Riemann tensor

Equations (627) and (628) are special cases of the usual definition of the Riemann tensor in curvilinear geometry... 

Equation (643) is also a Bianchi identity in the theory of gravitation because G v is derived from the antisymmetric part of the Riemann tensor, whose symmetric part can be contracted to the Einstein tensor. 

Similarly, Eq. (643) can be developed into an inhomogeneous equation of the unified field. First, raise indices in the Riemann tensor and field tensor ... 

Consider the antisymmetric part of the Riemann tensor in Eqs. (647) by suitable contraction. In Eq. (647c), for example, the contraction is X = p. The result reduces to the 0(3) inhomogeneous field equation of electromagnetism in the form... 

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... 

If the superposition of point-source solutions is determined by a Green function , then by analogy with Wheeler-Feynman absorber theory the total interaction results from a symmetrical combination of retarded and advanced Green functions. On dehning these Green functions to be compatible with space-time geometry (Riemann tensor) the interaction is shown to be consistent with Einstein s field equations. 

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. 

One obtains the Riemann tensor fundamental to the theory of the tensor 9iy... 

The most important second order tensor is the metric tensor g, whose components in a Riemann space are defined by the relations... 

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. 

SA = 0 subject to the energy constraint restates the principle of least action. When the external potential function is constant, the definition of ds as a path element implies that the system trajectory is a geodesic in the Riemann space defined by the mass tensor m . This anticipates the profound geometrization of dynamics introduced by Einstein in the general theory of relativity. 

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 ... 

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

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