{ "id": "ep7ymvzthe", "slug": "bekenstein-hawking-entropy", "title": "Bekenstein-Hawking entropy", "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-phenomenology", "label": "Quantum-spacetime phenomenology and semiclassical gravity", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.relativity-and-gravity.quantum-gravity-unification.qgu-phenomenology" }, { "id": "physical.physics.relativity-and-gravity.quantum-gravity-unification.qgu-phenomenology.qgu-phen-quantum-thermodynamics-horizons", "label": "Horizon thermodynamics as a quantum-gravity constraint", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.relativity-and-gravity.quantum-gravity-unification.qgu-phenomenology.qgu-phen-quantum-thermodynamics-horizons" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.physics.relativity-and-gravity", "label": "Non-geographic · 1946 to 2000: Relativity and gravitation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.relativity-and-gravity", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics" }, { "id": "geo.nongeo.t1946.physical.physics.relativity-and-gravity", "label": "Relativity and gravitation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.relativity-and-gravity" } ] } ], "excerpt": "The Bekenstein-Hawking entropy is the entropy assigned to a black hole, proportional to its event-horizon area; Bekenstein proposed it in 1972 and Hawking fixed the coefficient in 1974.", "snippet": "The Bekenstein-Hawking entropy is the entropy assigned to a black hole, proportional to its event-horizon area; Bekenstein proposed it in 1972 and Hawking fixed the coefficient in 1974.", "node": "physical.physics.relativity-and-gravity.quantum-gravity-unification.qgu-phenomenology.qgu-phen-quantum-thermodynamics-horizons", "markdown": "# Bekenstein-Hawking entropy\n\nThe **Bekenstein-Hawking entropy** is the entropy assigned to a black hole, proportional to the area A of its event horizon: \\( S_{\\mathrm{BH}} = A/(4\\ell_{P}^{2}) \\) in units with Boltzmann's constant set to one, where \\( \\ell_{P} = \\sqrt{G\\hbar/c^{3}} \\) is the [Planck length](https://www.edgepedia.org/planck-length).[1](https://www.mdpi.com/2624-960X/8/3/83) Jacob D. Bekenstein proposed an area-proportional black-hole entropy in 1972 and 1973, and [Stephen Hawking](https://www.edgepedia.org/stephen-hawking) fixed the coefficient of 1/4 in 1974 through his calculation of quantum particle creation by black holes.[2](https://doi.org/10.1103/PhysRevD.7.2333) • [3](https://jila.colorado.edu/~ajsh/courses/phys7810_26/grbook_260329.pdf) Unlike ordinary thermodynamic entropy, which for a gas in a box scales with volume, black-hole entropy scales with area, a property that underlies holography and much of modern quantum-gravity research.[4](https://link.springer.com/article/10.12942/lrr-2011-8)\n\n| Key fact | Detail |\n|---|---|\n| Formula | \\( S_{\\mathrm{BH}} = A/(4\\ell_{P}^{2}) \\), with \\( \\ell_{P} = \\sqrt{G\\hbar/c^{3}} \\); equivalently \\( S + A/4 \\) appears in the generalized second law in Planck units[1](https://www.mdpi.com/2624-960X/8/3/83) • [5](https://doi.org/10.1007/BF02345020) |\n| Origin | Bekenstein's 1972 note proposed area-proportional entropy and a generalized second law; his 1973 Physical Review D paper gave the information-theoretic definition[6](https://blackhole.org/reference/history/black-hole-thermodynamics-information/) • [2](https://doi.org/10.1103/PhysRevD.7.2333) |\n| Hawking's contribution | The 1974/1975 calculation of particle creation fixed the proportionality constant at 1/4 and gave the temperature \\( T \\approx 10^{-6}(M_{\\odot}/M) \\) K[3](https://jila.colorado.edu/~ajsh/courses/phys7810_26/grbook_260329.pdf) • [5](https://doi.org/10.1007/BF02345020) |\n| Generalized second law | The sum of ordinary entropy outside black holes and total black-hole entropy never decreases[7](https://www.scholarpedia.org/article/Bekenstein-Hawking_entropy) |\n| Scaling | Proportional to horizon area, not volume, which distinguishes it from the entropy of ordinary thermal systems[4](https://link.springer.com/article/10.12942/lrr-2011-8) |\n| Evaporation timescale | A primordial black hole of mass less than about \\( 10^{15} \\) g would have evaporated by the present epoch[5](https://doi.org/10.1007/BF02345020) |\n\n## Bekenstein's proposal\n\nBekenstein, then at Princeton's Joseph Henry Laboratories and later the Center for Relativity Theory at the [University of Texas at Austin](https://www.edgepedia.org/university-of-texas-at-austin), built his 1973 paper \"Black Holes and Entropy\" on a similarity between black-hole physics and thermodynamics: both black-hole area and entropy tend to increase irreversibly.[2](https://doi.org/10.1103/PhysRevD.7.2333) He defined black-hole entropy as the measure of information about the black-hole interior that is inaccessible to an exterior observer, and dimensional arguments indicated the entropy equals the black-hole area divided by the square of the Planck length times a dimensionless constant of order unity.[2](https://doi.org/10.1103/PhysRevD.7.2333) An independent approach using Kerr black holes and information theory led to the same conclusion and suggested a definite value for the constant.[2](https://doi.org/10.1103/PhysRevD.7.2333)\n\nThe proposal's physical significance comes from an extended statement of the second law: if ordinary entropy falls into a black hole, the sum of that entropy outside the hole and the hole's own entropy fails to diminish.[2](https://doi.org/10.1103/PhysRevD.7.2333) Without an entropy assigned to the hole, throwing matter into it would destroy entropy and violate the ordinary second law; including the black-hole term removes the violation.[7](https://www.scholarpedia.org/article/Bekenstein-Hawking_entropy) Bekenstein also conjectured that \\( A/(4G_{N}) \\) is an upper bound on the entropy that can be contained in a region of space before a black hole forms, an early hint of the connection between information and energy density in quantum gravity.[8](https://pmc.ncbi.nlm.nih.gov/articles/PMC12840427/)\n\n## Hawking radiation and the coefficient 1/4\n\nClassically, black holes are perfect absorbers that emit nothing, so their physical temperature was taken to be absolute zero.[9](https://link.springer.com/article/10.12942/lrr-2001-6) In 1974 Hawking announced, and in 1975 published in *Communications in Mathematical Physics*, a derivation demonstrating that quantum effects make black holes produce and radiate particles in the manner of heated objects, at a temperature of \\( \\hbar\\kappa/(2\\pi k) \\approx 10^{-6}(M_{\\odot}/M) \\) K, with \\( \\kappa \\) denoting the surface gravity.[5](https://doi.org/10.1007/BF02345020) This thermal emission slowly decreases the black hole's mass and leads to its eventual disappearance, and it supplied the constant of proportionality 1/4 in [Planck units](https://www.edgepedia.org/planck-units) that Bekenstein's dimensional analysis could not fix.[3](https://jila.colorado.edu/~ajsh/courses/phys7810_26/grbook_260329.pdf)\n\nThe radiation violates the classical law that a black hole's event-horizon area cannot decrease, since an evaporating hole shrinks. Hawking showed that a generalized second law nevertheless survives: \\( S + A/4 \\) never decreases, where S is the entropy of matter outside black holes and A is the sum of the surface areas of the event horizons.[5](https://doi.org/10.1007/BF02345020) The Scholarpedia account notes that the generalized second law was formulated two years before [Hawking radiation](https://www.edgepedia.org/hawking-radiation) was put in evidence and has been verified amply in subsequent analyses.[7](https://www.scholarpedia.org/article/Bekenstein-Hawking_entropy)\n\n## Why area, not volume\n\nThe area scaling is the formula's most distinctive feature and the source of its influence. Entanglement entropy provides one line of explanation: tracing over the degrees of freedom of a quantum field inside the horizon leaves an exterior observer with a density matrix whose entropy is dominated by short-distance correlations across the horizon, and this entanglement entropy is proportional to the area of the entangling surface.[4](https://link.springer.com/article/10.12942/lrr-2011-8) Mark Srednicki's 1993 calculation showed the area law holds even in flat spacetime with no black hole present, and 't Hooft's earlier 1985 brick-wall model, which computed the entropy of the thermal gas of Hawking particles just outside the horizon, had already found area scaling.[4](https://link.springer.com/article/10.12942/lrr-2011-8)\n\nInterpreting the Bekenstein-Hawking entropy *entirely* as entanglement entropy faces obstacles. The entanglement entropy is ultraviolet divergent and depends on the number of matter species and the cutoff used to regulate short-distance correlations, whereas the black-hole entropy should be universal.[4](https://link.springer.com/article/10.12942/lrr-2011-8) One resolution, proposed by Susskind and Uglum, treats the divergence as a quantum contribution renormalizing Newton's constant; Jacobson's induced-gravity models aim to derive the entropy along with the gravitational coupling.[4](https://link.springer.com/article/10.12942/lrr-2011-8) A major goal of quantum-gravity research remains a direct derivation of the entropy formula from microscopic degrees of freedom.[9](https://link.springer.com/article/10.12942/lrr-2001-6)\n\n## Consequences and current status\n\nThe area law suggested that a black hole admits a lower-dimensional holographic description, and the entropy formula became the anchor for the [AdS/CFT correspondence](https://www.edgepedia.org/ads-cft-correspondence) and the holographic principle.[10](https://arxiv.org/abs/1305.3757) It also generated the black-hole information paradox: Hawking's calculation produces thermal radiation, raising the possibility that information about what fell in is permanently lost, in tension with quantum mechanics.[11](https://doi.org/10.21275/SR26821110153) Recent work on the entropy of Hawking radiation, using quantum extremal surfaces, islands, and [replica wormholes](https://www.edgepedia.org/replica-wormhole), revealed large corrections needed for the entropy to be consistent with unitary evaporation and reproduced Page-curve behavior in semiclassical and holographic models.[12](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.93.035002) • [6](https://blackhole.org/reference/history/black-hole-thermodynamics-information/)\n\nThe formula's domain has also been reexamined. In 2025, Abhay Ashtekar of Penn State's Eberly College of Science and colleagues extended the first and second laws of black-hole mechanics to situations out of equilibrium, arguing that for dynamical black holes that are growing, evaporating, or merging, the event-horizon area cannot serve as the measure of physical entropy and that a dynamical horizon should replace it.[13](https://www.space.com/astronomy/black-holes/stephen-hawkings-famous-leaky-black-hole-theory-gets-much-needed-update) The Bekenstein-Hawking entropy remains the benchmark against which such extensions, and every proposed microscopic counting of black-hole states, are tested.[9](https://link.springer.com/article/10.12942/lrr-2001-6)\n\n## References\n\n1. [Quantum Entropic Relationalism (QER): Contemporary Debates and Theoretical Frontiers, MDPI Entropy](https://www.mdpi.com/2624-960X/8/3/83)\n2. [J. D. Bekenstein (1973). Black Holes and Entropy. Phys. Rev. D 7, 2333.](https://doi.org/10.1103/PhysRevD.7.2333)\n3. [General Relativity, Black Holes, and Cosmology (course text, JILA, University of Colorado)](https://jila.colorado.edu/~ajsh/courses/phys7810_26/grbook_260329.pdf)\n4. [S. N. Solodukhin (2011). Entanglement Entropy of Black Holes. Living Rev. Relativity 14, 8.](https://link.springer.com/article/10.12942/lrr-2011-8)\n5. [S. W. Hawking (1975). Particle creation by black holes. Commun. Math. Phys. 43, 199–220.](https://doi.org/10.1007/BF02345020)\n6. [Black-Hole Thermodynamics and Information, blackhole.org reference history](https://blackhole.org/reference/history/black-hole-thermodynamics-information/)\n7. [Bekenstein-Hawking entropy, Scholarpedia](https://www.scholarpedia.org/article/Bekenstein-Hawking_entropy)\n8. [Matrix Quantum Mechanics and Entanglement Entropy: A Review, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC12840427/)\n9. [R. M. Wald (2001). The Thermodynamics of Black Holes. Living Rev. Relativity 4, 6.](https://link.springer.com/article/10.12942/lrr-2001-6)\n10. [Thermodynamics in Black-hole/CFT Correspondence, arXiv:1305.3757](https://arxiv.org/abs/1305.3757)\n11. [The Black Hole Information Paradox: Origins, Battles, and Where We Stand Now](https://doi.org/10.21275/SR26821110153)\n12. [A. Almheiri et al. (2021). The entropy of Hawking radiation. Rev. Mod. Phys. 93, 035002.](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.93.035002)\n13. [Stephen Hawking's famous 'leaky' black hole theory gets much-needed update, Space.com](https://www.space.com/astronomy/black-holes/stephen-hawkings-famous-leaky-black-hole-theory-gets-much-needed-update)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › Quantum gravity and unification › Quantum-spacetime phenomenology and semiclassical gravity › Horizon thermodynamics as a quantum-gravity constraint*\n\n*Initially written 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": [ "https://jila.colorado.edu/~ajsh/courses/phys7810_26/grbook_260329.pdf" ], "url": "https://www.edgepedia.org/bekenstein-hawking-entropy", "markdown_url": "https://www.edgepedia.org/bekenstein-hawking-entropy.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. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Bekenstein-Hawking entropy\", Edgepedia (EdgeChat), https://www.edgepedia.org/bekenstein-hawking-entropy. Edgepedia Community License 1.0.", "credit_md": "\"[Bekenstein-Hawking entropy](https://www.edgepedia.org/bekenstein-hawking-entropy)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/bekenstein-hawking-entropy](https://www.edgepedia.org/bekenstein-hawking-entropy). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Bekenstein-Hawking entropy\", Edgepedia (EdgeChat), https://www.edgepedia.org/bekenstein-hawking-entropy. Edgepedia Community License 1.0.", "speakable": "The Bekenstein-Hawking entropy is the entropy assigned to a black hole, proportional to its event-horizon area; Bekenstein proposed it in 1972 and Hawking fixed the coefficient in 1974." }