On 4-Dimensional Lorentzian Manifolds with Pseudo W2-Curvature Tensor in General Relativity

The present paper is devoted to the study of Lorentzian spacetimes endowed with the pseudo W2-curvature tensor. We investigate the geometric and symmetry properties of pseudo W2-flat spacetimes and establish several results concerning their curvature structure and associated energy–momentum tensors. It is shown that a pseudo W2-flat spacetime is a spacetime of constant curvature under suitable conditions on the defining parameters of the pseudo W2-curvature tensor. Further, for a spacetime satisfying Einstein’s field equation with vanishing pseudo W2-curvature tensor, it is proved that a vector field generates a matter collineation if and only if it is a Killing vector field. We also establish that the Lie inheritance property of the energy–momentum tensor is equivalent to the existence of a conformal Killing vector field. Moreover, it is shown that if the energy–momentum tensor is of Codazzi type, then the pseudo W2-curvature tensor is divergence-free. To illustrate the obtained results, two explicit Lorentzian spacetime models are examined in detail, namely an isotropic warped spacetime and an FLRW-type exponential (de Sitter) spacetime. For each model, the metric structure, curvature tensors, Ricci operator, scalar curvature, and pseudo W2-flatness condition are verified explicitly. These examples provide concrete realizations of the theoretical results and demonstrate the significance of the pseudo W2-curvature tensor in the geometric study of Lorentzian manifolds and general relativity.