  In this paper, we perform a parallel analysis to the model proposed in [25].
By considering the central co-tetrad (instead of the central metric), we
investigate the modifications in the gravitational metrics coming from the
noncommutative spacetime of the $\kappa$-Minkowski type in four dimensions. The
differential calculus corresponding to a class of Jordanian
$\kappa$-deformations provides metrics, which lead either to cosmological
constant or spatial curvature type solutions of non-vacuum Einstein equations.
Among vacuum solutions, we find pp-wave type.
