Let x : M n → M ¯ n + 1 be an n-dimensional spacelike hypersurface of a constant sectional curvature Lorentz manifold M ¯ . Based on previous work of S. Montiel, L. Alías, A. Brasil and G. Colares studied what can be said about the geometry of M when M ¯ is a conformally stationary spacetime, with timelike conformal vector field K. For example, if M n has constant higher order mean curvatures H r and H r + 1 , they concluded that M n is totally umbilical, provided H r + 1 ≠ 0 on it. If div( K ) does not vanish on M n they also proved that M n is totally umbilical, provided it has, a priori, just one constant higher order mean curvature. In this paper, we compute L r ( S r ) for such an immersion, and use the resulting formula to study both r-maximal spacelike hypersurfaces of M ¯ , as well as, in the presence of a constant higher order mean curvature, constraints on the sectional curvature of M that also suffice to guarantee the umbilicity of M. Here, by L r we mean the linearization of the second order differential operator associated to the r-th elementary symmetric function S r on the eigenvalues of the second fundamental form of x.