Noncommutativity of four-dimensional axisymmetric spacetime in polar coordinate

It is shown that in the noncommutative spacetime defined by the generalized Moyal product, consistent noncommutativity can be obtained independent of the coordinate system such as Cartesian or polar one. In addition, based on the fact that the generalized Moyal product can be applied to arbitrary spacetime with non-trivial curvature, the effect of noncommutativity in the axisymmetric spacetime with central mass is investigated. The results demonstrate how the noncommutativity of spacetime modify the shape of the stationary limit surface in the noncommutative Kerr spacetime, which implies that the gravitational interaction seems to be effectively softened.