Immirzi parameter
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The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. As a result, its value is currently fixed by matching the semiclassical black hole entropy, as calculated by Stephen Hawking, and the counting of microstates in loop quantum gravity.
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The reality conditions
The Immirzi parameter arises in the process of expressing a Lorentz connection with concompact group SO(3,1) in terms of a complex connection with values in a compact group of rotations, either SO(3) or its double cover SU(2). Although named after Giorgio Immirzi, the possibility of including this parameter was first pointed out by Fernando Barbero. The significance of this parameter remained obscure until the spectrum of the area operator in LQG was calculated. It turns out that the area spectrum is proportional to the Immirzi parameter.
Black hole thermodynamics
In the 1970s Hawking, motivated by the analogy between the law of increasing area of black hole event horizons and the second law of thermodynamics, carried out a semiclassical calculation showing that black holes are in equilibrium with thermal radiation outside them, and that black hole entropy (more properly, the entropy of the radiation in equilibrium with the black hole) equals
- <math>S=A/4<math> (in Planck's units)
In 1997, Ashtekar, Baez, Corichi and Krasnov quantized the classical phase space of the exterior of a black hole in vacuum General Relativity. They showed that the geometry of spacetime outside a black hole is described by spin networks some of whose edges 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 group SO(2), which is isomorphic to U(1). The relationship between area and rotations is explained by the Gauss relationship between the area of spherical triangles and their angular excess.
By counting the number of spin-network states corresponding to a horizon of area A, the entropy of black holes is seen to equal
- <math>S=A/4\gamma.<math>
Here <math>\gamma <math> is the Immirzi parameter. It was thought to take the values
- <math>\ln(2) / \sqrt{3}\pi<math>
or
- <math>\ln(3) / \sqrt{8}\pi,<math>
depending on the gauge group used in loop quantum gravity. However, Krzysztof Meissner [1] (http://arxiv.org/abs/gr-qc/0407052) and Marcin Dogamala with Jerzy Lewandowski [2] (http://arxiv.org/abs/gr-qc/0407051) have fixed an incorrect assumption that only the minimal values of the spin contributes. 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.
Interpretation
The parameter may be viewed as a renormalization of Newton's constant. Various speculative proposals to explain this parameter have been suggested: for example, an argument due to Olaf Dreyer based on quasinormal modes. As of today, no alternative calculation of this constant exists. If a second match with experiment or theory (for example, the value of Newton's force at long distance) were found requiring a different value of the Immirzi parameter, it would constitute evidence that loop quantum gravity cannot reproduce the physics of general relativity at long distances. On the other hand, the Immirzi parameter seems to be the only free parameter of vacuum LQG, and once it is fixed by matching one calculation to an "experimental" result, it could in principle be used to predict other experimental results. Unfortunately, no such alternative calculations have been made so far.
External links
- Quantum Geometry and Black Hole Entropy (http://xxx.lanl.gov/abs/gr-qc/9710007), the original vacuum LQG calculation of BH entropy.
- Quantum Geometry of Isolated Horizons and Black Hole Entropy (http://xxx.lanl.gov/abs/gr-qc/0005126), a calculation incorporating matter and the theory of isolated horizons from General Relativity.