Riemann curvature tensor

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... 

If 0(3) electromagnetism [denoted e.m. in Eq. (640)] and gravitation are both to be seen as phenomena of curved spacetime, then both fields are derived ultimately from the same Riemann curvature tensor as follows ... 

The 0(3) field equations can be obtained from the fundamental definition of the Riemann curvature tensor, Eq. (631), by defining the 0(3) field tensor using covariant derivatives of the Poincare group. 

There are well known similarities between the Riemann curvature tensor of general relativity and the field tensor in non-Abelian electrodynamics. The Riemann tensor is... 

The mathematical detail of TGR depends on complicated tensor analysis which will not be considered here. The important result for purposes of the present discussion is the relationship, which is found to exist between two fundamental tensors1 The symmetric Riemann curvature tensor Rjlv (with... 

This defines the Riemann curvature tensor, RKppx, wherein qp could be any covariant 4-vector. 

Substituting (38) and (39) into (36), the relation between the spin curvature tensor and the Riemann curvature tensor follows ... 

Relativity theory has equally dramatic implications on the nature of the vacuum, which is shown not to be a void, but a medium that supports wave motion and carries electromagnetic fields. A new perspective on the nature of the vacuum is provided by the principle of equivalence. Space-time curvature can be described mathematically by a Riemann tensor, which the principle implies, should balance the gravitational field, which is sourced in the distribution of matter. This reciprocity indicates that Euclidean space-time is free of matter, which only emerges when curvature sets in. This is interpreted to mean that the homogeneous wave field of Euclidean vacuum generates matter when curved. Like a flat sheet that develops wrinkles when wrapped arormd a curved surface, the wave field generates non-dispersive persistent wave packets in the curved vacuum. 

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