Discussion of 385(5)

Thanks again. The antisymmetry conditions (8) to (10) of Note 385(1) originate in the magnetic part of the electromagnetic field tensor. Eqs. (11) to (13) come from Eq. (7), defining the absence of a magnetic flux density. The antisymmetry laws are written out in tensor notation in Eq. (22) of Note 379(5). The electric antisymmetry law is Eq. (23), and the vector potential is “electric” in the sense that it defines an electric field strength as in Eq. (22). If there is an electric field strength E present and no magnetic flux density B, then F sub 0i, i = i, 2, 3 are non zero, but F sub ij j, i , j = 1,2,3 are all zero. The former are the electric elements of the field tensor and the latter are the magnetic elements. The vector potential is called “electric” if there is no magnetic flux density present. It is called “magnetic”, if there is no electric field strength present. For Note 385(5) I had in mind simply working out Eqs. (17) to get the spin connections, and to check my hand derivation of Eq. (18) by computer algebra. I had in mind using firstly the constant omega sub 0 of Eq. (4), and secondly the omega sub 0 of Eq. (20). Eq. (18) of Note 385(5) comes from Eqs. (8) to (13) of Note 385(1). For the electric dipole field and an assumed B = 0 and partial A / partial t = 0, the vector potential is given the name “electric vector potential” because there is no magnetic flux density present. There is no incompatibilty between Eqs. (8) to (10) of Note 385(1), and Eqs. (11) to (13) of Note 385(1) because they use elements of the same overall four potential A sub mu, defined by

F sub mu nu = D sub mu A sub nu – D sub nu A sub mu

where F is the field tensor and D the covariant derivative. The antisymmetry laws originate in:

F sub mu nu = – F sub nu mu

which implies

D sub mu A sub nu = – D sub nu A sub mu

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