On mixed quasi-Einstein spacetimes

Abstract

The object of the present paper is to study mixed quasi-Einstein spacetimes, briefly M(QE)4 spacetimes. First we prove that every Z Ricci pseudosymmetric M(QE)4 spacetimes is a Z Ricci semisymmetric spacetime. Then we study Z flat spacetimes. Also we consider Ricci symmetric M(QE)4 spacetimes and among others we prove that the local cosmological structure of a Ricci symmetricM(QE)4 perfect fluid spacetime can be identified as Petrov type I, DorO. We show that such a spacetime is the Robertson-Walker spacetime. Moreover we deal with mixed quasi-Einstein spacetimes with the associated generators U and V as concurrent vector fields. As a consequence we obtain some important theorems. Among others it is shown that a perfect fluid M(QE)4 spacetime of non zero scalar curvature with the basic vector field of spacetime as velocity vector field of the fluid is of Segr?e characteristic [(1,1,1),1]. Also we prove that a M(QE)4 spacetime can not admit heat flux provided the smooth function b is not equal to the cosmological constant k. This means that such a spacetime describe a universe which has already attained thermal equilibrium. Finally, we construct a non-trivial Lorentzian metric of M(QE)4.