Metric perturbations in Noncommutative Gravity

TL;DR

This paper develops a noncommutative gravity framework using Hopf algebra and differential geometry, deriving corrections to classical gravity and analyzing perturbations around Schwarzschild spacetime.

Contribution

It introduces a general formalism for noncommutative gravity via Drinfeld twist and applies it to linearized perturbations, including a noncommutative Regge-Wheeler potential.

Findings

Derived a noncommutative correction to the Regge-Wheeler potential.

Presented a novel approach for noncommutative Einstein manifolds.

Performed first-order perturbation analysis in noncommutative gravity.

Abstract

We use the framework of Hopf algebra and noncommutative differential geometry to build a noncommutative (NC) theory of gravity in a bottom-up approach. Noncommutativity is introduced via deformed Hopf algebra of diffeomorphisms by means of a Drinfeld twist. The final result of the construction is a general formalism for obtaining NC corrections to the classical theory of gravity for a wide class of deformations and a general background. This also includes a novel proposal for noncommutative Einstein manifold. Moreover, the general construction is applied to the case of a linearized gravitational perturbation theory to describe a NC deformation of the metric perturbations. We specifically present an example for the Schwarzschild background and axial perturbations, which gives rise to a generalization of the work by Regge and Wheeler. All calculations are performed up to first order in perturbation of the metric and noncommutativity parameter. The main result is the noncommutative Regge-Wheeler potential. Finally, we comment on some differences in properties between the Regge-Wheeler potential and its noncommutative counterpart.