Gauging the twisted Poincaré symmetry as a noncommutative theory of gravitation

Abstract

Einstein's Theory of General Relativity was formulated as a gauge theory of Lorentz symmetry by Utiyama in 1956, while the Einstein-Cartan gravitational theory was formulated by Kibble in 1961 as the gauge theory of Poincaré transformations. In a noncommutative space-time with canonical commutation relations between the coordintes, Lorentz symmetry is violated and field theories constructed on such space-times have instead the so-called twisted Poincaré invariance. In this paper a gauge theory formulation of noncommutative gravity is proposed based on the twisted Poincaré symmetry together with the requirement of covariance under the general coordinate transformations, an essential ingredient of the theory of general relativity. The advantages of such a formulation as well as the related problems are discussed and possible ways out are outlined.

Quantum gravity, field theory and signatures of noncommutative spacetime

“…For example, certain dimensional reductions of noncommutative Yang-Mills theory from ten dimensions to four dimensions naturally induce deformations of a Poincaré gauge theory of gravity, owing to the occurrence of teleparallism in noncommutative gauge theory [3,71]. General relativity on noncommutative spacetime has also been constructed by gauging the twist-deformed Poincaré symmetry [72]. Deformations of gravity can moreover be induced from a noncommutative gauge theory with position-dependent noncommutativity θ µν (x) using the Seiberg-Witten map [73].…”

The noncommutative BTZ black hole in polar coordinates

“…There might be several factors for this but if we just count two of them: One is that noncommutative gravity itself is not quite established yet, and the other is that gravity is not exactly a gauge theory thus one cannot use the Seiberg-Witten map to noncommutative gravity directly. One way of evading this is to regard the Einstein's gravity as the Poincaré gauge theory and apply the Seiberg-Witten map for its noncommutative extension [5] or take the twisted Poincaré algebra approach [6,7,8] based on [9]. Only in the three dimensional case one can directly deal with the gravity using the Seiberg-Witten map in the conventional Einstein's framework thanks to the equivalence between the three dimensional gravity theory and the Chern-Simons theory [10,11].…”

Quasinormal modes and quantization of area/entropy for noncommutative BTZ black hole

“…One way of evading this is to regard the Einstein's gravity as the Poincaré gauge theory and apply the SeibergWitten map for its noncommutative extension [34] or take the twisted Poincaré algebra approach in Refs. [35][36][37] based on Refs. [38,39].…”