Editing Loop quantum gravity (section)

=== Black hole entropy ===
{{main|Black hole thermodynamics|Isolated horizon| Immirzi parameter}}
[[File:Black Hole Merger.jpg|thumb|upright=1.3|An artist depiction of two [[black hole]]s merging, a process in which the [[laws of thermodynamics]] are upheld]]

Black hole thermodynamics is the area of study that seeks to reconcile the [[laws of thermodynamics]] with the existence of [[black hole]] [[event horizon]]s. The [[no hair theorem|no hair conjecture]] of general relativity states that a black hole is characterized only by its [[mass]], its [[charge (physics)|charge]], and its [[angular momentum]]; hence, it has no [[entropy]]. It appears, then, that one can violate the [[second law of thermodynamics]] by dropping an object with nonzero entropy into a black hole.{{sfn|Bousso|2002|pp=825–874}} Work by [[Stephen Hawking]] and [[Jacob Bekenstein]] showed that the second law of thermodynamics can be preserved by assigning to each black hole a ''black-hole entropy''
<math display="block">S_{\text{BH}} = \frac{k_{\text{B}}A}{4\ell_{\text{P}}^2},</math>
where <math>A</math> is the area of the hole's event horizon, <math>k_{\text{B}}</math> is the [[Boltzmann constant]], and <math display="inline">\ell_{\text{P}} = \sqrt{G\hbar/c^{3}}</math> is the Planck length.{{sfn|Majumdar|1998|p=147}} The fact that the black hole entropy is also the maximal entropy that can be obtained by the [[Bekenstein bound]] (wherein the Bekenstein bound becomes an equality) was the main observation that led to the [[holographic principle]].{{sfn|Bousso|2002|pp=825–874}}

An oversight in the application of the [[no-hair theorem]] is the assumption that the relevant degrees of freedom accounting for the entropy of the black hole must be classical in nature; what if they were purely quantum mechanical instead and had non-zero entropy? This is what is realized in the LQG derivation of black hole entropy, and can be seen as a consequence of its background-independence – the classical black hole spacetime comes about from the semiclassical limit of the [[quantum state]] of the gravitational field, but there are many quantum states that have the same semiclassical limit. Specifically, in LQG<ref name="ReferenceA">See [[List of loop quantum gravity researchers]]</ref> it is possible to associate a quantum geometrical interpretation to the microstates: These are the quantum geometries of the horizon which are consistent with the area, <math>A</math>, of the black hole and the topology of the horizon (i.e. spherical). LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon.{{sfn|Rovelli|1996|pp=3288–3291}}{{sfn|Ashtekar|Baez|Corichi|Krasnov|1998|pp=904–907}} These calculations have been generalized to rotating black holes.{{sfn|Ashtekar|Engle|Broeck|2005|pp=L27}}

[[File:LQG black hole Horizon.jpg|thumb|left|upright=1.3|Representation of quantum geometries of the horizon. Polymer excitations in the bulk puncture the horizon, endowing it with quantized area. Intrinsically the horizon is flat except at punctures where it acquires a quantized [[Defect (geometry)|deficit angle]] or quantized amount of curvature. These deficit angles add up to <math>4 \pi</math>.]]

It is possible to derive, from the covariant formulation of full quantum theory ([[Spinfoam]]) the correct relation between energy and area (1st law), the [[Unruh temperature]] and the distribution that yields Hawking entropy.{{sfn|Bianchi|2012}} The calculation makes use of the notion of [[dynamical horizon]] and is done for non-extremal black holes.

A recent success of the theory in this direction is the computation of the [[entropy]] of all non singular black holes directly from theory and independent of [[Immirzi parameter]].{{sfn|Bianchi|2012}}{{sfn|Frodden|Ghosh|Perez|2013|p=121503}} The result is the expected formula <math>S=A/4</math>, where <math>S</math> is the entropy and <math>A</math> the area of the black hole, derived by Bekenstein and Hawking on heuristic grounds. This is the only known derivation of this formula from a fundamental theory, for the case of generic non singular black holes. Older attempts at this calculation had difficulties. The problem was that although Loop quantum gravity predicted that the entropy of a black hole is proportional to the area of the event horizon, the result depended on a crucial free parameter in the theory, the above-mentioned Immirzi parameter. However, there is no known computation of the Immirzi parameter, so it was fixed by demanding agreement with Bekenstein and Hawking's calculation of the [[black hole thermodynamics|black hole entropy]].