The mean curvature vector field of a submanifold in the Eu-
clidean n-space is said to be proper if it is an eigenvector of the Laplace
operator ∆. It is proven that every hypersurface with proper mean cur-
vature vector field in the Euclidean 4-space E4 has constant mean cur-
vature. In this paper, we study an extended version of the mentioned
subject on timelike (i.e., Lorentz) hypersurfaces of Minkowski 4-space
E4
1. Let x : M 3
1 → E4
1 be the isometric immersion of a timelike hyper-
surface M 3
1 in E4
1. The second mean curvature vector field H2 of M 3
1 is
called 1-proper if it is an eigenvector of the Cheng-Yau operator C (which
is the natural extension of ∆). We show that each M 3
1 with 1-proper H2
has constant scalar curvature. By a classification theorem, we show that
such a hypersurface is C-biharmonic, C-1-type or null-C-2-type. Since the
shape operator of M 3
1 has four possible matrix forms, the results will be
considered in four different cases.