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Cited by 355Black Holes and Theoretical PhysicsAstrophysical Phenomena and ObservationsStatistical Mechanics and Entropy

Time dependence of Hawking radiation entropy

Open access
Page, Don N·2013·Journal of Cosmology and Astroparticle Physics
DOI 10.1088/1475-7516/2013/09/028
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Abstract

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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Crossref · 10.1088/1475-7516/2013/09/028

Cites

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  • For black holes in our universe, both events occur when the black hole has approximately 90% evaporated [7].

    “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]

    The main text (JCAP09(2013)028; page unavailable) places the Page time, when Hawking-radiation entropy peaks, at 53.81% of the black hole's lifetime. Its reported remaining mass of approximately 0.7730 times the initial mass also means only about 23% of the mass has evaporated, not 90%. The citing sentence does not identify what its ‘both events’ are, but the paper's quantified entropy milestone contradicts the stated 90% figure. (LLM, conf=0.76)

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  • The factor of two in equation (3.3) arises because the Hawking radiation is not adiabatic and it generates coarse-grained entropy.

    “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]

    In the JCAP09(2013)028 discussion of evaporation entropy (page not available in our copy), Page says the excess semiclassical entropy in Hawking radiation is generated by a nonequilibrium emission process. That supports the explanation that emission is not adiabatic and generates entropy, but the paper gives a radiation-to-black-hole entropy-loss ratio of about 1.4847 for the case studied, not a factor of two; it also does not address the citing paper’s equation (3.3). (LLM, conf=0.87)

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  • The inclusion of the island leads to reproduction of the Page curve for the time-dependence of the von Neumann entropy of Hawking radiation in consistency with unitarity without invalidating the effective semiclassical description of bulk physics.

    “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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  • Page's theorem states that for a typical pure state in a bipartite system with one system much smaller than the other, the trace distance between the density matrix of the smaller subsystem and the microcanonical ensemble is of the order e^{-S/2} with S being the number of qubits in the larger one.

    “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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  • If evaporation is unitary, the fine-grained entropy of Hawking radiation must eventually fall back to zero following the Page curve.

    “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]

    The conclusion (JCAP09(2013)028) says the radiation’s von Neumann entropy rises and then falls to near zero under the assumptions of an initially nearly pure black hole and complete, unitary evaporation. Unitarity alone does not establish a fall to exactly zero: the paper says the final entropy approaches the original black hole’s entropy, which it assumes is small. (LLM, conf=0.87)

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Research works printed but not linked: 4

  1. [50]50. G. 't Hooft, The Holographic principle: Opening lecture, hep-th/0003004 [INSPIRE]. · arXiv
  2. [66]66. Gen. Rel. Grav. 36 (2004) 2419] [hep-th/0405160] [INSPIRE]. · arXiv
  3. [71]71. S.N. Solodukhin, Restoring unitarity in BTZ black hole, Phys. Rev. D 71 (2005) 064006 [hep-th/0501053] [INSPIRE]. · arXiv
  4. [174]174. B.D. Chowdhury and A. Puhm, Is Alice burning or fuzzing?, arXiv:1208.2026 [INSPIRE]. · arXiv

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Related

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
DOI
10.1088/1475-7516/2013/09/028
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
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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