Curved spacetime

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. 

There exist generally covariant four-valued 4-vectors that are components of q, and these can be used to construct the basic structure of 0(3) electrodynamics in terms of single-valued components of the quaternion-valued metric q1. Therefore, the Sachs theory can be reduced to 0(3) electrodynamics, which is a Yang-Mills theory [3,4]. The empirical evidence available for both the Sachs and 0(3) theories is summarized in this review, and discussed more extensively in the individual reviews by Sachs [1] and Evans [2]. In other words, empirical evidence is given of the instances where the Maxwell-Heaviside theory fails and where the Sachs and 0(3) electrodynamics succeed in describing empirical data from various sources. The fusion of the 0(3) and Sachs theories provides proof that the B(3) held [2] is a physical held of curved spacetime, which vanishes in hat spacetime (Maxwell-Heaviside theory [2]). 

The energy density in curved spacetime is given in the Sachs theory by the quaternion-valued expression... 

Therefore, we conclude that electromagnetic energy density exists in curved spacetime under all conditions, and devices can be constructed [8] to extract this energy density. 

The 0(3) electrodynamics developed by Evans [2], and its homomorph, the SU(2) electrodynamics of Barrett [10], are substructures of the Sachs theory dependent on a particular choice of metric. Both 0(3) and SU(2) electrodynamics are Yang-Mills structures with a Wu-Yang phase factor, as discussed by Evans and others [2,9]. Using the choice of metric (17), the electromagnetic energy density present in the 0(3) curved spacetime is given by the product... 

The curved spacetime 4-current is also generally covariant and has components such as... 

So, in general, in curved spacetime, there exist longitudinal and transverse components under all conditions. In 0(3) electrodynamics, the upper indices ((1),(2),(3)) are defined by the unit vectors... 

Indeed, classical U(l) electrodynamics is modeled as a field theory on a flat spacetime. In the more rigorous and general Sachs-Evans unified field approach, this is falsified. In that more fundamental model, EM waves and fields can propagate only through curved spacetime. 

M. Open System far from Equilibrium, Multiple Subprocesses, and Curved Spacetime... 

The system process is also a general relativistic process [33,36,37,41,45,63] whereby electromagnetic energy is utilized to curve local spacetime, and then the locally curved spacetime continuously acts back on the system and process by furnishing excess energy to the system and process directly from the curved spacetime the excess energy is continually input to the system from the imaginary plane (time domain) [1,16,20]. 

Producing driving and driven coils both in curved spacetime and with their magnetic flux inside a nanocrystalline core inside said coils and with a field-free magnetic vector potential in the space in which the coil is embedded... 

The present notion in EM theory that EM energy travels through a flat spacetime is in fact an oxymoron, since to have had an energy density change in spacetime at all is a priori to curve spacetime. 

To whet your appetite again energy is curved spacetime. 

In elementary differential geometry, therefore, the electromagnetic helix produces a nonzero T, and tangent vectors are characteristic of curved spacetime in general relativity. The scalar curvature in elementary differential geometry is... 

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 ... 

In Part 1 of this three-volume set, Sachs [117] has demonstrated that electromagnetic energy is available from curved spacetime by using the irreducible... 

In general, all the off-diagonal elements of the quaternion-valued commutator term [the fifth term in Sachs Eq. (4.19)] exist, and in this appendix, it is shown, by a choice of metric, that one of these components is the Ba> field discussed in the text. The B<3) field is the fundamental signature of 0(3) electrodynamics discussed in Vol. 114, part 2. In this appendix, we also give the most general form of the vector potential in curved spacetime, a form that also has longitudinal and transverse components under all conditions, including the vacuum. In the Maxwell-Heaviside theory, on the other hand, the vector... 

---