Some Remarks on Gravity in Noncommutative Spacetime and a New Solution to the Structure Equations

TL;DR

This paper explores various formulations of gravity in noncommutative spacetime based on Connes' noncommutative geometry, proposing a minimal set of constraints to determine geometric structures and emphasizing the role of torsion in the Connes-Lott model.

Contribution

It introduces a minimal set of constraints on torsion and connection to uniquely determine geometric notions from the metric in noncommutative gravity theories.

Findings

Diversity in noncommutative gravity arises from construction arbitrariness.

A minimal set of constraints can determine all geometric notions from the metric.

Including torsion is necessary to retain the full spectrum in the Connes-Lott model.

Abstract

In this paper, starting from the common foundation of Connes' noncommutative geometry (NCG) [1,2,3,4], various possible alternatives in the formulation of a theory of gravity in noncommutative spacetime are discussed in detail. The diversity in the final physical content of the theory is shown to the the consequence of the arbitrariness in each construction steps. As an alternative in the last step, when the staructure 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 metric. In the Connes-Lott model of noncommutative spacetime, in order to keep the full spectrum of the discretized Kaluza-Klein theory [5], it is necessary to include the torsion in the generalized Einstein-Hilbert-Cartan action.