Immirzi parameter
Encyclopedia
The 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.
s 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
(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 Girard's theorem
relating the area of a spherical triangle to its 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 be equal
Here
is the Immirzi parameter and
or
depending on the gauge group used in loop quantum gravity
. So, by choosing the Immirzi parameter to be equal to
one recovers Bekenstein-Hawking entropy formula
. This computation appears independent on the kind of black hole one considers being the given Immirzi parameter always the same. However, Krzysztof Meissner and Marcin Domagala with Jerzy Lewandowski 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.
the eigenvalues of area operator are symmetric by the ladder symmetry. Corresponding to each eigenvalue there are a finite number of degenerate states. One application could be if the classical null character of a horizon is disregarded in the quantum sector, in the lack of energy condition and presence of gravitational propagation the Immirzi parameter tunes to:
by the use of Olaf Dreyer
's conjecture for identifying the evaporation of minimal area cell with the corresponding area of the highly damping quanta. This proposes a kinematical picture for defining a quantum horizon via spin foam
models, however the dynamics of such a model has not yet initiated to be studied.
based on quasinormal mode
s.
Another more recent interpretation is that it is the measure of the value of parity
violation in quantum gravity, and its positive real value is necessary for the Kodama state
of loop quantum gravity. 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.
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
appearing in loop quantum gravity
Loop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
, a nonperturbative theory of quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
. The Immirzi parameter measures the size of the quantum of area in Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
. As a result, its value is currently fixed by matching the semiclassical black hole entropy, as calculated by Stephen Hawking
Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
, and the counting of microstates in loop quantum gravity.
The reality conditions
The Immirzi parameter arises in the process of expressing a Lorentz connection with noncompact 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 horizonEvent horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
s and the second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
, carried out a semiclassical calculation showing that black holes are in equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
with thermal radiation
Thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....
outside them, and that black hole entropy (more properly, the entropy of the radiation in equilibrium with the black hole) equals
(in Planck's unitsPlanck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
)
In 1997, Ashtekar
Abhay Ashtekar
Abhay Vasant Ashtekar is an Indian theoretical physicist. He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and Geometry at Pennsylvania State University. As the creator of Ashtekar variables, he is one of the founders of loop quantum gravity and its...
, Baez, Corichi
Alejandro Corichi
Alejandro Corichi is a theoretical physicist working at the Quantum Gravity group of the National Autonomous University of Mexico . He obtained his bachelor degree at UNAM and his PhD at Pennsylvania State University ....
and Krasnov quantized the classical phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
of the exterior of a black hole in vacuum General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. 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
Chern-Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type, introduced by Edward Witten. It is so named because its action is proportional to the integral of the Chern–Simons 3-form....
. 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 Girard's theorem
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
relating the area of a spherical triangle to its 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 be equal
Here
is the Immirzi parameter andor
depending on the gauge group used in loop quantum gravity
Loop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
. So, by choosing the Immirzi parameter to be equal to
one recovers Bekenstein-Hawking entropy formulaBlack hole thermodynamics
In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons...
. This computation appears independent on the kind of black hole one considers being the given Immirzi parameter always the same. However, Krzysztof Meissner and Marcin Domagala with Jerzy Lewandowski have fixed an incorrect assumption that only the minimal values of the spin contributes. Their result involves the logarithm of a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
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.
Immirzi parameter in Spin Foam theory
In late 2006, independent from the definition of isolated horizon theory, it was reported that in loop quantum gravityLoop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
the eigenvalues of area operator are symmetric by the ladder symmetry. Corresponding to each eigenvalue there are a finite number of degenerate states. One application could be if the classical null character of a horizon is disregarded in the quantum sector, in the lack of energy condition and presence of gravitational propagation the Immirzi parameter tunes to:
by the use of Olaf Dreyer
Olaf Dreyer
Olaf Dreyer is a German theoretical physicist and postdoctoral fellow in the Department of Applied Mathematics at the University of Waterloo whose research interests include quantum gravity and the quantum measurement problem. Dreyer received his Ph.D. in quantum gravity in 2001 from the...
's conjecture for identifying the evaporation of minimal area cell with the corresponding area of the highly damping quanta. This proposes a kinematical picture for defining a quantum horizon via spin foam
Spin foam
In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral description of quantum gravity...
models, however the dynamics of such a model has not yet initiated to be studied.
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 DreyerOlaf Dreyer
Olaf Dreyer is a German theoretical physicist and postdoctoral fellow in the Department of Applied Mathematics at the University of Waterloo whose research interests include quantum gravity and the quantum measurement problem. Dreyer received his Ph.D. in quantum gravity in 2001 from the...
based on quasinormal mode
Quasinormal mode
-Wave Mechanics:Quasinormal modes are the modes of energy dissipation of aperturbed object or field. A familiar example is theperturbation of a wine glass with a knife: the glass begins to...
s.
Another more recent interpretation is that it is the measure of the value of parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
violation in quantum gravity, and its positive real value is necessary for the Kodama state
Kodama state
In 1988, Hideo Kodama wrote down the equations of the Kodama state, but as it described a positive spacetime, which was believed to be inconsistent with observation, it was largely ignored....
of loop quantum gravity. 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
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
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 of Isolated Horizons and Black Hole Entropy, a calculation incorporating matter and the theory of isolated horizons from General RelativityGeneral relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. - Area, Ladder Symmetry, and Degeneracy in Loop Quantum Gravity, a brief review on the quantum of area ladder symmetry and area degeneracy in loop quantum gravityLoop quantum gravityLoop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
and the application of these two in the calculation incorporating the modifications of black hole radiation.