Special relativity field tensors

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. 

When we rotate a contravariant nxl column vector (for position, velocity, momentum, electric field, etc.) we premultiply it by an n x n rotation tensor R. When, instead, we transform the coordinate system in which such vectors are defined, then the coordinate system and, for example, the V operator are covariant 1 x n row vectors, which are transformed by the tensor R 1 that is the reciprocal of R. A "dot product" or inner product a b must be the multiplication of a row vector a by a column vector b, to give a single number (scalar) as the result. This will be expanded further in the discussion of special relativity (Section 2.13) and of crystal symmetry (Section 7.10). 

General relativity is the theory that gave physical content to Riemaim s formulation of curved mathematical space and identifies the four-dimensional metric tensor with the gravitational field. The four dimensions of general relativity are the same as in the Minkowski space of special relativity. The velocity of light remains a constant in free space and the inability to specify simultaneous events remains in force. 

Energy-momentum conservation is expressed by dvT = 0 for a closed system. If Tfi were a symmetric tensor (when converted to 7 /x"), this would be assured because i f Tfi = 0 by construction. Since the gauge field part of the tensor deduced from Noether s theorem is not symmetric, this requires special consideration, as discussed below. A symmetric energy-momentum tensor is required for any eventual unification of quantum field theory and general relativity [422], The fermion field energy and momentum are... 

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