Living Reviews in Relativity

"Entanglement Entropy of Black Holes"
Sergey N. Solodukhin  

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1 Introduction
2 Entanglement Entropy in Minkowski Spacetime
2.1 Definition
2.2 Short-distance correlations
2.3 Thermal entropy
2.4 Entropy of a system of finite size at finite temperature
2.5 Entropy in (1+1)-dimensional spacetime
2.6 The Euclidean path integral representation and the replica method
2.7 Uniqueness of analytic continuation
2.8 Heat kernel and the Sommerfeld formula
2.9 An explicit calculation
2.10 Entropy of massive fields
2.11 An expression in terms of the determinant of the Laplacian on the surface
2.12 Entropy in theories with a modified propagator
2.13 Entanglement entropy in non-Lorentz invariant theories
2.14 Arbitrary surface in curved spacetime: general structure of UV divergences
3 Entanglement Entropy of Non-Degenerate Killing Horizons
3.1 The geometric setting of black-hole spacetimes
3.2 Extrinsic curvature of horizon, horizon as a minimal surface
3.3 The wave function of a black hole
3.4 Reduced density matrix and entropy
3.5 The role of the rotational symmetry
3.6 Thermality of the reduced density matrix of a Killing horizon
3.7 Useful mathematical tools
3.8 General formula for entropy in the replica method, relation to the Wald entropy
3.9 UV divergences of entanglement entropy for a scalar field
3.10 Entanglement Entropy of the Kerr–Newman black hole
3.11 Entanglement entropy as one-loop quantum correction
3.12 The statement on the renormalization of the entropy
3.13 Renormalization in theories with a modified propagator
3.14 Area law: generalization to higher spin fields
3.15 Renormalization of entropy due to fields of different spin
3.16 The puzzle of non-minimal coupling
3.17 Comments on the entropy of interacting fields
4 Other Related Methods
4.1 Euclidean path integral and thermodynamic entropy
4.2 ’t Hooft’s brick-wall model
5 Some Particular Cases
5.1 Entropy of a 2D black hole
5.2 Entropy of 3D Banados–Teitelboim–Zanelli (BTZ) black hole
5.3 Entropy of d-dimensional extreme black holes
6 Logarithmic Term in the Entropy of Generic Conformal Field Theory
6.1 Logarithmic terms in 4-dimensional conformal field theory
6.2 Logarithmic terms in 6-dimensional conformal field theory
6.3 Why might logarithmic terms in the entropy be interesting?
7 A Holographic Description of the Entanglement Entropy of Black Holes
7.1 Holographic proposal for entanglement entropy
7.2 Proposals for the holographic entanglement entropy of black holes
7.3 The holographic entanglement entropy of 2D black holes
7.4 Holographic entanglement entropy of higher dimensional black holes
8 Can Entanglement Entropy Explain the Bekenstein–Hawking Entropy of Black Holes?
8.1 Problems of interpretation of the Bekenstein–Hawking entropy as entanglement entropy
8.2 Entanglement entropy in induced gravity
8.3 Entropy in brane-world scenario
8.4 Gravity cut-off
8.5 Kaluza–Klein example
9 Other Directions of Research
9.1 Entanglement entropy in string theory
9.2 Entanglement entropy in loop quantum gravity
9.3 Entropy in non-commutative theories and in models with minimal length
9.4 Transplanckian physics and entanglement entropy
9.5 Entropy of more general states
9.6 Non-unitary time evolution
10 Concluding remarks
11 Acknowledgments
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