Ricci tensor

The Ricci tensor that represents the geometry of space is next equated with the so-called energy-momentum (stress) tensor of the matter field that defines the influence of matter and field energy... 

Therefore R, is an antisymmetric Ricci tensor obtained from the index contraction from the Riemann curvature tensor. Further contraction of R leads to the scalar curvature R, which, for electromagnetism, is k2. The contraction must be... 

The principle of equivalence between electromagnetism and the antisymmetric Ricci tensor is... 

Electromagnetism can therefore be defined geometrically in curvilinear coordinates, and has vacuum properties such as scalar curvature, metric coefficient, affine connection, and Ricci tensor that manifest themselves fully on the 0(3) level ... 

In both cases, the A5 v factor is a hypersurface. This suggests that the Ricci tensor is in general complex, and given by... 

Next, contracting RKppL with the contravariant metric tensor yields the correspondence with the Ricci tensor RKp ... 

Finally, the scalar curvature field R follows from the further contraction of the Ricci tensor (43) with the metric tensor, giving... 

This latter expression allows us to verify easily the symmetry of the Ricci tensor, and to notice that if A is constant then half of the terms of equation (5) vanish. 

The first term, known as the Ricci tensor, is obtained from the 4-index Riemann-Christoffel tensor on contraction with the mixed fundamental tensor ... 

The right-hand side of (6.4) is implied by the fact that the Ricci tensor has zero divergence for any metric, which requires that the stress-energy tensor should likewise have V T > = 0. 

As in affine theory one can also define a curvature tensor for the connection r. By contraction one obtains an analog of the Ricci tensor ... 

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