Laws
of Black-Hole Thermodynamics |

**In General** > s.a. black-hole
thermodynamics and specific black-hole
types; gravitational thermodynamics.

* __In various theories__:
Laws of black-hole mechanics can be derived in any theory of gravity by
varying the expression that gives their energy as a function of various
parameters; If the theory is diffeomorphism-invariant, the entropy term
will be proportional to the horizon area; The specific form of the field
equations enters in the coefficient of d*S* in the first law – the
expression for *T* – and in the greybody factors for the radiation
spectrum (as Visser pointed out, the field equations are not directly
involved in the fact that there is radiation); Similarly, obtaining the
right form of the entropy or radiation spectrum in the semiclassical
sector of a theory of quantum gravity only indicates that the limit is
consistent with classical gravity.

@ __Intros, reviews__: Compčre gq/06-ln.

**Zeroth Law**

* __Idea__: The surface
gravity *κ* is
constant on the horizon, like temperature; For a Kerr-Newman black hole,

*κ* = 4π (*r*_{+}*c*^{2}
– *GM*)/*A*, *A*
= 4π*Gc*^{–4}[2*GM*^{2}
– *Q*^{2} + 2 (*G*^{2}*M*^{4
}–* J*^{2}*c*^{2}
– *GM*^{2}*Q*^{2})^{1/2}]
.

**First Law** > s.a. isolated
horizons; Smarr Formula.

* __Idea__: The
relationship usually called the "first law of black-hole thermodynamics"
is actually the black-hole version of the fundamental identity of
thermodynamics, analogous to d*E* = –*p* d*V*+ *T*
d*S* (rather than the first law d*E* = δ*W* + δ*Q*,
which is a more general expression of the conservation of energy),

d*M* = Ω · d**J** + (*κ*/4π) d*A*
+ Φ d*Q* ,

with Ω:= **a**/*α* = ** L**/4*M*^{3}
appearing in the expression for the Killing vector field tangent to the
black-hole horizon *l*^{a}
= *k*^{a} + Ω* m*^{a}
(*k* and *m* are the timelike and spacelike Killing
vectors, respectively), Ω = 4π*J*/*MA* is constant for a
stationary black hole, and Φ = 4π*Qr*_{+}/*A*,
where *Q* is here the black hole electric charge.

* __Other backgrounds__:
Has been shown to hold in AdS black holes, but the correct results are
from around 2005.

@ __General references__: Wald in(93)gq;
Sorkin & Varadarajan CQG(96)gq/95;
Iyer PRD(97)gq/96;
Fursaev PRD(99)ht/98
[energy vs Hamiltonian]; Fatibene et al AP(99)ht/98;
Hayward CQG(98)gq/97
[and relativistic thermodynamics]; Mukohyama PRD(99)gq/98
[Noether charge form]; Amsel et al PRD(08)-a0708
[physical process version, bifurcate Killing horizons]; Wall JHEP(09)-a0901
[critique of attempts at proof]; Ropotenko a1105;
Dolan CQG(11)-a1106
[pressure and volume]; Corda JHEP(11)-a1107
[effective temperature and corrections]; Dolan in(12)-a1209
[*p*d*V* term]; Kelly JHEP(14)-a1408
[without entanglement]; Ma & Zhao CQG(14)-a1411
[corrected form]; Armas et al a1512 [gravitational tension and black-hole volume].

@ __Quasilocal first law__: Mukohyama & Hayward CQG(00)gq/99;
Frodden et al PRD(13)-a1110;
Chatterjee & Ghosh a1511
[from local Lorentz transformations]

@ __Special types of black holes__: Gao & Wald PRD(01)gq
[charged, rotating]; Le Tiec et al PRD(12)-a1111,
Blanchet et al PRD(13)-a1211
[binary black holes]; McCormick ATMP(14)-a1302
[Einstein-Yang-Mills black holes]; Johnstone et al PRD(13)-a1305
[extremal black holes]; Viaggiu GRG(15)-a1506
[for dynamical apparent horizons, black holes in FLRW universes]; Prabhu a1511
[matter fields with internal gauge freedom]; > s.a. kerr
spacetime; specific black-hole types.

@ __Isolated, dynamical horizons__:
Ashtekar
et al PRD(00)gq,
PRD(01)gq
[rotating]; Allemandi et al gq/01;
Booth & Fairhurst
PRL(04)gq/03;
Hayward PRD(04)gq;
Chatterjee & Ghosh PRD(09)-a0812.

@ __Black rings__: Copsey & Horowitz PRD(06)ht/05
[dipole
charges]; Astefanesei & Radu PRD(06)ht/05
[quasilocal];
Rogatko PRD(05)ht.

@ __Modified theories__: Rogatko PRD(98)ht
[Einstein-Maxwell-axion-dilaton]; Sermutlu
CQG(98)
[strings];
Gao PRD(03)
[Einstein-Maxwell
and Einstein-Yang-Mills]; Koga PRD(05)ht
[higher-order,
AdS black holes]; Kastor & Traschen JHEP(06)
[Kaluza-Klein
black holes]; Rogatko PRD(07)-a0705
[for
black saturns]; Wu et al NPB(08)-a0711
[including braneworld]; Miao et al JCAP(11)-a1107
[violation in *f*(*T*) gravity]; Kunduri & Lucietti CQG(14)-a1310
[5D]; Fan & Lü PRD(15)-a1501
[quadratically extended theories]; > s.a. Smarr
Formula.

**(Generalized) Second Law** (Area law) > s.a. black-hole
entropy; entropy bounds; horizons;
Penrose Process; specific
black-hole types.

* __Idea__: For any
process, d*A* > 0 (conjecture by Floyd and Penrose, proved
by Christodoulou for some processes, and as a general theorem by Hawking,
assuming
the weak energy condition holds), which influences the amount of energy we
can extract
from a black hole, *A* ~ black-hole
entropy; The proof of this has been reduced to
the cosmic censorship conjecture.

@ __General references__: Bekenstein PRD(73),
PRD(74);
Hawking
PRD(76);
Unruh
& Wald
PRD(82);
Sewell
PLA(87);
Frolov
& Page PRL(93)gq
[quasistationary];
Mukohyama
PRD(97)gq/96
[non-eternal];
Sung gq/97;
Bekenstein
PRD(99)gq
[quantum
buoyancy]; Shimomura & Mukohyama
PRD(00)gq/99
[charged
particles]; Gao & Wald PRD(01)gq
[charged,
rotating];
Davies & Davis FP(02)
[cosmological];
Davis et al CQG(03)ap;
Matsas & Rocha da Silva PRD(05)gq
[thought
experiment]; Saida CQG(06)gq
[and radiation as non-equilibrium process]; He & Zhang JHEP(07)-a0712
[dynamical
horizons]; Kabe a1003/PRD;
Chakraborty et al EPL(10)-a1009
[and nature of the entropy function]; Hod PLB-a1511
[and the hoop conjecture].

@ __And entropy bounds__: Pelath & Wald PRD(99)gq;
Flanagan et al PRD(00)gq/99.

@ __Related topics__: Giulini JMP(98)gq
[cusps
on horizon]; Song & Winstanley IJTP(08)gq/00
[and
information theory]; Park IJMPA(09).

@ __In other theories__: Sadjadi PRD(07)-a0709
[*f*(*R*)
gravity]; Akbar IJTP(09)-a0808
[Gauss-Bonnet
and Lovelock gravity]; Sadjadi PS(11)-a1009
[Gauss-Bonnet gravity]; Sarkar & Wall PRD(11)-a1011
[Lovelock gravity, violation in black-hole merger]; Capela & Tinyakov
JHEP(10)-a1102
[massive gravity]; Abdolmaleki et al PRD(14)-a1401
[scalar-tensor gravity]; Wall IJMPD(15)-a1504-GRF
[higher-curvature gravity].

@ __Possible violations__: Shimomura et al PRD(00)gq/99;
Park CQG(08)-ht/06;
Eling & Bekenstein PRD(09)-a0810
[mechanisms
that make it work].

**Third Law** > s.a. specific
black-hole types.

* __Idea__: There
cannot be an equilibrium black hole with vanishing *κ*; Like *T*
in the third law of thermodynamics.

* __Remark__: The
Nernst formulation does not apply to rotating black holes.

@ __References__: Carter in(79);
Israel PRL(86);
Roman
GRG(88);
Dadhich
& Narayan PLA(97)gq
[and
gravitational charge]; Wald PRD(97)gq;
Rácz
CQG(00)gq;
Lowe
PRL(01)gq/00
[semiclassical];
Liberati et al IJMPD(01)gq/00
[extremal].

**Related Topics**

@ __Fourth law__: Loustó NPB(93)gq
[scaling laws in critical transitions].

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