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Time dependence of Hawking radiation entropy
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If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. Here numerical results are given for an approximation to the time dependence of the radiation entropy under an assumption of fast scrambling, for large nonrotating black holes that emit essentially only photons and gravitons. The maximum of the von Neumann entropy then occurs after about 53.81% of the evaporation time, when the black hole has lost about 40.25% of its original Bekenstein-Hawking (BH) entropy (an upper bound for its von Neumann entropy) and then has a BH entropy that equals the entropy in the radiation, which is about 59.75% of the original BH entropy 4π M 0 2 , or about 7.509 M 0 2 ≈ 6.268 × 10 76 ( M 0 / M ⊙ ) 2 , using my 1976 calculations that the photon and graviton emission process into empty space gives about 1.4847 times the BH entropy loss of the black hole. Results are also given for black holes in initially impure states. If the black hole starts in a maximally mixed state, the von Neumann entropy of the Hawking radiation increases from zero up to a maximum of about 119.51% of the original BH entropy, or about 15.018 M 0 2 ≈ 1.254 × 10 77 ( M 0 / M ⊙ ) 2 , and then decreases back down to 4π M 0 2 = 1.049 × 10 77 ( M 0 / M ⊙ ) 2 .
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“For black holes in our universe, both events occur when the black hole has approximately 90% evaporated [7].”
From Entanglement wedge reconstruction and the information paradox (2020) · cites this paper as [7]
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“The factor of two in (3.3) arises because the Hawking radiation is not adiabatic and it generates coarse grained entropy [4] (see also [8]).”
From The Page curve of Hawking radiation from semiclassical geometry (2020) · cites this paper as [4]
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“The inclusion of this spatially disconnected island leads to reproduction the Page curve [47,48] for the timedependence of the von Neumann entropy of the Hawking radiation in consistency with unitarity without invalidating the effective semiclassical description of bulk”
From Holographic spacetime, black holes and quantum error correcting codes: a review (2022) · cites this paper as [48]
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“This is a consequence of Page's theorem [47,48] which states that for a typical pure state in a bipartite system with one system being much smaller than the other, the trace distance between the density matrix of the smaller subsystem and the microcanonical ensemble is of the order of e -S/2 with S being the number of qubits in the larger one.”
From Holographic spacetime, black holes and quantum error correcting codes: a review (2022) · cites this paper as [48]
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“But if the evaporation is unitary, then it must eventually fall back to zero following the Page curve [2, 3].”
From Replica wormholes and the entropy of Hawking radiation (2020) · cites this paper as [3]
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“Despite its simplicity, this setup exhibits Hawking's information paradox, and the corresponding puzzle with the Page curve [2, 3].”
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“Using this we compute the entropy of Hawking radiation and argue that it follows the Page curve, as suggested by recent computations of the entropy and entanglement wedges for old black holes.”
From The Page curve of Hawking radiation from semiclassical geometry (2020) · cites this paper as [4]
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“This means that the Page time, which we recall is defined by S rad = A hor 4G N , (2.4) occurs when the horizon area A hor > A 0 hor /2, despite commonly being called the halfway point of the evaporation [7].”
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“The Page time is commonly called the 'halfway point' in the black hole evaporation, although, because of the thermodynamic irreversibility of the evaporation and the time dependence of the black hole temperature, it does not occur halfway through the evaporation either by time or by horizon area/entropy[7].”
From Entanglement wedge reconstruction and the information paradox (2020) · cites this paper as [7]
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Twenty Years of Debate with StephenPreprint
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“occurs when the horizon area Ahor > Ahor/2, despite commonly being called the halfway point of the evaporation [6].”
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“Then, as suggested by D. Page [60,61], the entropy of the radiation would need to follow the curve indicated in figure 7, as opposed to the Hawking curve.”
Jerusalem lectures on black holes and quantum information
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Metadata
- Authors
- Page, D. N.
- Affiliations
- University of Alberta (CA)
- Corresponding author
- Page, Don N
- ORCID
- Page, Don N
- Venue
- Journal of Cosmology and Astroparticle Physics
- Publisher
- IOP Publishing
- Published
- 2013-09-24
- Volume
- 2013
- Issue
- 09
- Pages
- 028-028
- ISSN
- 1475-7508 (ISSN-L)
- 1475-7516
- OpenAlex
- W2110969537
- Cited by
- 354
- References
- 243
- Citations by year
- 2026: 19 · 2025: 56 · 2024: 50 · 2023: 40 · 2022: 52 · 2021: 45 · 2020: 22 · 2019: 8 · 2018: 6 · 2017: 14 · 2016: 9 · 2015: 14 · 2014: 15 · 2013: 3
- FWCI
- 6.22
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Journal of Cosmology and Astroparticle Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Astrophysical Phenomena and Observations
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
- Keywords
- Hawking radiation, black hole evaporation, von Neumann entropy, Bekenstein-Hawking entropy, black hole information paradox, quantum scrambling, unitary evolution
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