## A study of generalized quasi Einstein spacetimes with applications in general relativity.(English)Zbl 1337.83010

Summary: The aim of this paper is to investigate some geometric and physical properties of the generalized quasi Einstein spacetime $$G(QE)_4$$ under certain conditions. Firstly, we prove the existence of $$G(QE)_4$$ by constructing a non trivial example. Then it is proved that the $$G(QE)_4$$ spacetime with the conditions $$\mathcal B\cdot S=L_SQ(g,S)$$, where $$\mathcal B$$ denotes the Ricci tensor or the concircular curvature tensor is an $$N(\frac{a-b}{3})$$-quasi Einstein spacetime and in a $$G(QE)_4$$ spacetime with $$C\cdot S=0$$, where $$C$$ is the conformal curvature tensor, $$a-b$$ is an eigenvalue of the Ricci operator. Then, we deal with the Ricci recurrent $$G(QE)_4$$ spacetime and prove that in this spacetime, the acceleration vector and the vorticity tensor vanish; but this spacetime has the non-vanishing expansion scalar and the shear tensor. Moreover, it is shown that every Ricci recurrent $$G(QE)_4$$ is Weyl compatible, purely electric spacetime and its possible Petrov types are I or D.

### MSC:

 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory 53Z05 Applications of differential geometry to physics 83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
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