We apply Bochner’s technique combined with a mild Dirichlet integral condition to show that a complete constant mean curvature (CMC) spacelike hypersurface immersed with positive semi-definite modified second fundamental form in a Lorentzian product space \(\mathbb {R}_1\times M^n\) , whose Riemannian base \(M^n\) has nonnegative sectional curvature, must be a slice \(\{t\}\times M^n\) . In particular, we conclude that the spacelike hyperplanes of the Lorentz–Minkowski space \(\mathbb {L}_1^{n+1}\) are the only complete CMC spacelike hypersurfaces immersed in \(\mathbb {L}_1^{n+1}\) , whose modified second fundamental form is positive semi-definite and the height function with respect to a spacelike hyperplane satisfies such a Dirichlet integral condition. A version of the classical Calabi–Bernstein Theorem to the context of Lorentzian product spaces is also given.
