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.
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.
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
and that if [OMEGA] = 0 then the Lorentz tensor is simply the negative of the Einstein tensor
From this covariant metric tensor, we can then construct our field equations for the gravitational field after formulating the Coefficients of affine connection, Riemann Christoffel tensor, Ricci tensor and the Einstein tensor
will therefore represent the generalized Einstein tensor
, such that we may have a corresponding geometric object given by
Both models start by calculating the Einstein tensor
G] is the symmetric Einstein tensor
, T is the energy-momentum tensor, and k = [+ or -] 8[pi]G/[c.
In the last of the above set of equations, we have introduced the generalized Einstein tensor
abc] of the Riemann-Christoffel curvature tensor (the Ricci tensor), one defines the regular Einstein tensor
The field has distributed stresses which are expressed by an addition to the electromagnetic field stress-tensor (see the Einstein tensor