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__:
Compère 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}] .

(While the other laws are analogous to the corresponding ones of ordinary thermodynamics, this law is not really analogous to the zeroth law.)

**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__: It 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 JHEP(16)-a1512 [gravitational tension and black-hole volume].

@ __Quasilocal first law__:
Mukohyama & Hayward CQG(00)gq/99;
Frodden et al PRD(13)-a1110;
Chatterjee & Ghosh EPJC(18)-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 CQG(17)-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];
Arderucio-Costa a1905 [generic semiclassical theory];
> 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];
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];
Cabero et al PRD(18)-a1711 [observational tests].

@ __Modified versions__:
Davies & Davis FP(02) [cosmological];
Bernamonti et al JHEP(18)-a1803 [holographic];
Azuma & Kato a2001 [in terms of quantum conditional entropy].

@ __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];
Hu et al PRD(19)-a1906 [refutation of claim].

**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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