- Black hole thermodynamics
In

physics ,**black hole thermodynamics**is the area of study that seeks to reconcile thelaws of thermodynamics with the existence ofblack hole event horizon s. Much as the study of the statistical mechanics ofblack body radiation led to the advent of the theory ofquantum mechanics , the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding ofquantum gravity , leading to the formulation of theholographic principle .**Black hole entropy****Black hole entropy**is theentropy carried by ablack hole .If black holes carried no entropy, it would be possible to violate the

second law of thermodynamics by throwing mass into the black hole. The only way to satisfy the second law is to admit that the black holes have entropy whose increase more than compensates for the decrease of the entropy carried by the object that was swallowed.Starting from theorems proved by

Stephen Hawking ,Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of itsevent horizon divided by the Planck area. Later, Stephen Hawking showed that black holes emit thermalHawking radiation corresponding to a certain temperature (Hawking temperature). Using thethermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at 1/4::$S\_\{BH\}\; =\; frac\{kA\}\{4ell\_\{mathrm\{P^2\}$

where "k" is

Boltzmann's constant , and $ell\_\{mathrm\{P=sqrt\{Ghbar\; /\; c^3\}$ is thePlanck length . The black hole entropy is proportional to its area $A$. The fact that the black hole entropy is also the maximal entropy that can be squeezed within a fixed volume was the main observation that led to theholographic principle . The subscript BH either stands for "black hole" or "Bekenstein-Hawking".Although Hawking's calculations gave further thermodynamic evidence for black hole entropy, until 1995 no one was able to make a controlled calculation of black hole entropy based on

statistical mechanics , which associates entropy with a large number of microstates. In fact, so called "no hair" theorems appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 whenAndrew Strominger andCumrun Vafa calculated the right Bekenstein-Hawking entropy of a supersymmetric black hole instring theory , using methods based onD-branes . Their calculation was followed by many similar computations of entropy of large classes of other extremal andnear-extremal black hole s, and the result always agreed with the Bekenstein-Hawking formula.Loop quantum gravity , viewed as the main competitor of string theory, also offered a slightly more heuristic calculation of the black hole entropy. This calculation confirms that the entropy is proportional to the surface area, with the proportionality constant dependent on the only free parameter in LQG,Immirzi parameter .**The laws of black hole mechanics**The four

**laws of black hole mechanics**are physical properties thatblack hole s are believed to satisfy. The laws, analogous to the laws ofthermodynamics , were discovered byBrandon Carter ,Stephen Hawking and James Bardeen.**tatement of the laws**The laws of black hole mechanics are expressed in

geometrized units .**The Zeroth Law**The horizon has constant

surface gravity for a stationary black hole.**The First Law**We have

:$dM\; =\; frac\{kappa\}\{8pi\},dA+Omega,\; dJ+Phi,\; dQ,$

where $M$ is the

mass , $A$ is the horizon area, $Omega$ is theangular velocity , $J$ is theangular momentum , $Phi$ is theelectrostatic potential , $kappa$ is thesurface gravity and $Q$ is theelectric charge .**The Second Law**The horizon area is, assuming the weak energy condition, a non-decreasing function of time,

:$dA\; geq\; 0$

**The Third Law**It is not possible to form a black hole with vanishing surface gravity.$kappa$ = 0 is not possible to achieve.

**Discussion of the laws****The Zeroth Law**The zeroth law is analogous to the

zeroth law of thermodynamics which states that the temperature is constant throughout a body inthermal equilibrium . It suggests that the surface gravity is analogous totemperature ."T" constant for thermal equilbrium for a normal system is analogous to $kappa$ constant over the horizon of a stationary black hole.**The First Law**The left hand side, "dM", is the change in mass/energy. Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right hand side represent changes in energy due to rotation and

electromagnetism . Analogously, thefirst law of thermodynamics is a statement ofenergy conservation , which contains on its right hand side the term "T dS".**The Second Law**The second law is the statement of Hawking's area theorem. Analogously, the

second law of thermodynamics states that theentropy of a isolated system is a non-decreasing function of time, suggesting a link between entropy and the area of a black hole horizon. However, this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. Generalised second law introduced as total entropy = black hole entropy + outside entropy**The Third Law**Extremal black holes have vanishing surface gravity. Stating that $kappa$ cannot go to zero is analogous to the

third law of thermodynamics which, in its weak formulation, states that it is impossible to reachabsolute zero temperature in a physical process. The strong version of the third law of thermodynamics, which states that as the temperature approaches zero, the entropy also approaches zero, does not have an analogue for black holes. However, the strong version is violated by many known systems in condensed matter physics, and has therefore been rejected as a law.**Interpretation of the laws**The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the

no hair theorem , zero entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emitthermal radiation (Hawking radiation ) at temperature:$T\_H\; =\; frac\{kappa\}\{2pi\}$.

From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein-Hawking entropy which is

:$S\_\{BH\}\; =\; frac\{A\}\{4\}$.

**Beyond black holes**Hawking and Page showed that black hole thermodynamics is more general than black holes, that cosmological event horizons also have an entropy and temperature.

More fundamentally, t'Hooft and Susskind used the laws of black hole thermodynamics to argue for a general

Holographic Principle of nature, which asserts that consistent theories of gravity and quantum mechanics must be lower dimensional. Though not yet fully understood in general, the holographic principle has led to the only complete theories of quantum gravity, such as theAdS/CFT correspondence .**ee also***

Stephen Hawking

*Jacob Bekenstein **References***cite journal |last=Bardeen |first=J. M. |authorlink= |coauthors=Carter, B.; Hawking, S. W. |year=1973 |month= |title=The four laws of black hole mechanics |journal=Communications in Mathematical Physics |volume=31 |issue=2 |pages=161–170 |doi=10.1007/BF01645742 |url= |accessdate= |quote=

*cite journal |last=Bekenstein |first=Jacob D. |authorlink= |coauthors= |year=1973 |month= |title=Black holes and entropy |journal=Physical Review D |volume=7 |issue=8 |pages=2333–2346 |doi=10.1103/PhysRevD.7.2333 |url= |accessdate= |quote=

*cite journal |last=Hawking |first=Stephen W. |authorlink= |coauthors= |year=1974 |month= |title=Black hole explosions? |journal=Nature |volume=248 |issue=5443 |pages=30–31 |doi=10.1038/248030a0 |url= |accessdate= |quote=

*cite journal |last=Hawking |first=Stephen W. |authorlink= |coauthors= |year=1975 |month= |title=Particle creation by black holes |journal=Communications in Mathematical Physics |volume=43 |issue=3 |pages=199–220 |doi=10.1007/BF02345020 |url= |accessdate= |quote=

*cite book |title=The Large Scale Structure of Space-time |last=Hawking |first=S. W. |authorlink= |coauthors=Ellis, G. F. R. |year=1973 |publisher=Cambridge University Press |location=New York |isbn=0521099064 |pages=

*cite journal |last=Hawking |first=Stephen W. |authorlink= |coauthors= |year=1994 |month= |title=The Nature of Space and Time |journal=arΧiv e-print |volume= |issue= |pages= |id=arXiv|hep-th|9409195v1 |url= |accessdate= |quote=

*cite journal |last=Meyer |first=A. J., II |authorlink= |coauthors= |year=2006 |month= |title=Black Holes, Entropy and the Third Law |journal=arΧiv e-print |volume= |issue= |pages= |id=arXiv|physics|0608080v1 |url= |accessdate= |quote=**External links*** [

*http://nrumiano.free.fr/Estars/bh_thermo.html Black Hole Thermodynamics*]

* [*http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=1&index1=3281361 Black hole entropy on arxiv.org*]

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