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To: EMyrone@aol.com, Sent: 11/12/2017 14:22:02 GMT Standard Time

Subj: Re: Eq. (42) of UFT298I will give some examples for non-vanishing diagonal elements of the Christoffel connection in the text book I am preparing. I will see that I can finish the first chapter over the holidays and then will send it to you as a draft.

Horst

Am 10.12.2017 um 13:14 schrieb EMyrone:

This is where the trace antisymmetry constraint first appears, and uses the fact that the trace of an antisymmetric tensor or matrix is zero. The excellent Eckardt Lindstrom papers UFT292 – UFT299 are by now all heavily studied classics. This can be seen in the daily report I send out every morning. In the case of ECE2 field tensors each element of the trace is zero as is well known. The fundamental property of a square matrix is that it is made up of a sum of symmetric and antisymmetric square matrices. This is described by Stephenson, “Mathematical Methods for Science Students”, pp 292 ff. (fifth impression 1968). Every one of the antisymmetric matrices used by Stephenson has diagonal elements which are all zero. This was my undergraduate text book and it is signed by me with the address: “Brig y Don”, Sea View Place, Aberystwyth. I would like to ask Horst where the antisymmetric connection with non zero elements but zero trace first makes an appearance. I guess that it is the classic UFT354, but I cannot find it there. This is an interesting idea which I would like to study further. If the connection is defined by the commutator as starting point, instead of metric compatibility, the connection is antisymmetric with all diagonal elements zero, so is the curvature, so is the commutator of covariant derivatives.

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[D sub mu, D sub mu] = – [D sub mu , D sub mu] = 0

It is fine to use the original Lindstrom constraint.

To: EMyrone@aol.com, dwlindstrom@gmail.com

Sent: 11/12/2017 14:17:38 GMT Standard Time

Subj: Re: Hayley Hamilton TheoremsThe fact that the Christoffel symbol is not a tensor should only play a role as far as tensor operations are involved. In the definiton of torsion, diagonal elements are allowed:

If Gamma ^rho _ {mu mu} not equal zero,

then Gamma ^rho _ {mu mu} – Gamma ^rho _ {mu mu} = 0,

that means T^rho_ {mu mu} = 0 as required by Cartan geometry.

The commutator is built from T ans R which are tensors.

One argument for staying at the original trace antisymmetry equation is that we otherwise would obtain too many equations. The spatial spin connections are determined completely by the antisymmetry equations of the field tensor.Horst

Am 11.12.2017 um 14:54 schrieb EMyrone:

Many thanks, it looks as if this is an exposition on the trace of a rank three object. The Christoffel connection is not a tensor as you know, because it does not transform as tensor. I think that it is safe to assume that the trace of the mixed index connection gamma sup a sub mu nu is zero, so I will revert to your original trace antisymmetry equation. .

Sent: 10/12/2017 17:01:49 GMT Standard Time

Subj: Re: Eq. (42) of UFT298Myron, I concur. This is the first instance of the trace being mentioned that I am aware of. The invariant nature of the Trace function I think stems from the Hayley-Hamilton theorems which looks at the eigenvalues for the characteristic equation associated with an equation set. There may be some ambiguity in applying trace invariance to a form structure such as Gamma sum a sub mu sub nu when summed along the a – nu axis. There should be no difficulty however in the tensorial representation, Gamma sup rho sub mu sub nu. In this case we should get twelve trace equations, as shown in the attached. Note that not all of the equations are independent.

Doug

=

On Dec 10, 2017, at 5:14 AM, EMyrone wrote:

This is where the trace antisymmetry constraint first appears, and uses the fact that the trace of an antisymmetric tensor or matrix is zero. The excellent Eckardt Lindstrom papers UFT292 – UFT299 are by now all heavily studied classics. This can be seen in the daily report I send out every morning. In the case of ECE2 field tensors each element of the trace is zero as is well known. The fundamental property of a square matrix is that it is made up of a sum of symmetric and antisymmetric square matrices. This is described by Stephenson, “Mathematical Methods for Science Students”, pp 292 ff. (fifth impression 1968). Every one of the antisymmetric matrices used by Stephenson has diagonal elements which are all zero. This was my undergraduate text book and it is signed by me with the address: “Brig y Don”, Sea View Place, Aberystwyth. I would like to ask Horst where the antisymmetric connection with non zero elements but zero trace first makes an appearance. I guess that it is the classic UFT354, but I cannot find it there. This is an interesting idea which I would like to study further. If the connection is defined by the commutator as starting point, instead of metric compatibility, the connection is antisymmetric with all diagonal elements zero, so is the curvature, so is the commutator of covariant derivatives.

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Sent: 10/12/2017 19:07:34 GMT Standard Time

Subj: more from the tetrad postulateMyron,Horst:

Here are some thoughts on metric compatibility, the tetrad postulate, and trace invariance that I’ve been working on. I will also append a section on total antisymmetry of the gamma connection and how that reduces equation complexity.Doug=

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Sent: 10/12/2017 17:01:49 GMT Standard Time

Subj: Re: Eq. (42) of UFT298Myron, I concur. This is the first instance of the trace being mentioned that I am aware of. The invariant nature of the Trace function I think stems from the Hayley-Hamilton theorems which looks at the eigenvalues for the characteristic equation associated with an equation set. There may be some ambiguity in applying trace invariance to a form structure such as Gamma sum a sub mu sub nu when summed along the a – nu axis. There should be no difficulty however in the tensorial representation, Gamma sup rho sub mu sub nu. In this case we should get twelve trace equations, as shown in the attached. Note that not all of the equations are independent.

Doug

=

On Dec 10, 2017, at 5:14 AM, EMyrone wrote:

This is where the trace antisymmetry constraint first appears, and uses the fact that the trace of an antisymmetric tensor or matrix is zero. The excellent Eckardt Lindstrom papers UFT292 – UFT299 are by now all heavily studied classics. This can be seen in the daily report I send out every morning. In the case of ECE2 field tensors each element of the trace is zero as is well known. The fundamental property of a square matrix is that it is made up of a sum of symmetric and antisymmetric square matrices. This is described by Stephenson, “Mathematical Methods for Science Students”, pp 292 ff. (fifth impression 1968). Every one of the antisymmetric matrices used by Stephenson has diagonal elements which are all zero. This was my undergraduate text book and it is signed by me with the address: “Brig y Don”, Sea View Place, Aberystwyth. I would like to ask Horst where the antisymmetric connection with non zero elements but zero trace first makes an appearance. I guess that it is the classic UFT354, but I cannot find it there. This is an interesting idea which I would like to study further. If the connection is defined by the commutator as starting point, instead of metric compatibility, the connection is antisymmetric with all diagonal elements zero, so is the curvature, so is the commutator of covariant derivatives.

]]>

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Doug

On Dec 10, 2017, at 5:14 AM, EMyrone wrote:

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<B> sub ij = (curl A sub 0 – <omega> x A sub 0) sub ij

is antisymmetric. In MZ theory, B, omega and A sub 0 are all known, so Eq. (38) is always true. I will give more details in the next note before proceeding to the MZ theory of the magnetic potential and field.

To: EMyrone@aol.com

Sent: 07/12/2017 14:37:12 GMT Standard Time

Subj: Re: 394(2): Application of AntisymmetryIt seems that a factor of cos(theta) has been lost in eq. (30). resolving for (delta r)^2 gives the two last equations of the protocol for omega_r and omega_theta.

Why is there a particular consideration of antisymmetry for <B> in eqs. (39-42)? I thought that the antisymmetry laws apply for (38) as usual.Horst

Am 28.11.2017 um 15:00 schrieb EMyrone:

This note applies conservation of scalar and vector antisymmetry to the shivering electric and magnetic dipole fields. In the given approximations the mean eletric dipole spin connections are given by Eqs. (16) and (17) and the mean magnetic dipole spin connections by Eqs. (34) and (35). They are both directly proportional to the mean square fluctuations in the vacuum. So there is a close similarity with Lamb shift theory.

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