Noncommutative Einstein Equations

TL;DR

This paper introduces a noncommutative deformation of general relativity where gravity is described by matrix-valued fields, leading to new equations of motion that extend Einstein's theory into a noncommutative framework.

Contribution

It presents a novel formulation of Einstein equations using matrix-valued tensors and derives their equations of motion within a gauge-invariant action, highlighting the noncommutative aspects.

Findings

Noncommutative Einstein equations are derived from a matrix-valued scalar curvature.

The noncommutative part of the equations is expressed via a tensor density that vanishes in the commutative limit.

A noncommutative energy-momentum tensor for matter is also formulated.

Abstract

We study a noncommutative deformation of general relativity where the gravitational field is described by a matrix-valued symmetric two-tensor field. The equations of motion are derived in the framework of this new theory by varying a diffeomorphisms and gauge invariant action constructed by using a matrix-valued scalar curvature. Interestingly the genuine noncommutative part of the dynamical equations is described only in terms of a particular tensor density that vanishes identically in the commutative limit. A noncommutative generalization of the energy-momentum tensor for the matter field is studied as well.