Gravity from Noncommutative Geometry

TL;DR

This paper develops a noncommutative geometric framework combining continuous manifolds with discrete spaces, deriving curvature and scalar curvature, and relating the results to the Standard Model and cosmological constant.

Contribution

It introduces a linear connection in noncommutative geometry for a product of manifold and discrete space, analyzing its metric properties and curvature, and connecting to physical models.

Findings

Scalar curvature differs from standard by a term akin to cosmological constant

No additional dynamical fields are found in the model

Solution with scaling dependence matches Standard Model features

Abstract

We introduce the linear connection in the noncommutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, define the metric connection and calculate the curvature. We define also the Ricci tensor and the scalar curvature. We find that the latter differs from the standard scalar curvature of the manifold by a term, which might be interpreted as the cosmological constant and apart from that we find no other dynamical fields in the model. Finally we discuss an example solution of flat linear connection, with the nontrivial scaling dependence of the metric tensor on the discrete variable. We interpret the obtained solution as confirmed by the Standard Model, with the scaling factor corresponding to the Weinberg angle.