Helicoidal Flow

I would guess that it is similar to helical motion, equation of the helix, but it must be a solution of curl v = alpha v. Donald Reed does not give an analytical solution as far as I can see.

To: EMyrone@aol.com
Sent: 29/01/2014 13:35:25 GMT Standard Time
Subj: Re: 257(2) : Extended Beltrami Electrodynamics for Vacuum Fields

Is there an analytical example known for a helicoidal flow? I tried to construct one in cylinder coordinates but it did not give an eigenvalue equation of the curl operator. It seems not to be so easy to find such examples. Maybe one has to use the complex circular basis for this.

Horst

EMyrone@aol.com hat am 29. Januar 2014 um 13:10 geschrieben:

These are illustrated with plane waves but more generally they are given by Eqs. (55) to (58), which are four equations in four unknowns which can be solved numerically in the general case (more general than vacuum plane waves). They ought to produce helicoidal vacuum flow of the type sketched in Figure (1), so that the B(3) field emerges as the field line along Z, well known in Beltrami hydrodynamics and aerodynamics. This is yet another way of proving the B(3) field and its hugely successful ECE theory.

Remark by Dr. Horst Eckardt : Numerical Solutions of Beltrami Flow

This is an excellent example by Horst, I just sent over the extended Beltrami theory based on UFT256.

In a message dated 29/01/2014 11:32:13 GMT Standard Time, writes:

I found a simple example: the vector

v = [ r, 0 , 0 ]

is rotated with the z axis. Then we have to define

v = [ r cos(kappa z), r sin(kappa z) , 0 ]

The curl of v in cartesian coordinates is

This means the “eigenvalue” is -kappa.

Horst

EMyrone@aol.com hat am 29. Januar 2014 um 10:05 geschrieben:

This is a very interesting idea. In the case of the vacuum plane wave the solutions are analytical, transverse plane waves and the B(3) field, which is a propagating field,related by the B Cyclic Theorem. I would guess that numerical solution and animation of Beltrami flow is by now a highly developed subject area in hydrodynamics and plasma physics, to give just two examples. The Moses / Reed / Evans decomposition into (1), (2), (3) allows difficult terms in the Navier Stokes equation to be eliminated. So it should be important in aerodynamics and hydrodynamics. I have no doubt that the motion of a whirlpool galaxy can be solved numerically and animated to produce helicoidal flow if the galaxy ever obeys curl v = alpha v. This would produce the mythical “black hole” in a rigorously correct manner. Of course there is no “black hole”, and the analysis in galaxies must be correctly relativistic. We have already solved all these problems of how to explain the velocity curve and so on. The media have massively damaged physics by their slavish repetition of absurd dogma, but we can salvage some real physics.

PS: Beltrami equation

PS:
Doug, is there a chance to solve the Beltrami equation numerically by the FEM method? I guess it has to be in 3D.

Horst

Horst Eckardt <mail@horst-eckardt.de> hat am 29. Januar 2014 um 08:42 geschrieben:

Myron, Doug,
The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com
Sent: 28/01/2014 13:45:47 GMT Standard Time
Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields
Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:
B(1) x B(2) = i B(o) B(3)*, (93)
A conjugate product which relates three basic magnetic field components in vacuo defined as:
B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)
B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)
B(3) = B(0)k, (93c)
where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.
Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].
It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone wrote:

On 28/01/2014, at 7:00 PM, EMyrone wrote: 
To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag To: EMyrone Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf 

On 28/01/2014, at 7:00 PM, EMyrone wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?

257(2) : Extended Beltrami Electrodynamics for Vacuum Fields

These are illustrated with plane waves but more generally they are given by Eqs. (55) to (58), which are four equations in four unknowns which can be solved numerically in the general case (more general than vacuum plane waves). They ought to produce helicoidal vacuum flow of the type sketched in Figure (1), so that the B(3) field emerges as the field line along Z, well known in Beltrami hydrodynamics and aerodynamics. This is yet another way of proving the B(3) field and its hugely successful ECE theory.

a257thpapernotes2.pdf

Numerical Solutions of Beltrami Flow

This is a very interesting idea. In the case of the vacuum plane wave the solutions are analytical, transverse plane waves and the B(3) field, which is a propagating field,related by the B Cyclic Theorem. I would guess that numerical solution and animation of Beltrami flow is by now a highly developed subject area in hydrodynamics and plasma physics, to give just two examples. The Moses / Reed / Evans decomposition into (1), (2), (3) allows difficult terms in the Navier Stokes equation to be eliminated. So it should be important in aerodynamics and hydrodynamics. I have no doubt that the motion of a whirlpool galaxy can be solved numerically and animated to produce helicoidal flow if the galaxy ever obeys curl v = alpha v. This would produce the mythical “black hole” in a rigorously correct manner. Of course there is no “black hole”, and the analysis in galaxies must be correctly relativistic. We have already solved all these problems of how to explain the velocity curve and so on. The media have massively damaged physics by their slavish repetition of absurd dogma, but we can salvage some real physics.

PS: Beltrami equation

PS:
Doug, is there a chance to solve the Beltrami equation numerically by the FEM method? I guess it has to be in 3D.

Horst

Horst Eckardt <mail@horst-eckardt.de> hat am 29. Januar 2014 um 08:42 geschrieben:

Myron, Doug,
The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com
Sent: 28/01/2014 13:45:47 GMT Standard Time
Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields
Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:
B(1) x B(2) = i B(o) B(3)*, (93)
A conjugate product which relates three basic magnetic field components in vacuo defined as:
B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)
B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)
B(3) = B(0)k, (93c)
where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.
Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].
It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone wrote:

On 28/01/2014, at 7:00 PM, EMyrone wrote: 
To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag To: EMyrone Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf 

On 28/01/2014, at 7:00 PM, EMyrone wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?

The Beltrami Equation

An example is given in UFT256, curl q = kappa q, the curl is proportional to q, so is parallel to it, kappa being the scalar magnitude of wavevector. This was first used by Eugenio Beltrami in the eighteen eighties with curl v = alpha v, where v is velocity and alpha a scalar. It is Eq. (5) of

Donald Reed, pp. 525 ff. of M. W. Evans Ed., “Modern Nonlinear Optics”, volume 119(3) of “Advances in Chemical Physics”, (Wiley Interscience, New York, 2001), second edition.

This is available in all good libraries and there ought to be several libraries in the Munich area which have it. It can be purchased from Amazon as a hardback or e book. I am just about to write up Note 257(2) which will explain things in all detail. Beltrami flow has no Magnus force, and its helicoidal structure was discovered by Beltrami. It is illustrated in Figure 3 of Reed’s article, which also illustrates real Beltrami flows. Helicoidal means that the field lines start as transverse and end up as longitudinal. It is an axisymmetric sheared vortex. The B(3) field is related to the longitudinal field line. We have:

curl B(1) sub L = kappa B(1) sub L = curl B(2) sub L
curl B(1) sub R = – kappa B(1) sub R = curl B(2) sub R
curl B(3) = 0 B(3)

The curl eigenvalues are 1, 0, -1 . These are also the eigenvalues of helicity in quantum field theory. In UFT256 these were related directly to the spin connection of Cartan geometry. Here subscript L denotes anticlockwise rotating, subscript R denotes clockwise rotating. Beltrami flow is known loosely as “eigenfunctions of the curl operator”. Moses (cited by Reed) has shown that any vector field can be decomposed into (1), (2) and (3) modes (not just (1) and (2), a point of key importance because it infers the B(3) field immediately). In electrodynamics these lead to the B(1), B(2) and B(3) fields (H. F. Moses, SIAM J. Applied Mech., 21(1), 114 (1971)). This leads to the B Cyclic Theorem of O(3) electrodynamics:

B(1) x B(2) = i B(0) B(3)*

et cyclicum

which I inferred in the mid nineties independently of Moses and Reed. The formalism can also be related to Hamilton’s quaternions and to Cartan’s spinors in SU(2) rep., homomorphic with O(3) rep. So there is plenty of mileage here. The new ingredient post UFT256 is the spin connection. This is recognized now as being related to the alpha factor of Beltrami hydrodynamics, aerodynamics, electrodynamics, and possibly galactic dynamics. I have reviewed Beltrami electrodynamics in UFT100 Section 7, and in another UFT paper post UFT100. That can be found with google or keywords. Note that there is a small typo in UFT100 Section 7, the correct equations are as above.

Sent: 29/01/2014 07:42:05 GMT Standard Time
Subj: Re: Extract from Paper by Donald Reed in ACP vol 119(3), Modern Nonlinear Optics

Myron, Doug,
The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com
Sent: 28/01/2014 13:45:47 GMT Standard Time
Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields
Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:
B(1) x B(2) = i B(o) B(3)*, (93)
A conjugate product which relates three basic magnetic field components in vacuo defined as:
B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)
B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)
B(3) = B(0)k, (93c)
where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.
Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].
It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone wrote:

On 28/01/2014, at 7:00 PM, EMyrone wrote: 
To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag To: EMyrone Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf 

On 28/01/2014, at 7:00 PM, EMyrone wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?

Galaxies and Beltrami Flow

Many thanks. I will send out note 257(2) today on the B(3) field and Beltrami flow. The latter is helicoidal and is illustrated by Reed in his article. Reed mentions the Cartan calculus in his article in Evans and Kielich, “Modern Nonlinear Optics” and I have all the volumes here. In helicoidal, forceless, Beltrami flow there is a force line along the Z axis. In O(3) electrodynamics, a development of Beltrami electrodynamics, this is the B(3) field within a factor. The higher topology relates the ((1), (2), (3)) frame to the Cartesian frame. Beltrami flow has no Magnus force, and is defined by curl v = alpha v, where v is the velocity and where alpha is a constant. In UFT256 it was shown that curl q = kappa q where q is the tetrad vector. So if there is Beltrami flow in a galaxy, then a field line would exist at its centre akin to the B(3) field. The field line would be infinitesimal in transverse dimension. Therefore the first thing to do is to find whether a galaxy obeys the equation curl v = alpha v under any circumstances. The latest work on galaxies in UFT238 ff explained the velocity curve straightforwardly, so ECE is preferred to Einsteinian relativity, which fails completely in this context, and is rejected by AIAS because its mathematics are totally wrong due to neglect of torsion. There are also galactic animations from our new equations by Bernhard Foltz on www.aias.us, and a meticulous study and animation by Robert Cheshire of the whirlpool galaxy. See also postings on this blog. So we are well ahead of anything that Hawking has ever produced. He always neglected torsion so his entire output is incorrect. The media do not understand these mathematics at all, but Reed has a grasp of them.

To: EMyrone@aol.com
Sent: 28/01/2014 15:15:53 GMT Standard Time
Subj: Additional Note re galaxies.

Myron if y’all do getbround to synthesising these ideas – also see Fig
2 at http://arxiv.org/ftp/arxiv/papers/1012/1012.1384.pdf
This connects theory to actual data. Regards Norman.

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Extract from Paper by Donald Reed in ACP vol 119(3), Modern Nonlinear Optics

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com
Sent: 28/01/2014 13:45:47 GMT Standard Time
Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields
Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:
B(1) x B(2) = i B(o) B(3)*, (93)
A conjugate product which relates three basic magnetic field components in vacuo defined as:
B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)
B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)
B(3) = B(0)k, (93c)
where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.
Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].
It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone wrote:

On 28/01/2014, at 7:00 PM, EMyrone wrote: 
To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag To: EMyrone Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf 

On 28/01/2014, at 7:00 PM, EMyrone wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?

Some Remarks by Horst Eckardt

These are interesting remarks as usual, and this type of imaginative hypothesis is badly needed in modern physics.

Subj: Re: FOR POSTING: New vacuum interactions paper by Eckardt and Lindstrom

Many thanks for these interesting comments. Actually I found eq.(34) before we used it in UFT255, this is an interesting coincidence. There is also another connection between the “soft electrons” postulated by Cater and our findings. Our momentum analysis of Compton scattering resulted in photon masses in the range of electron masses. Cater postulates that each photon is surrounded with a “soft electron”, a weak non-Maxwellian ether structure. It could be possible that these structures are effective during scattering experiments, thus resulting in the high mass values of photons we computed. All this is more a hypothesis than settled physics, but it is interesting to see that several independent developments run together under the roof of ECE theory.

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 09:55 geschrieben:

This is a very original and interesting paper. It can certainly be published open source on www.aias.us and I also accept it for the journal. I have a few remarks as follows.

1) Eq. (17) has a Beltrami type structure – longitudinal waves, in this case longitudinal vacuum waves.
2) Eq. (34) is the same in mathematical structure as the Euler Bernoulli type equation derived in UFT255, and so two entirely different derivations come to the same conclusion for spin connection resonance. It would be interesting to apply this to low energy nuclear reactors (LENR).
3) I like Figure 1, which is an insightful illustration of how vortices can develop around dipoles in ECE theory. These do not exist in the Maxwell Heaviside field theory of standard physics.
4) The phenomenon attributed reproducibly on many occasions to certain observers is very remarkable. Subject to correction by the authors this means that some observers can pick up, or sense, structures not present in Maxwell Heaviside theory. They can do this without instruments. These are described by Balck in reference nine, and confirm the existence of the four polarization indices of the rigorous ECE unified field theory. These come from a higher order topology in electrodynamics. The theory can be written equivalently as four index and no index. This fundamental topological point is made clear in UFT266, introducing the new engineering model. The B(3) field and inverse Faraday effect indicate this higher topoology.
5) The scaling of the Bohr radius in Eq. (47) is also of fundamental interest.
6) Spatial structures are illustrated for glass and water material. Gareth J. Evans and Trevor Morris have also reported interesting structures around objects which they interpret in terms of a non MH theory and photon mass. They mention that they are preparing a review article on this subject, and that article can also be posted and published.
7) The spacetiome flux of ECE theory is interpreted in terms of Tesla’s ether flux. So ECE theory can be interpreted in terms of an ether flux. This answers some recent remarks by Kerry Pendergast on this blog.
8) There is a minor typo in ref. (8).

Subj: for publication: vacuum interactions

Doug and myself have finished a paper that laid in the stack for some time. It describes interactions with the ECE vacuum and proposes an explanation for difficile physical effects. Can we publish the paper on the AIAS web site?

Horst

Beltrami fields Forgot the link – Norman

On 28/01/2014, at 7:00 PM, EMyrone wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?