Gravity in Non-Commutative Geometry

TL;DR

This paper explores a novel approach to gravity using non-commutative geometry, modeling spacetime as a product of a four-dimensional manifold and a two-point space, leading to a scalar field coupled to Einstein gravity.

Contribution

It introduces a gravity action within non-commutative differential geometry for a spacetime modeled as a product space, linking scalar fields to geometric distances.

Findings

Derives a scalar field coupled to Einstein gravity from non-commutative geometry.

Provides a geometric interpretation of the scalar field as inter-point distance.

Models gravity on a product of a manifold and a two-point space.

Abstract

We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.