## PS

OK thanks, we did some similar work in UFT150 to UFT155. You may still have the code for that work. This is all very interesting.

Date: Mon, Mar 26, 2018 at 1:02 PM
Subject: Re: PS
To: Myron Evans <myronevans123>

I just realized this. The integrand then diverges at the boundaries, I am investigating the effects. The results are completely different now.

Horst

Am 26.03.2018 um 14:01 schrieb Myron Evans:

The factor 1 / a must also be negative in Eqs. (60) and (61), and there must be a factor 4 / 3 in front of the first term on the LHS of Eq. (61)

## PS

The factor 1 / a must also be negative in Eqs. (60) and (61), and there must be a factor 4 / 3 in front of the first term on the LHS of Eq. (61)

## Section 3 of UFT 403

Many thanks, this is an excellent result and it would be very helpful to add it to the Section 3, making clear that 1 / a should be negative. I will also fix this erratum in Sections 1 and 2. It is clear from the apsidal method that the change in the ellipse is a precession, because the apsidal angle is no longer pi. So it has been proven that the ECE orce equation produces orbital precession, Q. E. D.

To: Myron Evans <myronevans123>

I tried to integrate eq.(53). For omega_r=0 this should give the result phi=pi when integrated from u_min to u_max. There seems to be a problem with the Constant C1. From (55) it is seen that the term must be negative since 2mH is negative. However in (60) you used +1/a for this term. Numerical tests show that the result phi=pi comes indeed out if the constant term is set to -1/a. Otherwise it is -0.43 or so.
The effect of the term 2 omega_r log(u) is seen by plotting the integrand for three omega_r values (see Figure). The additional term shifts the minimum of the u range to right, i.e. the ellipse becomes smaller in a non-linear way. Therefore the integration bounds would have to be modified.
I could add this diagram to section 3 of the paper.

Horst

Am 26.03.2018 um 09:45 schrieb Myron Evans:

FOR POSTING : Section 3 of UFT403

Many thanks. Incisive numerical results as usual. I would say that reversing the sign of the spin connection would reverse the sign of the integral and the precession. This can seen from Eq. (41) if the apsidal method is used. so both forward and retrograde precessions can be described, whereas EGR can describe only forward precessions. The sign in eq. (53) is correct because of the change of variable r to 1 / u. It would be very intersting to try to integrate Eq. (53) numerically, although I realize that it is a difficult procedure. That would give the orbit itself. Wen my computer returns from the workshop today I will repost sections 1 and 2 with errors corrected.

To: Myron Evans <myronevans123>, Dave Burleigh <burleigh.personal>

This is section 3 of the paper. The Integral in eq.(60) comes out negative for the planet Mercury. Can we be sure that the sign of the integral is correct? The sign changes when going from the r to the u coordinate, but I think this was rightly changed in the notes.

Horst

## Section 3 of UFT 403

FOR POSTING : Section 3 of UFT403

Many thanks. Incisive numerical results as usual. I would say that reversing the sign of the spin connection would reverse the sign of the integral and the precession. This can seen from Eq. (41) if the apsidal method is used. so both forward and retrograde precessions can be described, whereas EGR can describe only forward precessions. The sign in eq. (53) is correct because of the change of variable r to 1 / u. It would be very intersting to try to integrate Eq. (53) numerically, although I realize that it is a difficult procedure. That would give the orbit itself. Wen my computer returns from the workshop today I will repost sections 1 and 2 with errors corrected.

To: Myron Evans <myronevans123>, Dave Burleigh <burleigh.personal>

This is section 3 of the paper. The Integral in eq.(60) comes out negative for the planet Mercury. Can we be sure that the sign of the integral is correct? The sign changes when going from the r to the u coordinate, but I think this was rightly changed in the notes.

Horst

paper403-3.pdf

## Note 403(10)

This looks very interesting. With a theoretical understanding as in UFT396 the gyroscope could be used for lifting in heavy engineering.

Myron Evans <myronevans123>

Thanks, we are in contact with Sandy Kidd who built the first lifting gyroscopic device.

Horst

Am 24.03.2018 um 11:56 schrieb Myron Evans:

Errata in UFT403

Many thanks for this computer check. This means that there the following errata in UFT403

1) Factor of 4 / 3 in front of the first term of the right hand side of Eq. (42).
2) Replace 3 b 4 in Eq. (47).
3) Factor of 4 / 3 in front of the first term on the LHS of Eq. (61).

I will rescan and repost sections 1 and 2 when my main computer returns from maintenance work on its hard disk.

Congratulations on the high interest in the gyroscope paper UFT396! There is much more to the gyroscope than meets the eye.

From: Horst Eckardt <mail>
Date: Sat, Mar 24, 2018 at 9:47 AM
Subject: Note 403(10)
To: Myron Evans <myronevans123>

Obviously there is a typo in eq.(4), a factor of 4/3 is missing in front of the first term. This seems to have some consequences including the final paper, but only on a typo correction level.
I am working out section 3 now.

horst

## Note 403(10)

Errata in UFT403

Many thanks for this computer check. This means that there the following errata in UFT403

1) Factor of 4 / 3 in front of the first term of the right hand side of Eq. (42).
2) Replace 3 b 4 in Eq. (47).
3) Factor of 4 / 3 in front of the first term on the LHS of Eq. (61).

I will rescan and repost sections 1 and 2 when my main computer returns from maintenance work on its hard disk.

Congratulations on the high interest in the gyroscope paper UFT396! There is much more to the gyroscope than meets the eye.

From: Horst Eckardt <mail>
Date: Sat, Mar 24, 2018 at 9:47 AM
Subject: Note 403(10)
To: Myron Evans <myronevans123>

Obviously there is a typo in eq.(4), a factor of 4/3 is missing in front of the first term. This seems to have some consequences including the final paper, but only on a typo correction level.
I am working out section 3 now.

horst

403(10).pdf

## 403(9): Analytical Orbit in the Near Circular Approximation

OK many thanks, this is useful and I will look in to different solutions. This should be straightforward, and the apsidal method is very useful.

Date: Sun, Mar 18, 2018 at 5:55 PM
Subject: Re: Fwd: 403(9): Analytical Orbit in the Near Circular Approximation
To: Myron Evans <myronevans123>

I rechecked the note 403(6). All is o.k. up to eq.(41), first line. Then you replaced 1/r^2 by omega*r. With omega in the denominator, the limit omega –> 0 gives an expression 0/0 which is undefined. Can this be the reason for the wrong limit of delta_phi?
Furthermore, in (44) you assume <drdr> = const. According to the definition of omega in (40), then omega is constant and its derivative vanishes, in contrast to (44). This may be a second problem.

Horst

Am 18.03.2018 um 15:42 schrieb Myron Evans:

I will have another look at this problem of asymptotic behaviour. There must be some minor error or misconception in going from Eq.. (380 to (47) of Note 403(6) because we have already agreed on the basic apsidal method, which you checked to be correct. I will check this tomorrow. In the meantime the numerical solution of Eq. (1) could perhaps be compared with the circular approximation (11) to see if the latter is meaningful,but the numerical integration of Eq. (2) is already sufficient. I think that the Maxima result might be the more reliable. The circular approximation is useful because it gives an analytical orbit. The apsidal method is useful to show the existence of precession for any force law. The apsidal method and your numerical integration are sufficient to prove precession.

Date: Sun, Mar 18, 2018 at 11:11 AM
Subject: Re: 403(9): Analytical Orbit in the Near Circular Approximation
To: Myron Evans <myronevans123>

If we assume that the approximation in eq.(10) is justified, the bigger problem seems to be to equate it to the RHS because this term has not the right asymptotic behaviour of delta_phi for epsilon –> 0 as commented for note 8.
The integral in (11) gives a somewhat different result in Maxima, see %o3 of the protocol. Maybe both results are identical when transformed.

Horst

Am 17.03.2018 um 14:06 schrieb Myron Evans:

This is given by Eq. (13) from the Wolfram online integrator. This result can be checked with Maxima, which is probably more accurate than Wolfram. At the perihelion, the constant of integration A is adjusted to give exact agreement with the apsidal method, also used in the near circular approximation. This known to give precession at the perihelion. The analytical orbit is given by Eq. (15). The elliptical orbit is given by Eq. (17), which can be compared graphically with Eq. (15). The overall result is that the origin of any precession is vacuum fluctuation, a major advance in cosmology of all kinds.

## 403(9): Analytical Orbit in the Near Circular Approximation

I will have another look at this problem of asymptotic behaviour. There must be some minor error or misconception in going from Eq.. (380 to (47) of Note 403(6) because we have already agreed on the basic apsidal method, which you checked to be correct. I will check this tomorrow. In the meantime the numerical solution of Eq. (1) could perhaps be compared with the circular approximation (11) to see if the latter is meaningful,but the numerical integration of Eq. (2) is already sufficient. I think that the Maxima result might be the more reliable. The circular approximation is useful because it gives an analytical orbit. The apsidal method is useful to show the existence of precession for any force law. The apsidal method and your numerical integration are sufficient to prove precession.

Date: Sun, Mar 18, 2018 at 11:11 AM
Subject: Re: 403(9): Analytical Orbit in the Near Circular Approximation
To: Myron Evans <myronevans123>

If we assume that the approximation in eq.(10) is justified, the bigger problem seems to be to equate it to the RHS because this term has not the right asymptotic behaviour of delta_phi for epsilon –> 0 as commented for note 8.
The integral in (11) gives a somewhat different result in Maxima, see %o3 of the protocol. Maybe both results are identical when transformed.

Horst

Am 17.03.2018 um 14:06 schrieb Myron Evans:

This is given by Eq. (13) from the Wolfram online integrator. This result can be checked with Maxima, which is probably more accurate than Wolfram. At the perihelion, the constant of integration A is adjusted to give exact agreement with the apsidal method, also used in the near circular approximation. This known to give precession at the perihelion. The analytical orbit is given by Eq. (15). The elliptical orbit is given by Eq. (17), which can be compared graphically with Eq. (15). The overall result is that the origin of any precession is vacuum fluctuation, a major advance in cosmology of all kinds.

403(9).pdf

## 403(8): Final Version of Note 403(7)

Many thanks,numerical integration of Eq.(25)succeeds in solving the problem, so congratulations! Eq. (27) comes from the apsidal method of note 403(6), Eq. (47), which you checked to be correct some days ago. I used the value of r at the perihelion. It may be better to compare the result of the direct integration with Eq. (42) of Note 403(6). In any case i will go through the note again.

Date: Sun, Mar 18, 2018 at 10:59 AM
Subject: Re: 403(8): Final Version of Note 403(7)
To: Myron Evans <myronevans123>

Numerical integration of (24) for a model system with alpha=1, epsilon=0.3 gives

Delta phi = 0.0119 * <drdr>.

From eq. (27), which is the near-curcular approximation, I obtain

Delta phi = 0.6533 * <drdr>

which is quite different. For epsilon=0 the integral gives Delta phi=0 as expected while (27) does not. I do not see that (27) is a meaningful approximation.
BTW, (26) is not the minimal but maximal u. In (27) there should be a minus sign in front of epsilon.
It is possible to construct a graph delta_phi/<drdr> in dependence of epsilon which could be interesting.

Horst

Am 16.03.2018 um 14:33 schrieb Myron Evans:

This leads to the integral (25), which has no analytical solution, but at the perihelion for nearly circular orbits leads to the result (27). It is seen that teh precesion at the perihelion is due to vacuum fluctuations. The result (25) is a small correction to an ellipse, and this correction is a precession.

403(8).pdf

## note 403(6)

Very interesting integrands.

Date: Fri, Mar 16, 2018 at 9:49 AM
Subject: Re: Fwd: note 403(6)
To: Myron Evans <myronevans123>

The integral of delta phi with exponent 3/2 is not analytically integrable according to Maxima. I have plottet the integrands I(u) and I(phi), see attachments. Both have a pole, we will see if we have to integrate up to this value for realistic orbital parameters.

Horst

Am 16.03.2018 um 09:41 schrieb Myron Evans:

Many thanks again, I plan to use this note as the main part of UFT403, Sections 1 and 2. Your check is very interesting, and I will include it in the final paper.. I will rework Note 403(7) it is probably power 3/2 in the denominator, a minor error of hand calculation. Use of the computer to mend these errors is obviously of key importance. Maxima may have an analytical or numerical solution for the correction function delta phi, giving the complete orbit phi + delta phi as well as the precession of the perihelion for orbits of small eccentricity from Note 403(6). Einstein’s famous or infamous result for the precession of the perihelion did not attempt to give the orbit. The major advance in knowledge is that precession is due to vacuum fluctuations.

Subject: note 403(6)
To: Myron Evans <myronevans123>

The Schwarzschild radius is 3*10^3 m, not 3*10^5 m, a small typo.
In eq. (34) there is a sign typo but it has been corrected in (35).

A check for the correctness of the result (37) is as follows:
assume
omega = a/r
as in previous papers. Then eq.(37) gives

psi = pi (1+a).

For a small a, a<<1, we have psi ~ pi which is the result for vanishing precession.

Horst