WKB Approximation in Noncommutative Gravity

Abstract

Abstract. We consider the quasi-commutative approximation to a noncommutative geometry defined as a generalization of the moving frame formalism. The relation which exists between noncommutativity and geometry is used to study the properties of the highfrequency waves on the flat background.Key words: noncommutative geometry; models of quantum gravity 2000 Mathematics Subject Classification: 46L87; 83C35 Preliminary formalismOur purpose in this paper is to analyze the relation which exists between the noncommutativity of the local coordinates and the gravitational field on a given space-time. In particular model of noncommutative gravity which we develop, in the noncommutative frame formalism, this relation is expressed as consistency between the algebraic and the differential-geometric structures, or in other language as generalized Jacobi identities.Let µ be a typical 'large' source mass with 'Schwarzschild radius' G N µ. We have two length scales, determined by respectively G N , the square of the Planck length and byk, the scale of noncommutativity. The gravitational field is weak if the dimensionless parameter GF = G N −1 µ 2 is small; the space-time is almost commutative if the dimensionless parameter =kµ 2 is small. These two parameters are not necessarily related but we shall here assume that they are of the same order of magnitude,If noncommutativity is not directly related to gravity then it makes sense to speak of ordinary gravity as the limitk → 0 with G N µ nonvanishing. On the other hand if noncommutativity and gravity are directly related then both should vanish withk. We wish here to consider an expansion in the parameter , which we have seen is a measure of the relative dimension of a typical 'space-time cell' compared with the Planck length of a typical quantity of gravitational energy. Our motivation for considering noncommutative geometry as an 'avatar' of gravity is the belief that it sheds light on the role of the gravitational field as the universal regulator of ultra-violet divergences. We mention here only some elements of the approach we use to This paper is a contribution to the

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“…For another approach see Refs. [32][33][34], where a solid indication that the degrees of freedom or basic modes of the resulting theory of gravity can be put in correspondence with those of the noncommutative structure has been presented. In this case, the usual symmetries such as coordinate invariance are built-in, and the commutator of coordinates can have arbitrary dependence on them.…”

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“…Besides the above approaches, construction of noncommutative gravitational models can be achieved using the noncommutative realization of matrix geometries [24,25,26,27,28,29,30]. Also, for alternatives see [31,32,33] (see also [34]). In general, it is understood that the formulation of noncommutative gravity is accompanied by the breaking of Lorentz invariance by noncommutative deformations.…”