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
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 ).
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]].