A Noncommutative Extension of Gravity

TL;DR

This paper extends classical gravity to a noncommutative setting using matrix algebra, linking it to Kaluza-Klein theory with an internal SU(n) space in a simplified approximation.

Contribution

It introduces a noncommutative geometric framework for gravity, connecting it to established Kaluza-Klein models with a novel algebraic approach.

Findings

Equivalent to Kaluza-Klein theory with SU(n) internal space

Formulates Einstein-Hilbert action in noncommutative geometry

Provides a truncated approximation linking noncommutative and classical gravity

Abstract

The commutative algebra of functions on a manifold is extended to a noncommutative algebra by considering its tensor product with the algebra of nxn complex matrices. Noncommutative geometry is used to formulate an extension of the Einstein-Hilbert action. The result is shown to be equivalent to the usual Kaluza-Klein theory with the manifold SUn as an internal space, in a truncated approximation.