Noncommutative general relativity

Abstract

We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations. Local Lorentz invariance is treated as a gauge theory with the spin connection field taken in the so(3,1) enveloping algebra. The resulting theory appears to be a noncommutative extension of the unimodular theory of gravitation. We compute the leading order noncommutative correction to the action and derive the noncommutative correction to the equations of motion of the weak gravitation field.

Self-dual road to noncommutative gravity with twist: A new analysis

“…(1.26) is important because, from the condition of vanishing torsion 2-form, i.e. 27) one finds a solution of the vacuum Einstein equations, since the latter read as 28) while from dT = 0 and from R = dω − ω ∧ ω and associativity of the exterior product one finds R ∧ e = 0, (1.29) which coincides with Eq. (1.28) upon imposing self-duality or anti-self-duality: ( * ) R = ±iR according to Eq.…”

“…We find it encouraging that the self-dual option can be pursued to the extent shown in our paper, without any use of the Seiberg-Witten map or yet other techniques applied in the previous literature [24][25][26][27][28][29][30][31][32][33][34][35][36][37] Appendix A: Twist differential geometry…”

Hawking radiation as quantum tunneling from a noncommutative Schwarzschild black hole

“…To inquire into this issue, one would require to prosperously build the noncommutativity corresponding to the General Relativity. Although this issue has been considered in the literature [10], but no perfect and wholly convincing theory of this model yet exists. There are plenty formulations of noncommutative field theory established upon the Weyl-Wigner-Moyal ⋆-product [11] that conduct to downfall in finding a solution to the some prominent difficulties, such as Lorentz invariance breaking, defeat of unitarity and UV divergences of quantum field theory.…”

“…The authors in Ref. [10] have really demonstrated that the leading noncommutativity amendments to the form of the Einstein-Hilbert action are at least second order in the noncommutativity parameter, θ. The generalization of quantum field theory by noncommutativity based on coordinate coherent state formalism is also interestingly curing the short distance behavior of pointlike structures [6][7][8][9] (see also [12]).…”

Second order noncommutative corrections to gravity

“…Another approach has been proposed based on true physical symmetries [6,7]. In that approach one restricts the noncommutative action to symmetries of the noncommutative algebra:…”