  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.
