Characterizations of Pseudo-Symmetric Space–Times in Gray’s Subspaces and f(R)-Gravity Vacuum Solutions

This paper investigates pseudo-symmetric space–times within two interrelated frameworks: vacuum f(R)-gravity and Gray’s seven canonical decomposition subspaces. First, it is established that any conformally flat pseudo-symmetric space–time satisfying the vacuum field equations of f(R)-gravity necessarily corresponds to a perfect fluid. Subsequently, a detailed analysis of Gray’s subspaces reveals the following structural results: In the trivial and 𝒜 subspaces, pseudo-symmetric space–times are Ricci-simple and Weyl-harmonic, and thus are necessarily generalized Robertson–Walker space–times. In the B and 𝒜⊕B subspaces, the associated time-like vector field ξl is shown to be an eigenvector of the Ricci tensor with the eigenvalue −R/2. Furthermore, for a perfect fluid pseudo-symmetric space–time obeying f(R)-gravity and belonging to the trivial, 𝒜, B, or 𝒜⊕B subspaces, the isotropic pressure p and energy density σ are proven to be constants. Additionally, it is demonstrated that Gray’s I subspace reduces to the B subspace in the pseudo-symmetric setting. Finally, under specific geometric conditions, pseudo-symmetric space–times in the I⊕𝒜 and I⊕B subspaces are also shown to admit perfect fluid representations. These results collectively clarify the geometric and physical constraints imposed by pseudo-symmetry within f(R)-gravity and Gray’s classification scheme.