Many thanks for going through these notes, and also to Doug.
1) Agreed, Eq. (5) is the covariant unit vector e sub mu in the Cartesian basis, an Eq. (6) in the complex circular basis.
2) Agreed, the relation of W to A is developed in note 316(6). As defined they both have the same units of tesla metres. Eqs. (22) and (24) are some of the first results of the curvature based ECE theory. Eq. (22) looks like MH but this is due to cancellation by removal of the tangent indices. The c indices in the last two terms of Eq. (23) and (24) run from (0) to (3), but they can be removed leading to a Maxwell Heaviside result for the electric field also. The c indices are removed by multiplying by e sub c e sup c = – 2. So we are all three agreed. The result is that the curvature based ECE theory reduces to a generally covariant MH theory in a space with torsion and curvature both non zero. This seems to be an important result.
3) In deriving Eq. (26) I used Eq. (24) with
A sup (1) sub 1 = – A sup (1) sub X,
epsilon sup 1 = 1 = epsilon sub X
A sup (1) sub 2 = – A sup (1) sub Y,
epsilon sup 2 = 1 = epsilon sub Y
The minus signs in Eqs. (24) and (25) are definitions. Yes I used Eq. (26) for the unit vector instead of for A sup (1).
In a message dated 30/05/2015 15:07:08 GMT Daylight Time, writes:
I went through the first three notes.
Thanks, Doug, for doing so too, I have not read the email comments to
these yet.note (1): Eqs. 5 and 6 obviously are compact notations, a diagonal
matrix of the matrix of unit vectors. The curvature 2-form reminds to
the Ricci tensor but is used completely differently here and has a
different symmetry.note (2): The magnetic flux potential seems to be an alternative
potential definition. How is it related to the vector potential A? Eq.
22 suggests that it is the potential leading directly to the
Maxwell-Heaviside form.
Do the c indices in eqs. 23/24 run from 1 to 3? Then the two last terms
should cancel out due to the covariant/contravariant sign change. Then
these equations take the Maxwell form too. This would mean that W and
omega_0 potential are the direct counterparts of the Mawell-Heaviside
form and include curvature/torsion directly. Reduction to Maxwell as
derived earlier by Doug.note (3): According to eqs. 24,25, sign changes are to be introduced
into eqs. (21,22) to obtain (26,27). I understand thatepsilon sup 1 = epsilon sub X
(for greek index 1). Then there is only one sign change in 26/27 but
there is none written.Another point: You have used a term A sub (1) in eq. 40. This is nowhere
defined, only a vector A sup (1) in (35). Did you use the covariant
version of (24) to define A sub (1) ?Horst