Editing Loop quantum gravity (section)

== Physical applications of LQG ==
=== 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]].

===Hawking radiation in loop quantum gravity===
{{main|Hawking radiation}}
A detailed study of the quantum geometry of a black hole horizon has been made using loop quantum gravity.{{sfn|Ashtekar|Baez|Corichi|Krasnov|1998|pp=904–907}} Loop-quantization does not reproduce the result for [[black hole entropy]] originally discovered by Bekenstein and Hawking, unless one chooses the value of the [[Immirzi parameter]] to cancel out another constant that arises in the derivation. However, it led to the computation of higher-order corrections to the entropy and radiation of black holes.

Based on the fluctuations of the horizon area, a quantum black hole exhibits deviations from the Hawking spectrum that would be observable were [[X-ray]]s from Hawking radiation of evaporating [[primordial black holes]] to be observed.{{sfn|Ansari|2007|pp=179–212}} The quantum effects are centered at a set of discrete and unblended frequencies highly pronounced on top of Hawking radiation spectrum.{{sfn|Ansari|2008|pp=635–644}}

=== Planck star ===
{{main|Planck star|black hole firewall|black hole information paradox}}

In 2014 Carlo Rovelli and [[Francesca Vidotto]] proposed that there is a [[Planck star]] inside every black hole.{{sfn|Rovelli|Vidotto|2014|p=1442026}} Based on LQG, the theory states that as stars are collapsing into black holes, the energy density reaches the Planck energy density, causing a repulsive force that creates a star. Furthermore, the existence of such a star would resolve the [[black hole firewall]] and [[black hole information paradox]].

=== Loop quantum cosmology ===
{{main|Loop quantum cosmology|Big bounce|inflation (cosmology)}}
The popular and technical literature makes extensive references to the LQG-related topic of loop quantum cosmology. LQC was mainly developed by Martin Bojowald. It was popularized in ''[[Scientific American]]'' for predicting a [[Big Bounce]] prior to the [[Big Bang]].{{sfn|Bojowald|2008}} Loop quantum cosmology (LQC) is a symmetry-reduced model of classical general relativity quantized using methods that mimic those of loop quantum gravity (LQG) that predicts a "quantum bridge" between contracting and expanding cosmological branches.

Achievements of LQC have been the resolution of the big bang [[Singularity (mathematics)|singularity]], the prediction of a Big Bounce, and a natural mechanism for [[inflation (cosmology)|inflation]].

LQC models share features of LQG and so is a useful toy model. However, the results obtained are subject to the usual restriction that a truncated classical theory, then quantized, might not display the true behaviour of the full theory due to artificial suppression of degrees of freedom that might have large quantum fluctuations in the full theory. It has been argued that singularity avoidance in LQC are by mechanisms only available in these restrictive models and that singularity avoidance in the full theory can still be obtained but by a more subtle feature of LQG.{{sfn|Brunnemann|Thiemann|2006a|pp=1395–1428}}{{sfn|Brunnemann|Thiemann|2006b|pp=1429–1484}}

=== Loop quantum gravity phenomenology ===
Quantum gravity effects are difficult to measure because the Planck length is so small. However recently physicists, such as Jack Palmer, have started to consider the possibility of measuring quantum gravity effects mostly from astrophysical observations and gravitational wave detectors. The energy of those fluctuations at scales this small cause space-perturbations which are visible at higher scales.

=== Background-independent scattering amplitudes ===
{{main|Scattering amplitude}}

Loop quantum gravity is formulated in a background-independent language. No spacetime is assumed a priori, but rather it is built up by the states of theory themselves – however scattering amplitudes are derived from <math>n</math>-point functions ([[Correlation function (quantum field theory)|Correlation function]]) and these, formulated in conventional quantum field theory, are functions of points of a background spacetime. The relation between the background-independent formalism and the conventional formalism of quantum field theory on a given spacetime is not obvious, and it is not obvious how to recover low-energy quantities from the full background-independent theory. One would like to derive the <math>n</math>-point functions of the theory from the background-independent formalism, in order to compare them with the standard perturbative expansion of quantum general relativity and therefore check that loop quantum gravity yields the correct low-energy limit.

A strategy for addressing this problem has been suggested;{{sfn|Modesto|Rovelli|2005|p=191301}} by studying the boundary amplitude, namely a path integral over a finite spacetime region, seen as a function of the boundary value of the field.{{sfn|Oeckl|2003a|pp=318–324}}{{sfn|Oeckl|2003b|pp=5371–5380}} In conventional quantum field theory, this boundary amplitude is well–defined{{sfn|Conrady|Rovelli|2004|p=4037}}{{sfn|Doplicher|2004|p=064037}} and codes the physical information of the theory; it does so in quantum gravity as well, but in a fully background–independent manner.{{sfn|Conrady|Doplicher|Oeckl|Rovelli|2004|p=064019}} A generally covariant definition of <math>n</math>-point functions can then be based on the idea that the distance between physical points – arguments of the <math>n</math>-point function is determined by the state of the gravitational field on the boundary of the spacetime region considered.

Progress has been made in calculating background-independent scattering amplitudes this way with the use of spin foams. This is a way to extract physical information from the theory. Claims to have reproduced the correct behaviour for graviton scattering amplitudes and to have recovered classical gravity have been made. "We have calculated Newton's law starting from a world with no space and no time." – Carlo Rovelli.