The Ricci tensor and its trace, the scalar curvature R = [R.sup.s.sub.s], are necessary to define the

Einstein tensor, [G.sub.ij] = [R.sub.ij] - (1/2)R[g.sub.ij].

where [G.sub.ij] is the

Einstein tensor. Suppose ([bar.M], [bar.g]) is an ARS-spacetime with an ARS vector field V and satisfies the hypothesis of Theorem 3.

This is because the conformal

Einstein tensor splits neatly into the original

Einstein tensor and a conformally related part.

Finally, the first-order correction to the

Einstein tensor, i.e.

Further we have defined a semi-Einstein space, given an example and proved that in a Ricci-recurrent Riemannian manifold energy momentum tensor T is generalized recurrent if and only if the

Einstein tensor G is generalized Ricci-recurrent and G will be generalized Ricci-recurrent if and only if the manifold is semi-Einstein.

Since the scalar curvature is zero, the

Einstein tensor components are given by the Ricci tensor components, as

Inspection shows that the matter-gravity tensor must be identified with the Rosenfeld-Belinfante symmetric tensor [3,4], thus complying with the intrinsic conservation property of the

Einstein tensor as it should be.

Here [G.sub.[mu][nu]] = [R.sub.[mu][nu]] - (1/2)R[g.sub.[mu][nu]] is the

Einstein tensor and [T.sup.m.sub.[mu][nu]] = ([[rho].sub.m] + [p.sub.m])[u.sub.[mu]][u.sub.[nu]] + [p.sub.m][g.sub.[mu][nu]] is the energy-momentum tensor of the ordinary matter.

In this respect, it is shown that the gravitational field of a massive body is no longer described by a pseudo-tensor, but appears as a true tensor in the field equations as it should be, in order to balance the conceptually conserved property of the

Einstein tensor.

This immediately suggests looking for second derivatives of the metric which is a second rank tensor as well as divergence free, paving the way to the

Einstein tensor. On the other hand, for the right hand side the natural choice being the matter energy momentum tensor [T.sub.ab], one ends up equating Einstein's tensor with the matter energy momentum tensor resulting in Einstein's equations (for a more detailed discussion, see [33]).

where [G.sub.[micro]v] is the

Einstein tensor and [L.sub.[micro]v] is a tensor we propose to call the "Lorentz" tensor.

In the general relativity (GR) framework, where [G.sub.[mu][nu]] = 8[pi][T.sub.[mu][nu]], the satisfaction of BI by the

Einstein tensor ([G.sup.[nu].sub.[mu]]) is equivalent to the satisfaction of OCL by [T.sup.[nu].sub.[mu]].