Black Hole Entropy| Black Hole Entropy |
In General > s.a. black hole thermodynamics; gravitational
phenomenology [spacetime measurement]; 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: A Schwarzschild
black hole of classical surface area AH has
entropy (in any dimension)
SB = 
(kBc3/G
) AH .
* Reissner-Nordström:
An extremal black hole has S = 0 but A 
0.
* Remark: With matter
fields, in general S A/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-in, ht/01-in;
Mukohyama gq/99-in;
Damour ht/04-in;
Jacobson et al ht/05-in
[trialogue on interpretation]; Soloviev gq/05-in.
@ General references: Wald PRD(79);
Gould PRD(87);
Martinez & York PRD(89);
Hiscock PRD(89);
Frolov & Novikov PRD(93)gq;
Wald PRD(93)gq [Noether
charge]; Dowker CQG(94);
Teitelboim ht/94 [and
dimensional continuation];
Frolov ht/94-in,
ht/95-in;
Jacobson et al PRD(94)gq/93;
Brown & York gq/94-in
[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;
Lue & Weinberg GRG(00)gq [and
monopoles]; Major & Setter CQG(01)
[universality]; Medved gq/04 [re
interpretation]; Sorkin SHPMP(05)ht-in
[point of view]; Cadoni MPLA(06)ht/05;
Jacobson et al IJTP(05)
[views]; Carlip gq/07-in,
GRG(07)-a0705-GRF
[the problem of universality].
@ Properties: Iyer & Wald PRD(94)gq,
PRD(95)gq;
Lau gq/94;
Bekenstein gq/94-in;
Solodukhin PRD(96)
[approaches agree]; Corichi & Sudarsky MPLA(02)gq/00 [and
area];
Åman
et al GRG(03)gq [and
state space metric].
From Black Hole Microstates > s.a. 2D
gravity; quantum black holes.
@ Fluctuations: Gerlach PRD(76);
York PRD(83),
in(84); Pavón & Rubí PRD(88);
Frolov PRL(95); Sorkin & Sudarsky CQG(99)gq;
Gour & Medved CQG(03)gq;
Requardt a0708.
@ Spacetime microstructure: Scardigli CQG(97)gq [foam];
Padmanabhan PRL(98)ht;
Garattini
Ent(00)gq [foam,
Schwarzschild-de Sitter], IJMPD(02)gq/00;
Bergamin & Grumiller IJMPD(06)gq-GRF.
@ Strings, M-theory: Horowitz & Strominger PRL(96)
[near extremal];
Strominger & Vafa PLB(96);
Horowitz & Marolf PRD(97)ht/96;
Maldacena et al JHEP(97)ht,
Horowitz & Roberts PRL(07)
[extremal,
in M-theory];
Lowe PRL(98);
Dabholkar IJMPD(06).
@ Horizon cft: Carlip gq/95,
gq/96-in, PRD(97)gq/96 [3D],
NPPS(97)gq,
PRL(99)ht/98,
CQG(99)gq, gq/99-in;
Park NPB(02)ht/01 [deformations];
Carlip CQG(05)ht/04 [stretched
horizon], gq/05-in,
IJTP(07)gq/06-in
[horizon constraints and symmetry algebra].
@ Horizon microstates: Sfetsos & Skenderis NPB(98)ht/97;
Brown PRD(98);
Epp & Mann MPLA(98)gq,
Epp gq/98/PRD
[tetrad approach]; Cvetic & Larsen
PRL(99)
[rotating]; Soloviev PRD(00)ht/99;
Dou & Sorkin FP(03)gq,
Rideout & Zohren gq/06-in
[causal
links]; Das & Shankaranarayanan gq/07.
@ Loop quantum gravity:
Garay & Mena CQG(03),
Ghosh & Mitra PLB(05)gq/04 [and
Immirzi parameter]; Domagala & Lewandowski CQG(04)gq;
Meissner CQG(04)gq;
Swain IJMPD(05)gq-GRF;
Mitra a0705-in
[state counting]; Tamaki CQG(07)-a0707;
Jacobson CQG(07)-a0708 [and
renormalization]; Sahlmann PRD(07)-a0709.
Other Origin and Approaches > s.a. horizons.
* Ways of forming the
black hole: S is the log of the number of quantum mechanically
distinct ways that the black hole could have been made, or information lost in
the creation
of
the black hole; This leads to an estimate of the quantum black hole level spacing.
* Entanglement entropy:
Black hole entropy is semi-classical, in the sense that it comes from tracing
out
over
internal degrees of freedom of quantum fields in a classical geometry,
without
even backreaction – related to the "brick wall model" idea; Appears
to be proportional to A for fields in the ground state or in coherent/squeezed
states, not in excited states.
@ Ways of forming the black hole: Zurek & Thorne PRL(85); & Bekenstein; & Hawking;
Mukhanov FP(03);
Hsu & Reeb a0706 [and
monster states].
@ Entanglement entropy: Sorkin in(83); Bombelli et al PRD(86);
Srednicki
PRL(93);
Jacobson gq/94 [and
induced gravity]; Kabat NPB(95)ht [and
1-loop corrections]; Muller & Loustó PRD(95);
Ahmadi et al ht/05-in
[deviations
from area law]; Ansari NPB(08)gq/06 [in
lqg]; Emparan JHEP(06)
[holographic derivation]; Das & Shankaranarayanan gq/06-in,
Das et al a0705,
a0708-in [power-law
corrections in
different states]; Brout a0802 [independence
of number of species]; Das et al a0806-in
[rev].
@ And brick wall: Mukohyama et al PRD(97)gq;
Mukohyama gq/98-PhD, gq/99-in;
Jing & Yan PRD(99)gq [Kerr];
Garattini MPLA(04)
[and spacetime foam].
@ Other proposals: Yang MPLA(01)
[topological invariants]; Badiali JPA(06)gq/05 [spacetime
order]; Banerjee et al a0804 [covariant anomalies].
Corrections > s.a. 2D gravity;
3D black holes; 3D quantum
gravity; black
hole radiation; 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.
@ From partition function: Gibbons & Hawking PRD(77); York PRD(86).
@ Quantum / log corrections: Susskind & Uglum PRD(94)
[canonical quantum gravity and string theory]; Mitra gq/95-in;
Ghosh & Mitra
PRD(97);
Carlip CQG(00)gq [log,
Cardy formula];
Kaul & Majumdar PRL(00);
Das et al CQG(02)ht/01, ht/02-in;
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;
Medved & Vagenas PRD(04)ht [using
gup]; Zhao et al MPLA(07)gq/06;
Shankaranarayanan MPLA-a0805.
@ Renormalized: Odintsov & Yoon IJMPA(96)gq/95; Larsen & Wilczek
NPB(96).
@ Higher-derivative gravity:
Correa-Borbonet BJP(05)ht-in
[Lovelock
theory]; Kraus & Larsen JHEP(05)ht.
@ Related topics: Belgiorno & Liberati GRG(97)gq/96 [and
Casimir effect]; Liberati NCB(97)gq/96 ["intrinsic" thermodynamics];
Strominger JHEP(98)ht/97 [AdS
geometry]; Jacobson gq/99-in
[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]; > s.a.
entropy.
Other Related Topics > see fine structure constant [variation].
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23 jun 2008