Noncommutative General Relativity

TL;DR

This paper develops a noncommutative version of general relativity for canonical noncommutative spaces, extending unimodular gravity and analyzing gauge symmetries, with explicit leading-order corrections to the gravitational action and equations.

Contribution

It introduces a noncommutative general relativity framework with volume-preserving transformations and gauge treatment of local Lorentz invariance, extending unimodular gravity.

Findings

Identifies volume-preserving coordinate transformations in noncommutative spacetime.

Derives leading-order noncommutative corrections to gravitational action.

Calculates noncommutative modifications to weak-field equations.

Abstract

We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations. Local Lorentz invariance is treated as a gauge theory with the spin connection field taken in the so(3,1) enveloping algebra. The resulting theory appears to be a noncommutative extension of the unimodular theory of gravitation. We compute the leading order noncommutative correction to the action and derive the noncommutative correction to the equations of motion of the weak gravitation field.