In this paper, we study timelike hypersurfaces of the Minkowski 4-pace with 1-proper second mean curvature vector field. The second mean curvature vector field is 1-proper if it is an eigenvector of the Cheng-Yau operator. we prove that these hypersurfaces have constant scalar curvature. As a classification result, we show that such a hypersurface has to be finite type or 1-minimal.
