nLab Bekenstein-Hawking entropy





physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



In gravity, Bekenstein-Hawking entropy is an entropy assigned to black hole, on the basis of laws of thermodynamics and observers outside black hole. It striking property is that it is proportional to the surface area of the balck hole’s horizon.


Interpretation in AdS/CFT correspondence

In the context of string theory BH entropy is explained by a version of the AdS/CFT correspondence. Here every black brane solution in supergravity is the strong-coupling limit of a D-brane worldvolume QFT. After KK-reduction these black brane configurations become ordinary black holes. The entropy of the D-brane worldvolume theories on the event horizon turns out to coincide with the BH entropy of the corresponding black hole.

Detailed computations exist in particular for D1-brane/D5-brane systems. This is parts of the AdS/CFT correspondence. See (AGMOO, chapter 5).

See also

Interpretation by strong coupling limit of D-branes

Another way to derive Bekenstein-Hawking entropy in string theory is by computing the entropy of weakly coupled open strings on D-brane configurations in flat Minkowski space which transmute as the coupling constant is increased to given (supersymmetric) black hole configurations. More on this is at black holes in string theory.



Textbook account:

  • Valeri Frolov, Andrei Zelnikov, Introduction to black hole physics, Oxford 2011


Review form the point of view of thermal field theory:

  • S.A. Fulling, S.N.M. Ruijsenaars, Temperature, periodicity and horizons, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 (pdf, doi:10.1016/0370-1573(87)90136-0)

Basic introductory accounts:

and further review:

  • S. P. de Alwis, Comments on Entropy Calculations in Gravitational Systems [arXiv:2304.07885]

A more general discussion which identifies thermodynamic properties of all horizons appearing on gravity (not just black hole horizons) was given in

This article showed that under some assumptions the Einstein equations can even be derived from identifying gravitational horizon area with entropy and them imposing laws of thermodynamics.

For more comments and more references on this observation see

(Later authors tried to argue that derivations like this show that gravity is not a fundamental force of nature such as electromagnetism or the strong nuclear force, but rather an entropic force that arises only from more fundamental forces in a thermodynamic limit. This however remains at best unclear.)

Discussion of black hole entropy from entropy of conformal field theory associated with the horizon has been given in

and reviewed in

Further developments on black hole entropy are in

A discussion of the “black hole firewall problem”:

Via Wick-rotated thermal field theory

Discussion via Wick rotation to Euclidean field theory on spacetimes with compact/periodic Euclidean time (thermal field theory on curved spacetimes) is in

  • S.A. Fulling, S.N.M. Ruijsenaars, Temperature, periodicity and horizons, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 (pdf, doi:10.1016/0370-1573(87)90136-0)

  • Gary Gibbons, Malcolm J. Perry, Black Holes and Thermal Green Functions, Vol. 358, No. 1695 (1978) (jstor:79482)

Interpretation in string theory

Microscopic explanation of Bekenstein-Hawking entropy via geometric engineering of black holes in string theory as bound states of D-branes:

Review of interpretation of black holes in string theory includes

Discussion in view of higher curvature corrections includes

See also

Interpretation as entanglement entropy

Discussions of the interpreation of BH entropy as holographic entanglement entropy include

Computation of black hole entropy in 4d via AdS4-CFT3 duality from holographic entanglement entropy in the ABJM theory for the M2-brane is discussed in

  • Jun Nian, Xinyu Zhang, Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole (arXiv:1705.01896)

Entanglement island proposal for BH information paradox


Claim that the proper application of holographic entanglement entropy to the discussion of Bekenstein-Hawking entropy resolves the apparent black hole information paradox:

Review aimed at readers with minimal background:

See also:


  • Anna Karlsson, Concerns about the replica wormhole derivation of the island conjecture (arXiv:2101.05879)

  • Harvendra Singh, Islands and Icebergs contribute nothing to the Page curve (System with a symmetrical bath) [arXiv:2210.13970]

Argument that the would-be proof of the island conjecture secretly works only for massive gravity:

reviewed in:

  • Hao Geng, Recent Progress in Quantum Gravity: Karch-Randall Braneworld, Entanglement Islands and Graviton Mass, Washington (2022) [hdl:1773/49427]

and discussion of what to do about this problem:

Claim that the whole argument relies on the false assumption that the split property holds for quantum gravity:

Review in:

  • Nirmalya Kajuri, Of Islands, Holograms and Saving Quantum Physics From a Black Hole Paradox, Science – The Wire (Nov 2022)

    [][\ldots] a few sceptics have argued that while the calculations are correct, they don’t help resolve the black hole information-loss paradox. The troubles stem from the reservoir attached to the anti-de Sitter universe. The physicists who authored the island papers assumed that gravity stopped at the boundary of the anti-de Sitter space and didn’t enter the reservoir. This is not an innocuous assumption. [][\ldots] The key takeaway is that the island way to recover information and save unitarity works perfectly well – if you slightly modify Einstein’s theory of gravity. These criticisms have been around for some two years now, and physicists are yet to resolve them in print. [][\ldots] physicists continue to publish papers by the hundreds about the entanglement islands but few attempt to answer whether the islands are compatible with the Einsteinian gravity of our universe. [][\ldots]

See also:

  • Saskia Demulder, Alessandra Gnecchi, Ioannis Lavdas, Dieter Lüst, Islands and Light Gravitons in type IIB String Theory [arXiv:2204.03669]

  • Emil Martinec: “Thus far, picturesque evocations of quantum wormholes remain largely just that – picturesque evocations.” (2nd Nov 2022)

Argument that discussion of the issue in the toy example of 2d JT gravity does not actually generalize to higher dimensions:

  • Filip Landgren, Arvind Shekar, Islands and entanglement entropy in dd-dimensional curved backgrounds [arXiv:2401.01653]

Debate between a proponent and a critic :

See also:

Relation to weight systems on chord diagrams

An argument that dual observables on black hole thermodynamics are generically given by single trace operators that evaluate to weight systems on chord diagrams (such as observables on SYK model-like systems)

(for more see at weight systems on chord diagrams in physics):

Last revised on April 18, 2023 at 07:25:34. See the history of this page for a list of all contributions to it.