This is interesting and important progress towards a complete solution where all three components of Q (or A in electromagnetism) and the four components of the spin connection four-vector can be found from solving seven simultaneous equations. Then these interesting solutions would be examples of the general solution. The latter would be applicable to any situation in electrodynamics and gravitation, and a combination of both. Can Maxima be programmed to solve the seven equations simultaneously and in general? The three dimensional problem can be reduced to two dimensions for gravitation, thereby simplifying the computational problem considerably and producing precessing orbits in two dimensions in terms of the spin connection. This precession could be compared numerically with the very important results of UFT378, where forward and retrograde precessions were found from the ECE2 lagrangian. One could assume that the vector potential for electromagnetism is a plane wave, and use these equations to find the spin conenction for the plane wave. There is an effectively infinite number of things that could be done, so we can selct the most important for practical application. Finally it would be very interesting to graph these solutions by Horst Eckardt.
To: EMyrone@aol.com
Sent: 29/06/2017 15:11:42 GMT Daylight Time
Subj: Re: Discussion of 380(5)I extended the examples by a rotating Q vector in 2 and 3 components. With 3 oscillating components (section 2.1), eqs. o22-o24 are fulfilled with
beta = omega_0
bold k = bold omegaOne has to be cautious however because not all equations are compatible in all cases. In section 2.2 only 2 oscillating components are used. This leads to equations i37. The first two equations hold either if Q1=Q2=0 or if the cos and sin functions are equal. The latter is only the case for an argument of pi/4, therefore the solution set is restricted to a kind of line in the XY plane, parametrized by t.
Horst
Am 29.06.2017 um 15:19 schrieb EMyrone:
This is exactly what is needed, and congratulations! It is the required solution of seven equations in seven unknowns with sensible assumptions. This is the required general method that can be used for any problem and it will be very interesting to graph the solutions, and other types of solution. I think that this is another significant step forward in progress. It means that the spin connection is no longer an unknown, a key development.
To: EMyrone
Sent: 29/06/2017 11:32:54 GMT Daylight Time
Subj: Re: Discussion of 380(5)I programmed all these 7 equations. Then I made the assumptions
Q is a wave only in X direction
and
omega_0 is only a fuction of t.
(section 2.1). Then follows for exampleomega_z / k_z = omega_0 / beta,
omega_y / k_y = omega_0 / beta.In section 2.2, I assumed
bold omega = [0, 0, kappa] = const.
and
omega_0 is only a fuction of t.This reduces the equations significantly to the set i28/o28 (see protocol). The additional assumption of a rotating Q vector (i31) with a special Z dependence then leads to the equation set i32. Here we can set
omega_0 = beta
and the result for bold cap Omega can be rewritten with a trigonometric theorem to the form o35. This is a vector with phase difference of pi/2 between X and Y component, i.e. a rotating field as desired. Converted to electromagnetism, this should be a proof that it is possible to obtain a constant spin connection omega by a rotationg magnetic field. A very special vector potential is required, but this should not matter when the magnetic field is generated directly.
I will also check what comes out with 2 waves in X and Y direction.
Horst
Am 29.06.2017 um 10:02 schrieb EMyrone:
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
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.