This is very good progress by Maxima. Off the cuff, I should think that a combination of equations such as Eq. (65) would be needed for a rotating magnetic field. Can Maxima solve the general problem:
del B = 0
curl E + partial B / partial t = 0
B = curl A – omega x A
E = – partial A / partial t – omega sub 0 A
plus the antisymmetry equations (68) to (70)? Your problem would then be a special case of the general solution, and you could design any magnetic field of relevance. It may also be possible for Maxima to give the general solution of Eq. (6), which comes from the Faraday law of induction for gravitation. The Gauss law del omega = 0 leads to Eq. (5). So Eqs. (5) and (6) are the homogeneous laws of gravitation expressed in terms of potentials and spin connections. Eq. (6) for electromagnetism is found by replacing Q by A. You could try assuming a plane wave solution for A, and find the spin connection from the antisymmetry laws. I will have a look at this by hand.
To: EMyrone@aol.com
Sent: 29/06/2017 07:43:25 GMT Daylight Time
Subj: Re: 380(5): Example Solution, the Spin Connection as the Wave Four Vector.The assumed solution of eqs.(21-23) , eq.(27), is even the general solution. Maxima says:
The diff. eq.
has the general solution:
.
An interesting result is than a constant omega (eq.76) can be constituted from oscillating A and B fields (71) and (79/80). A technically more interesting case would be to have a rotating B field in the XY plane:
B_X = B_0 cos(omega t)
B_Y = B_0 sin(omega t)
B_Z = 0Is it possible to construct this in a similar way? If a Z dependence of type
cos (ometa t – kappa Z)
remains, it would be important to determine kappa in a way that the variation in Z is small. This requirement collides with eq.(39) where we need a large kappa to obtain a significang spin connection.
Horst
Am 28.06.2017 um 15:45 schrieb EMyrone:
This note gives an example solution with the spin connection as the wave four vector, both for electromagnetism and gravitation. The magnetic flux density is a travelling wave in the j axis, and the electric field strength is a travelling wave in the i axis. These hand calculations should be checked with computer algebra as usual. The computer can also be used to give the most general possible solution, and that would be very useful for an infinite number of problems in the physical sciences and engineering.
