Second order noncommutative corrections to gravity

It is difficult to formulate general relativity on noncommutative spaces, and there are thus different approaches in the literature. In [1] for example a deformation of Einstein’s gravity was studied using a construction based on gauging the noncommutative SO(4,1) de Sitter group and the Seiberg-Witten map with subsequent contraction to ISO(3,1). Most recently constructions of a noncommutative gravitational theory [2,3] were proposed based on a twisted Poincaré algebra [4,5]. The main problem in formulating a theory of gravity on noncommutative manifolds is that it is difficult to implement symmetries such as general coordinate covariance and local Lorentz invariance and to define derivatives which are torsion-free and satisfy the metricity condition. Another approach has been proposed based on true physical symmetries [6,7] (see also[8]). In that approach one restricts the noncommutative action to symmetries of the noncommutative algebra