It is shown that a general relativistic space-time with covariant-constant energymomentum tensor is Ricci symmetric. Two particular types of such general relativistic space-times are considered and the nature of each is determined.
I N T R O D U C T I O NGeneral relativity flows from the Einstein equation which implies that the energy-momentum tensor is of vanishing divergence. This requirement of the energy-momentum tensor is satisfied if this tensor is covariant-constant. It is therefore meaningful to ask whether the energy-momentum tensor of a given general relativistic space-time is covariant-constant. In this paper we first show that a general relativistic space-time with covariant-constant energy-momentum tensor is Ricci symmetric, i.e., it has covariant-constant Ricci tensor. Next we consider a special type of space-time which is called pseudo Ricci symmetric.A Riemannian manifold (M", g) is called pseudo Ricci symmetric if its Ricci tensor S of type (0, 2) satisfies the condition (Chaki, 1988) where A is a 1-form,for all vector fields X, and V denotes the operator of covaxiant differentiation with respect to the metric tensor g.
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