where [L.sub.W] = Lambert W(X), X = - 16[q.sup.2][l.sup.2][Y.sup.2]/[[beta].sup.2][r.sup.4], and [GAMMA] = [square root of 1 + X/4] and therefore the nonzero component of electromagnetic field tensor
Not enough, Rainich also saw that the electromagnetic field tensor
is performed from the congruences of two dual surfaces.
where the electromagnetic field tensor
can now be expressed by the extended form (given in Section 3)
(The Einstein summation convention employed throughout this work.) As in , for reasons that will be clear later, we define the electromagnetic field tensor
F via the torsion tensor of spacetime (the anti-symmetric part of the connection [GAMMA]) as follows:
to the electromagnetic field tensor
F = [F.sub.[micro]v][[omega].sup.[micro] [cross product] [[omega].sup.v]
Meanwhile, the electromagnetic field tensor
is given by
The electromagnetic field tensor
in our unified field theory is therefore given by
Let us first call the following expression for the covariant components of the electromagnetic field tensor
in terms of the covariant components of the canonical electromagnetic four-potential A:
An isotropic electromagnetic field is such where the field invariants [F.sub.[alpha][beta]][F.sup.[alpha][beta]] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], constructed from the electromagnetic field tensor
[F.sub.[alpha][beta]] and the field pseudo-tensor [F.sup.*[alpha][beta]] = 1/2 [[eta].sup.[alpha][beta][mu]v][F.sup.[mu]v] dual, are zero
where [F.sub.[alpha][beta]] = 1/2 ([partial derivative][A.sub.[beta]/[partial derivative][x.sup.a] - [partial derivative][A.sub.[alpha]]/[partial derivative][x.sub.beta]]) is Maxwell's electromagnetic field tensor
, while [A.sup.[alpha]] is the four-dimensional electromagnetic field potential given the observable chr.inv.-projections [phi] = [A.sub.0]/[square root of [g.sub.00] and [q.sup.i] = [A.sup.i] (the scalar and vector three-dimensional chr.inv-potentials).
is the determinant of the metric, [F.sub.b] =[[partial derivative].sub.a]Ab -[[partial derivative].sub.b]Aa is the electromagnetic field tensor
, and Rab is the Ricci tensor.
Emulating the way that Maxwell's electromagnetic field tensor
is introduced, we introduce the tensor of a time density field as the rotor of its four-dimensional vector potential