Math 2.0

Relativity and gravitational theory Black holes Proposition

Bekenstein bound

Builds on Bekenstein–Hawking entropy

Statement

Let a physical system be contained in a sphere of radius RR and have total energy EE, including its rest energy. Let ℏ\hbar be the reduced Planck constant, cc the speed of light and kBk_B the Boltzmann constant. Then its entropy SS satisfies S≤2πkBREℏc.S \le \frac{2\pi k_B R E}{\hbar c}.

Remarks

Bekenstein proposed the bound in 1981, arguing from the generalised second law: lowering the system into a black hole must not decrease the total entropy. In this form it is a physical principle, and the quantities SS, EE and RR need careful definitions for it to become a theorem. In 2008 Casini gave a precise version in quantum field theory and proved it from the positivity of relative entropy. In it, SS is a difference of entanglement entropies and EE is replaced by the expectation of a modular Hamiltonian.

Example: for a Schwarzschild black hole of mass MM, take R=2GM/c2R = 2GM/c^2 and E=Mc2E = Mc^2. The right side is 2πkB(2GM/c2)(Mc2)/(ℏc)=4πkBGM2/(ℏc)2\pi k_B (2GM/c^2)(Mc^2)/(\hbar c) = 4\pi k_B G M^2/(\hbar c). This is exactly the Bekenstein–Hawking entropy of the black hole, so black holes saturate the bound.

Going deeper

For a student

The bound connects information theory, quantum field theory and gravity. Its gravitational constant GG cancels, so it can be stated in flat spacetime, which is why it could be given a precise proof in quantum field theory. It is one of the ideas behind the holographic principle, which bounds the entropy of a region by its boundary area.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Cauchy surface, concept, no evidence levelCauchy surfaceconceptCongruence of geodesics, concept, no evidence levelCongruence of geodesi…conceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptEnergy conditions, concept, no evidence levelEnergy conditionsconceptEvent horizon, concept, no evidence levelEvent horizonconceptGeodesic, concept, no evidence levelGeodesicconceptKilling horizon, concept, no evidence levelKilling horizonconceptKilling vector field, concept, no evidence levelKilling vector fieldconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptRicci curvature, concept, no evidence levelRicci curvatureconceptWave equation, concept, no evidence levelWave equationconceptPenrose–Hawking singularity theorems, statement, no evidence levelPenrose–Hawking singu…statementRaychaudhuri equation, statement, no evidence levelRaychaudhuri equationstatementHawking area theorem, statement, no evidence levelHawking area theoremstatementLaws of black hole mechanics, statement, no evidence levelLaws of black hole me…statementHawking radiation, statement, no evidence levelHawking radiationstatementBekenstein–Hawking entropy, statement, no evidence levelBekenstein–Hawking en…statementBekenstein bound, statement, no evidence levelBekenstein boundstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cauchy surface concept
  2. Congruence of geodesics concept
  3. Einstein field equations concept
  4. Energy conditions concept
  5. Event horizon concept
  6. Geodesic concept
  7. Killing horizon concept
  8. Killing vector field concept
  9. Lorentzian manifold concept
  10. Ricci curvature concept
  11. Wave equation concept
  12. Penrose–Hawking singularity theorems statement
  13. Raychaudhuri equation statement
  14. Hawking area theorem statement
  15. Laws of black hole mechanics statement
  16. Hawking radiation statement
  17. Bekenstein–Hawking entropy statement
  18. Bekenstein bound statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bekenstein_bound
About this page
Tag7C2CSSHPKA
URIm2://7C2CSSHPKA
Kindstatement, proposition
WritingW0 stub
MSC83C47, 81T20, 94A17
StatusThis copy of the site has no build data. Run m2 status 7C2CSSHPKA.
AboutBekenstein–Hawking entropy
UsesBekenstein–Hawking entropy
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Changed3 November 2026

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