system for entropy calculations. This prevents the radiation entropy from increasing forever and reproduces the Page curve.[2][6][7]

The island and replica-wormhole results are widely regarded as major progress. However, there is still debate over how directly these model calculations apply to realistic black holes in our universe.

Fuzzballs and firewalls

The fuzzball proposal, developed in string theory, suggests that individual black-hole microstates do not have a conventional empty interior. Instead, the black hole is replaced by a complicated horizon-scale quantum structure. In this view, information is stored in the detailed microstate structure and can affect the outgoing radiation.[9][10]

The firewall proposal is another radical possibility. It suggests that an observer falling through the horizon might encounter high-energy quantum effects rather than empty space. Firewalls were proposed in response to tensions between unitarity, semiclassical geometry, and the equivalence principle.

Both fuzzballs and firewalls modify the conventional picture of a smooth black-hole horizon. They show how deeply the information paradox challenges the usual separation between quantum theory and classical spacetime geometry.

Remnants and final-stage release

Another possible resolution is that information remains inside the black hole until the final stages of evaporation. When the black hole becomes Planck-sized, quantum-gravity effects could release the information or leave behind a stable remnant.

This idea has the advantage that large black holes can remain close to the semiclassical picture for most of their lifetime. However, remnant scenarios face difficulties. A tiny remnant may need to store an arbitrarily large amount of information, and this can create problems for effective field theory and entropy bounds.[11][12]

Soft hair

Stephen Hawking, Malcolm Perry, and Andrew Strominger proposed that black holes may carry soft hair: low-energy gravitational or electromagnetic degrees of freedom associated with asymptotic symmetries.[13]

Soft hair may store some information about the black hole's history. Whether it is sufficient to solve the full information paradox remains debated.

Information loss as a real effect

A minority view is that information may genuinely be lost in black-hole evaporation. This would require modifying ordinary quantum mechanics or accepting non-unitary time evolution in gravitational processes.

Such proposals are controversial because unitarity is deeply built into quantum theory. They also raise questions about energy conservation, locality, and the foundations of quantum measurement.[11][12]

Relation to quantum gravity

The information paradox is not just a puzzle about black holes. It is a clue about the structure of quantum gravity. It suggests that at least one familiar principle must be revised or reinterpreted: locality, smooth horizons, semiclassical spacetime, or the way information is encoded in gravitational systems.

Modern resolutions often point toward some form of holography or nonlocal encoding of information. This does not necessarily mean that information can be transmitted faster than light, but it does suggest that gravitational theories may not factor into independent inside and outside degrees of freedom in the same way as ordinary quantum field theories on fixed backgrounds.

Status

The black hole information paradox remains an active field of research. Many researchers believe that information is preserved and that Hawking's original conclusion of information loss was incomplete. The Page curve, holography, replica wormholes, and island calculations provide strong evidence in this direction in certain models.

However, there is still no universally accepted, experimentally confirmed microscopic account of black-hole evaporation in realistic quantum gravity. The paradox therefore remains one of the most important open problems at the intersection of black holes, thermodynamics, quantum information, and spacetime.

See also

Table of contents (184 articles)

Index

Quantum Book II
Quantum Book III
Quantum Book IV

Full contents

1. Foundations (11)↑
  1. Physics:Quantum basics
  2. Physics:Quantum Postulates
  3. Physics:Quantum Hilbert space
  4. Physics:Quantum Observables and operators
  5. Physics:Quantum mechanics
  6. Physics:Quantum mechanics measurements
  7. Physics:Quantum state
  8. Physics:Quantum system
  9. Physics:Quantum superposition
  10. Physics:Quantum probability
  11. Physics:Quantum Mathematical Foundations of Quantum Theory
2. Conceptual and interpretations (13)↑
  1. Physics:Quantum Interpretations of quantum mechanics
  2. Physics:Quantum Wave–particle duality
  3. Physics:Quantum Complementarity principle
  4. Physics:Quantum Uncertainty principle
  5. Physics:Quantum Measurement problem
  6. Physics:Quantum Bell's theorem
  7. Physics:Quantum Hidden variable theory
  8. Physics:Quantum nonlocality
  9. Physics:Quantum contextuality
  10. Physics:Quantum Darwinism
  11. Physics:Quantum A Spooky Action at a Distance
  12. Physics:Quantum A Walk Through the Universe
  13. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together
  14. Physics:Quantum measurement problem
3. Mathematical structure and systems (13)↑
  1. Physics:Quantum Density matrix
  2. Physics:Quantum Exactly solvable quantum systems
  3. Physics:Quantum Formulas Collection
  4. Physics:Quantum A Matter Of Size
  5. Physics:Quantum Symmetry in quantum mechanics
  6. Physics:Quantum Angular momentum operator
  7. Physics:Quantum Runge–Lenz vector
  8. Physics:Quantum Approximation Methods
  9. Physics:Quantum Matter Elements and Particles
  10. Physics:Quantum Dirac equation
  11. Physics:Quantum Klein–Gordon equation
  12. Physics:Quantum pendulum
  13. Physics:Quantum configuration space
4. Atomic and spectroscopy (14)↑
    Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.
  1. Physics:Quantum Atomic structure and spectroscopy
  2. Physics:Quantum Hydrogen atom
  3. Physics:Quantum number
  4. Physics:Quantum Multi-electron atoms
  5. Physics:Quantum Fine structure
  6. Physics:Quantum Hyperfine structure
  7. Physics:Quantum Isotopic shift
  8. Physics:Quantum defect
  9. Physics:Quantum Zeeman effect
  10. Physics:Quantum Stark effect
  11. Physics:Quantum Spectral lines and series
  12. Physics:Quantum Selection rules
  13. Physics:Quantum Fermi's golden rule
  14. Physics:Quantum beats
5. Wavefunctions and modes (9)↑
    A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.
  1. Physics:Quantum Wavefunction
  2. Physics:Quantum Superposition principle
  3. Physics:Quantum Eigenstates and eigenvalues
  4. Physics:Quantum Boundary conditions and quantization
  5. Physics:Quantum Standing waves and modes
  6. Physics:Quantum Normal modes and field quantization
  7. Physics:Number of independent spatial modes in a spherical volume
  8. Physics:Quantum Density of states
  9. Physics:Quantum carpet
6. Quantum dynamics and evolution (17)↑
  1. Physics:Quantum Time evolution
  2. Physics:Quantum Schrödinger equation
  3. Physics:Quantum Time-dependent Schrödinger equation
  4. Physics:Quantum Stationary states
  5. Physics:Quantum Perturbation theory
  6. Physics:Quantum Time-dependent perturbation theory
  7. Physics:Quantum Adiabatic theorem
  8. Physics:Quantum Scattering theory
  9. Physics:Quantum S-matrix
  10. Physics:Quantum tunnelling
  11. Physics:Quantum speed limit
  12. Physics:Quantum revival
  13. Physics:Quantum reflection
  14. Physics:Quantum oscillations
  15. Physics:Quantum jump
  16. Physics:Quantum boomerang effect
  17. Physics:Quantum chaos
7. Measurement and information (9)↑
  1. Physics:Quantum Measurement theory
  2. Physics:Quantum Measurement operators
  3. Physics:Quantum Projective measurement
  4. Physics:Quantum POVM
  5. Physics:Quantum Weak measurement
  6. Physics:Quantum Measurement collapse
  7. Physics:Quantum entanglement
  8. Physics:Quantum Zeno effect
  9. Physics:Quantum limit
8. Quantum information and computing (10)↑
  1. Physics:Quantum information theory
  2. Physics:Quantum Qubit
  3. Physics:Quantum Entanglement
  4. Physics:Quantum Gates and circuits
  5. Physics:Quantum Computing Algorithms in the NISQ Era
  6. Physics:Quantum Noisy Qubits
  7. Physics:Quantum random access code
  8. Physics:Quantum pseudo-telepathy
  9. Physics:Quantum network
  10. Physics:Quantum money
9. Quantum optics and experiments (5)↑
    Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
  1. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
  2. Physics:Quantum optics beam splitter experiments
  3. Physics:Quantum Ultra fast lasers
  4. Physics:Quantum Experimental quantum physics
  5. Physics:Quantum optics
    6. Template:Quantum optics operators
10. Open quantum systems (9)↑
  1. Physics:Quantum Open systems
  2. Physics:Quantum Master equation
  3. Physics:Quantum Lindblad equation
  4. Physics:Quantum Decoherence
  5. Physics:Quantum dissipation
  6. Physics:Quantum Markov semigroup
  7. Physics:Quantum Markovian dynamics
  8. Physics:Quantum Non-Markovian dynamics
  9. Physics:Quantum Trajectories
11. Quantum field theory (20)↑
    Structural dependency map of quantum field theory.
  1. Physics:Quantum field theory (QFT) basics
  2. Physics:Quantum field theory (QFT) core
  3. Physics:Quantum Fields and Particles
  4. Physics:Quantum Second quantization
  5. Physics:Quantum Harmonic Oscillator field modes
  6. Physics:Quantum Creation and annihilation operators
  7. Physics:Quantum vacuum fluctuations
  8. Physics:Quantum Propagators in quantum field theory
  9. Physics:Quantum Feynman diagrams
  10. Physics:Quantum Path integral formulation
  11. Physics:Quantum Renormalization in field theory
  12. Physics:Quantum Renormalization group
  13. Physics:Quantum Field Theory Gauge symmetry
  14. Physics:Quantum Non-Abelian gauge theory
  15. Physics:Quantum Electrodynamics (QED)
  16. Physics:Quantum chromodynamics (QCD)
  17. Physics:Quantum Electroweak theory
  18. Physics:Quantum Standard Model
  19. Physics:Quantum triviality
  20. Physics:Quantum confinement problem
12. Statistical mechanics and kinetic theory (9)↑
  1. Physics:Quantum Statistical mechanics
  2. Physics:Quantum Partition function
  3. Physics:Quantum Distribution functions
  4. Physics:Quantum Liouville equation
  5. Physics:Quantum Kinetic theory
  6. Physics:Quantum Boltzmann equation
  7. Physics:Quantum BBGKY hierarchy
  8. Physics:Quantum Relaxation and thermalization
  9. Physics:Quantum Thermodynamics
13. Condensed matter and solid-state physics (13)↑
  1. Physics:Quantum Band structure
  2. Physics:Quantum Fermi surfaces
  3. Physics:Quantum Semiconductor physics
  4. Physics:Quantum Phonons
  5. Physics:Quantum Electron-phonon interaction
  6. Physics:Quantum Superconductivity
  7. Physics:Quantum Topological phases of matter
  8. Physics:Quantum well
  9. Physics:Quantum spin liquid
  10. Physics:Quantum spin Hall effect
  11. Physics:Quantum phase transition
  12. Physics:Quantum critical point
  13. Physics:Quantum dot
14. Plasma and fusion physics (8)↑
    Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
  1. Physics:Quantum Fusion reactions and Lawson criterion
  2. Physics:Quantum Plasma (fusion context)
  3. Physics:Quantum Magnetic confinement fusion
  4. Physics:Quantum Inertial confinement fusion
  5. Physics:Quantum Plasma instabilities and turbulence
  6. Physics:Quantum Tokamak core plasma
  7. Physics:Quantum Tokamak edge physics and recycling asymmetries
  8. Physics:Quantum Stellarator
15. Timeline (8)↑
  1. Physics:Quantum mechanics/Timeline
  2. Physics:Quantum mechanics/Timeline/Pre-quantum era
  3. Physics:Quantum mechanics/Timeline/Old quantum theory
  4. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
  5. Physics:Quantum mechanics/Timeline/Quantum field theory era
  6. Physics:Quantum mechanics/Timeline/Quantum information era
  7. Physics:Quantum mechanics/Timeline/Quantum technology era
  8. Physics:Quantum mechanics/Timeline/Quiz
16. Advanced and frontier topics (16)↑
  1. Physics:Quantum topology
  2. Physics:Quantum battery
  3. Physics:Quantum Supersymmetry
  4. Physics:Quantum Black hole thermodynamics
  5. Physics:Quantum Holographic principle
  6. Physics:Quantum gravity
  7. Physics:Quantum De Sitter invariant special relativity
  8. Physics:Quantum Doubly special relativity
  9. Physics:Quantum arithmetic geometry
  10. Physics:Quantum unsolved problems
  11. Physics:Quantum Yang-Mills mass gap
  12. Physics:Quantum gravity problem
  13. Physics:Quantum black hole information paradox
  14. Physics:Quantum dark matter problem
  15. Physics:Quantum neutrino mass problem
  16. Physics:Quantum matter-antimatter asymmetry problem

References

  1. ↑ Hawking, S. W. (1976). "Breakdown of predictability in gravitational collapse". Physical Review D 14 (10): 2460–2473. doi:10.1103/PhysRevD.14.2460. Bibcode: 1976PhRvD..14.2460H. 
  2. ↑ 2.0 2.1 Almheiri, Ahmed; Hartman, Thomas; Maldacena, Juan; Shaghoulian, Edgar; Tajdini, Amirhossein (21 July 2021). "The entropy of Hawking radiation". Reviews of Modern Physics 93 (3): 035002. doi:10.1103/RevModPhys.93.035002. Bibcode: 2021RvMP...93c5002A. 
  3. ↑ 3.0 3.1 Raju, Suvrat (January 2022). "Lessons from the information paradox". Physics Reports 943: 1–80. doi:10.1016/j.physrep.2021.10.001. Bibcode: 2022PhR...943....1R. 
  4. ↑ Hawking, Stephen (1 August 1975). "Particle Creation by Black Holes". Communications in Mathematical Physics 43 (3): 199–220. doi:10.1007/BF02345020. Bibcode: 1975CMaPh..43..199H. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-43/issue-3/Particle-creation-by-black-holes/cmp/1103899181.pdf. 
  5. ↑ Page, Don N. (6 December 1993). "Information in Black Hole Radiation". Physical Review Letters 71 (23): 3743–3746. doi:10.1103/PhysRevLett.71.3743. PMID 10055062. Bibcode: 1993PhRvL..71.3743P. 
  6. ↑ 6.0 6.1 Penington, Geoffrey (September 2020). "Entanglement wedge reconstruction and the information paradox". Journal of High Energy Physics 2020 (9): 2. doi:10.1007/JHEP09(2020)002. Bibcode: 2020JHEP...09..002P. 
  7. ↑ 7.0 7.1 Almheiri, Ahmed; Engelhardt, Netta; Marolf, Donald; Maxfield, Henry (December 2019). "The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole". Journal of High Energy Physics 2019 (12): 63. doi:10.1007/JHEP12(2019)063. Bibcode: 2019JHEP...12..063A. 
  8. ↑ Almheiri, A.; Hartman, T.; Maldacena, J.; Shaghoulian, E.; Tajdini, A. (2020). "Replica Wormholes and the Entropy of Hawking Radiation". Journal of High Energy Physics 2020 (5). doi:10.1007/JHEP05(2020)013. 
  9. ↑ Mathur, Samir D. (21 November 2009). "The information paradox: a pedagogical introduction". Classical and Quantum Gravity 26 (22): 224001. doi:10.1088/0264-9381/26/22/224001. Bibcode: 2009CQGra..26v4001M. 
  10. ↑ Skenderis, Kostas; Taylor, Marika (October 2008). "The fuzzball proposal for black holes". Physics Reports 467 (4–5): 117–171. doi:10.1016/j.physrep.2008.08.001. Bibcode: 2008PhR...467..117S. 
  11. ↑ 11.0 11.1 Giddings, Steven B. (1995). "The black hole information paradox". Johns Hopkins Workshop on Current Problems in Particle Theory 19 and the PASCOS Interdisciplinary Symposium 5. Bibcode: 1995hep.th....8151G. 
  12. ↑ 12.0 12.1 Preskill, John (1992). "Do Black Holes Destroy Information?". International Symposium on Black Holes, Membranes, Wormholes, and Superstrings. Bibcode: 1993bhmw.conf...22P. 
  13. ↑ Hawking, Stephen W.; Perry, Malcolm J.; Strominger, Andrew (2016-01-05). "Soft Hair on Black Holes". Physical Review Letters 116 (23): 231301. doi:10.1103/PhysRevLett.116.231301. PMID 27341223. Bibcode: 2016PhRvL.116w1301H. 

Further reading


Author: Harold Foppele





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