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A new solution to the structure equation in noncommutative spacetime

In this paper, starting from the common foundation of Connes’ noncommutative geometry (NCG) [1–4], various possible alternatives in the formulation of a theory of gravity in noncommutative spacetime are discussed in details. The diversity in the final physical content of the theory is shown to be the consequence of the arbitrary choices in each construction steps. | Communications in Physics, Vol. 24, No. 1 (2014), pp. 21-34 A NEW SOLUTION TO THE STRUCTURE EQUATION IN NONCOMMUTATIVE SPACETIME NGUYEN AI VIET Information Technology Institute, Vietnam National University, Hanoi, Vietnam E-mail: naviet@vnu.edu.vn Received 31 December 2013 Accepted for publication 17 February 2014 Abstract. In this paper, starting from the common foundation of Connes’ noncommutative geometry (NCG) [1–4], various possible alternatives in the formulation of a theory of gravity in noncommutative spacetime are discussed in details. The diversity in the final physical content of the theory is shown to be the consequence of the arbitrary choices in each construction steps. As an alternative in the last step, when the structure equations are to be solved, a minimal set of constraints on the torsion and connection is found to determine all the geometric notions in terms of the metric. In the Connes-Lott model of noncommutative spacetime, in order to keep the full spectrum of the discretized Kaluza-Klein theory [8], it is necessary to include the torsion in the generalized Einstein-Hilbert-Cartan action. Keywords: non-commutative geometry, gravity, general relativity, Kaluza-Klein theory, Cartan formalism. I. INTRODUCTION More than twenty years after A.Connes [1–3] suggested the idea of using NCG in physics, the contruction of the physical theory of gravity in noncommutative spacetime has not reached a completely satisfactory state. The basic ideas of generalizing Riemannian geometry in noncommutative context have been presented in [4–9]. It was Chamseddine and Fr¨ohlich who initiated the first real effort to construct an extended Einstein theory of gravity utilizing the noncommutative geometric model of Connes-Lott’s two- sheeted spacetime [10]. Following this pioneering paper, a great number of other papers have discussed different aspects of this interesting topic. However, these works do not agree on the physical content of the final action, where .