Editing Immirzi parameter (section)

==Black hole thermodynamics==
In the 1970s Stephen Hawking, motivated by the analogy between the law of increasing area of black hole [[event horizon]]s and the [[second law of thermodynamics]], performed a [[Semiclassical gravity|semiclassical]] calculation showing that black holes are in [[thermodynamic equilibrium|equilibrium]] with [[thermal radiation]] outside them, and that black hole entropy (that is, the entropy of the black hole itself, not the entropy of the radiation in equilibrium with the black hole, which is infinite) equals

:<math>\, S=A/4\!</math> (in [[Planck units]])

In 1997, [[Abhay Ashtekar|Ashtekar]], [[John Baez|Baez]], [[Alejandro Corichi|Corichi]] and [[Kirill Krasnov|Krasnov]] quantized the classical [[phase space]] of the exterior of a black hole in vacuum [[General Relativity]].<ref name="Ashtekar1997">{{cite journal |last=Ashtekar |first=Abhay |author2=Baez, John |author3=Corichi, Alejandro |author4= Krasnov, Kirill  |date=1998 |title=Quantum Geometry and Black Hole Entropy |journal=Physical Review Letters |volume=80 |issue=5 |pages=904&ndash;907 |doi=10.1103/PhysRevLett.80.904 |arxiv=gr-qc/9710007 |bibcode=1998PhRvL..80..904A|s2cid=18980849 }}</ref> They showed that the geometry of spacetime outside a black hole is described by [[spin network]]s, some of whose [[edge (graph theory)|edge]]s puncture the event horizon, contributing area to it, and that the quantum geometry of the horizon can be described by a [[U(1)]] [[Chern–Simons theory]]. The appearance of the group U(1) is explained by the fact that two-dimensional geometry is described in terms of the [[Rotation (mathematics)|rotation group]] SO(2), which is isomorphic to U(1). The relationship between area and rotations is explained by [[Girard's theorem]] relating the area of a [[spherical triangle]] to its angular excess.

By counting the number of spin-network states corresponding to an event horizon of area A, the entropy of black holes is seen to be
 
:<math>\, S=\gamma_0 A/4\gamma.\!</math>

Here <math>\gamma </math> is the Immirzi parameter and either

:<math>\gamma_0=\ln(2) / \sqrt{3}\pi</math>

or

:<math>\gamma_0=\ln(3) /  \sqrt{8}\pi,</math>

depending on the [[gauge group]] used in [[loop quantum gravity]]. So, by choosing the Immirzi parameter to be equal to <math>\,\gamma_0</math>, one recovers the [[Bekenstein–Hawking formula]].

This computation appears independent of the kind of black hole, since the given Immirzi parameter is always the same. However, Krzysztof Meissner<ref name="Meissner">{{cite journal |last=Meissner |first=Krzysztof A. |date=2004 |title=Black-hole entropy in loop quantum gravity |journal=Classical and Quantum Gravity |volume=21 |issue= 22|pages=5245&ndash;5251 |doi=10.1088/0264-9381/21/22/015 |arxiv=gr-qc/0407052 |bibcode = 2004CQGra..21.5245M |s2cid=12995629 }}</ref> and Marcin Domagala with Jerzy Lewandowski<ref name="Dogamala">{{cite journal |last=Domagala |first=Marcin |author2=Lewandowski, Jerzy |date=2004 |title=Black-hole entropy from quantum geometry |journal=Classical and Quantum Gravity |volume=21 |issue= 22|pages=5233&ndash;5243 |doi=10.1088/0264-9381/21/22/014 |arxiv=gr-qc/0407051|bibcode = 2004CQGra..21.5233D |s2cid=8417388 }}</ref> have corrected the assumption that only the minimal values of the spin contribute. Their result involves the logarithm of a [[transcendental number]] instead of the logarithms of integers mentioned above.

The Immirzi parameter appears in the denominator because the entropy counts the number of edges puncturing the event horizon and the Immirzi parameter is proportional to the area contributed by each puncture.