Black-Hole Entropy  

In General > s.a. black-hole thermodynamics; origin of black-hole entropy; specific types of black holes.
* Idea: Black-hole entropy was introduced in a formal way; Several interpretations of it have been proposed, some of which are variations on Bekenstein's original idea that it corresponds to the log of "distinct quantum internal states" of the black hole, S = kB ln W, but in general it is not clear what this W means physically; Classically, one has to use S = ∞ to avoid inconsistencies with the second law, but quantum mechanically there is black-hole radiation and all is ok.
* Schwarzschild spacetime: A Schwarzschild black hole of classical surface area AH has entropy (in any dimension)

SB = \(1\over4\)(kBc3/G\(\hbar\)) AH .

* Reissner-Nordström: An extremal black hole has S = 0 but A ≠ 0.
* Remark: With matter fields, in general SA/4; It is possible to form black holes without a global increase in entropy.
* Orders of magnitude: Compare SSun = 1058 with Ssolar mass bh = 1077.
@ Reviews: Mitra ht/96; Frolov & Fursaev CQG(98)ht; Majumdar gq/98-conf, ht/01-conf; Mukohyama gq/99-conf; Damour ht/04-in; Jacobson et al IJTP(05)ht-proc [trialogue on interpretation]; Soloviev gq/05-proc; Mitra a0902-conf.
@ General references: Wald PRD(79); Gould PRD(87); Martinez & York PRD(89); Hiscock PRD(89); Frolov & Novikov PRD(93)gq; Dowker CQG(94); Teitelboim ht/94 [and dimensional continuation]; Frolov ht/94-conf, ht/95-proc; Jacobson et al PRD(94)gq/93; Brown & York gq/94-conf [path integral]; Martinez PRD(95)gq/94 [microcanonical]; Brown PRD(95)gq [from Hamiltonian]; Pretorius et al PRD(98)gq/97 [operational]; Kay ht/98; Mäkelä & Repo gq/98; Medved gq/04 [re interpretation]; Sorkin SHPMP(05)ht-proc [point of view]; Cadoni MPLA(06)ht/05; Jacobson et al IJTP(05) [views]; Carlip JPCS(07)gq, GRG(07)-a0705-GRF, a0807-fs [the problem of universality]; Zhang et al GRG(11)-a1102 [interpretation]; Carlip Ent(11)-a1107 [effective "conformal dual" description]; Nomura & Weinberg PRD(14)-a1310 [as entropy of a vacuum]; Kupferman PhD(13)-a1603 [different concepts].
@ As Noether charge: Wald PRD(93)gq; Bodendorfer & Neiman PRD(14)-a1304 [from loop quantum gravity]; Halyo a1403 [and Rindler energy]; Jacobson & Mohd PRD(15)-a1507.
@ Properties: Iyer & Wald PRD(94)gq, PRD(95)gq; Lau gq/94; Bekenstein gq/94-MG7; Solodukhin PRD(96) [approaches agree]; Corichi & Sudarsky MPLA(02)gq/00 [and area]; Major & Setter CQG(01) [universality]; Åman et al GRG(03)gq [and state space metric]; Saida PTP(09)-a0910 [and area, violation of relationship].
@ Related topics: Lue & Weinberg GRG(00)gq [and monopoles]; Acquaviva et al PRD(15)-a1411, a1604-MG14 [and collapse]; > s.a. gravitational phenomenology [spacetime measurement]; Zhang PRD(15)-a1510 [entropy in the interior of a black hole, and thermodynamics].
> Related topics: see noether charge; thermodynamical systems [contribution to cosmological entropy]; fine-structure constant [variation].

Corrections > s.a. black-hole radiation ands thermodynamics; quantum black holes; specific types of black holes.
* Reasons: Can be due to thermal fluctuations around an equilibrium canonical ensemble, or quantum spacetime fluctuations within a microcanonical framework; Typically, higher-order gravity theories give corrections of the form AH R, R2, etc, while various approaches to effective quantum gravity corrections give terms of the form ln AH.
@ General references: Shankaranarayanan Ent(11)-a1101; Viaggiu MPLA(14) [in an expanding universe].
@ From partition function: Gibbons & Hawking PRD(77); York PRD(86).
@ And GUP: Medved & Vagenas PRD(04)ht; Ko et al IJTP(10); Majumder PLB(12); Gupta et al AHEP(14)-a1312; Bargueño & Vagenas PLB-a1501.
@ Quantum / log corrections: Susskind & Uglum PRD(94) [canonical quantum gravity and string theory]; Mitra gq/95-talk; Ghosh & Mitra PRD(97); Carlip CQG(00)gq [log, Cardy formula]; Kaul & Majumdar PRL(00); Das et al CQG(02)ht/01, ht/02-proc; Gour PRD(02)gq [log]; Mukherji & Pal JHEP(02)ht [AdS-cft]; Chatterjee & Majumdar gq/03, PRL(04)gq/03 [log, non-rotating]; Akbar & Das CQG(04)ht/03 [Schwarzschild and Reissner-Nordström, thermal]; Bhaduri et al PRD(04)gq/03 [microcanonical]; Ghosh & Mitra PRD(05)gq/04; Hod CQG(04)ht [higher-order]; Medved CQG(03) [2D], CQG(05)gq/04, CQG(05)gq/04; Zhao et al MPLA(07)gq/06; Shankaranarayanan MPLA(08)-a0805; Panković et al a0810; Garattini PLB(10) [with modified dispersion relations]; Das et al a1002-MG12 [from entanglement]; Aros et al JHEP(10)-a1003 [as Noether charge]; Yoon GRG(12)-a1211; Ghosh & Mitra CQG(15)-a1506.
@ Renormalized: Odintsov & Yoon IJMPA(96)gq/95; Larsen & Wilczek NPB(96); Jacobson & Satz PRD(13)-a1212; Satz & Jacobson a1301-MG13.
@ Related topics: Belgiorno & Liberati GRG(97)gq/96 [and Casimir effect]; Liberati NCB(97)gq/96 ["intrinsic" thermodynamics]; Strominger JHEP(98)ht/97 [anti-de Sitter geometry]; Jacobson AIP(99)gq [partial horizon]; Wu IJMPD(00)gq [topology]; Dreyer et al CQG(01)gq [symmetries]; Carlip PRL(02)gq [conformal symmetry]; More CQG(05)gq/04 [higher-order corrections]; Ahmad & Alam a1207-proc [subleading corrections]; Park FdP(14)-a1304 [from ADM reduction, and subleading corrections]; > s.a. entropy.

In Other Theories > see 2D gravity; 3D black holes; 3D quantum gravity; Shape Dynamics.
@ General references: Borsten et al PRP(09) [in string theory, and entanglement of qubits and qutrits]; Brustein & Medved JHEP(10)-a1003 [unitary theories]; Das MSc(10)-a1007 [for an arbitrary covariant theory of gravity, inputs and derivation].
@ Higher-derivative gravity: Correa-Borbonet BJP(05)ht-proc [Lovelock theory]; Kraus & Larsen JHEP(05)ht; Faraoni Ent(10)-a1005 [scalar-tensor and f(R) gravity]; Kolekar et al PRD(12)-a1111 [Lanczos-Lovelock models]; Conroy et al PRL(15)-a1503 [ghost-free infinite-derivative theories, Wald entropy]; Bhattacharyya et al JHEP(17)-a1612 [Lovelock theories, dynamical black holes].
> Other areas of physics: see high-temperature superconductors.


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