Curvature tensor

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

To consider magnetic flux density components of IAIV, Q must have the units of weber and R, the scalar curvature, must have units of inverse square meters. In the flat spacetime limit, R 0, so it is clear that the non-Abelian part of the field tensor, Eq. (6), vanishes in special relativity. The complete field tensor F vanishes [1] in flat spacetime because the curvature tensor vanishes. These considerations refute the Maxwell-Heaviside theory, which is developed in flat spacetime, and show that 0(3) electrodynamics is a theory of conformally curved spacetime. Most generally, the Sachs theory is a closed field theory that, in principle, unifies all four fields gravitational, electromagnetic, weak, and strong. 

The quaternion-valued vector potential and the 4-current J both depend directly on the curvature tensor. The electromagnetic field tensor in the Sachs theory has the form... 

Similarly, the 4-current J depends directly on the curvature tensor / [1], and there can exist no 4-current in the Heaviside-Maxwell theory, so the... 

The curvature tensor is defined in terms of covariant derivatives of the spin-affine connections fip, and according to Section ( ), has its equivalent in 0(3) electrodynamics. 

The curvature tensor can be written as a commutator of covariant derivatives... 

This result is consistent with the fact that the curvature tensor k0> must be minimized, which is a consistent result. The curvature is... 

Warning physicists use the word tensor to describe objects that arise in the theory of general relativity (such as the metric tensor or the curvature tensor), among other places. Although these objects are indeed tensors in the sense we will define below, they are also more complicated they involve multiple coordinate systems. We warn the reader that this section will not address the issues raised by multiple coordinate systems. Thus a reader who has been confused by such physicists tensors may not be fiilly satisfied by our discussion here." ... 

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... 

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

The local geometry of a surface is generally characterized by two surface tensors, the metric tensor and curvature tensor. Letting (C, ) = (C C )i the tangent vectors to each of the coordinate curves at a point P can be represented as the basis vectors Uq, = a = 1,2) where the partial derivatives are to be evaluated where the coordinate curves on the curves intersect. It follows that an element of arc length with respect to the surface coordinates is represented by ... 

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 ... 

This result is a consequence of the dependence of the definition of Fp% of the spin curvature tensor, Kpk, according to Eq. (49) (as well as qpk = 0). This is because the spin curvature tensor Kpk is the four-dimensional curl of a 4-vector in configuration space ... 

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