Spin connection resonance in magnetic motors

M.W. Evans, H. Eckardt

Abstract
We'll consider two sections from Evans' & Eckardt's article displayed in black. There are several evident typos marked in red.
The essentials:
The aim of the article under review is to show how Evans' ECE theory could explain some effects with so-called magnetic motors that allegedly rotate under appropriate conditions without exterior drive. However:
− There exists no physical foundation for assuming fourfold electromagnetic potential A^a and field F^a instead of the usual potential A and field F. But then - in usual Maxwell theory - there appears no spin connection ω^a_b in the theory of electromagnetism.
− Evans' derivation of the balance equations (18-20) from eq. (16) contains an erroneous conclusion beyond repair.
− Therefore the result of the article under review is the deep insight that a plane wave fulfils the wave equation, or, that - after separation of the exp(iωt)-factor - the Helmholtz equation is fulfilled.
Hence the aimed explanation for the alleged magnetic motor effects has failed.

2. The Balance condition

In ECE theory [1–12] the magnetic flux density in Tesla is defined by

B^a = Ñ × A^a − ω^a_b × A^b,         (3)

where A^a is the vector potential and where ω^a_b is the spin connection vector.

Eq. (3) is a conclusion from Evans' 4-dimensional version of the potential derivation equations of the electromagnetic field

F^a = d Ù A^a − ω^a_b Ù A^b,         (3')

where the indices a,b are running over 0,1,2,3 and A, F are 4-vector valued 1-forms and 2-forms respectively. ω^a_b is the (1,1) valued spin connection 1-form. For Cartesian coordinates x¹, x², x³ and x^o= ct we obtain:

F^a_μν dx^μÙdx^ν = (∂_μA^a_ν − ω^a_bμ A^b_ν) dx^μÙdx^ν = ½[(∂_μA^a_ν − ∂_νA^a_μ) − (ω^a_bμA^b_ν − ω^a_bνA^b_μ)] dx^μÙdx^ν ,         (3'')

The spatial part μ,ν = 1,2,3 of eq. (3'') can be rewritten as eq.(3) for 3-D-vectors as a 3-D-vector equation (3).

However, the problem with Evans' eq. (3') is that the electromagnetic potential form A and the electromagnetic field form F occur fourfold as A^a and F^a where a=0,1,2,3. Evans could never give a satisfying explanation for the relation of his field forms F^a to the one field form F the experimental physicists deal with. But on the other hand the appearance of the spin connection form ω^a_b in the derivation eq. (3'), which does not appear in the usual Maxwell theory, is caused just by that hypothesis of fourfold potential A^a and the fourfold field form A^a. See more in [3].

Thus, Evans' claim of resonance effects due to spin connection is not well founded physically.

The magnetic flux density may be expressed as an angular momentum [16] through

B^a = (^μ_oe/_4πMr³) L^a,         (4)

where μ_o is the S.I. permeability of the vacuum, e is charge, M is mass and

V = ⁴/₃ πr³                         (5)

is the volume for an assumed spherical symmetry.

In the standard model:

B = Ñ × A                 (6)

and the spin connection is missing. There is no possibility of spin connection resonance (SCR) in the standard model. Given the existence of a net magnetic dipole moment m in a magnet or assembly of magnets, there is present a force, torque and energy defined by

F^a = Ñ(m · B^a),         (7)

T q^a = m × B^a,         (8)

En^a = − m · B^a.         (9)

In ECE theory the torque is

T q^a = ¹/₃ (^μ_oe/_mV) m × L^a,                 (10)

where the angular momentum is

L^a = ^3mV/_μoe (Ñ × A^a − ω^a_b × A^b),         (11)

= 3 (^mVA(o)/_μoe) (Ñ × q^a − ω^a_b × q^b).         (12)

Here q^a is the vector part of the Cartan tetrad:

q^a_μ = (q^a_o, −q^a),         (13)

The quantity m of the eqs (10-12) is not defined here but is evidently identical with the quantity M given by eq. (5).
The meaning of the notation T q^a is unclear, but the r.h.s. of the eqs. (8),(10),(15) show that a 3D-vector is meant.

which defines the Cartan torsion through the first structure equation of Cartan:

T^a = d Ù q^a + ω^a_b Ù q^b.         (14)

So the origin of the torque Eq. (12) is the Cartan torsion of space-time itself. In the standard model A is considered classically as a convenient mathematical quantity introduced by Heaviside, and has no relation to the Cartan tetrad in the standard model (MH theory). Therefore in ECE theory the torque on a magnetic assembly of net magnetic dipole moment m is

T q^a = A^(o)m × (Ñ × q^a − ω^a_b × q^a).         (15)

In this expression m and A^(o) area properties of the magnetic assembly, and the rest of the expression is space-time itself. It is seen that if

Ñ × q^a = ω^a_b × q^b,         (16)

there is no torque, and the magnetic assembly does not spin. It is observed experimentally [14] that a magnetic assembly of a given critical design spins continuously with no electrical input of any kind. This is a reproducible and repeatable phenomenon that has no explanation in MH theory. In ECE theory the phenomenon is explained straightforwardly as a torque between m and a magnetic flux density:

B^a = A^(o)(Ñ × q^a − ω^a_b × q^b),         (17)

generated by space-time. Thus, ECE theory is preferred over MH theory.

The balance condition (eq. (16) ?) may be expressed in the complex circular basis [1–12] as

Ñ × q^(1)* = iκ₁q⁽²⁾ × q⁽³⁾,         (18)

Ñ × q^(2)* = iκ₂q⁽³⁾ × q⁽¹⁾,         (19)

Ñ × q^(3)* = iκ₃q⁽¹⁾ × q⁽²⁾,         (20)

in general.

Regrettably, the authors here failed to tell us which choice of the spin connection ω^a_b would yield the eqs. (18-20) from eq. (16). However, in Evans' GCUFT book [1](vol.1, p.183) a "second Evans duality equation" occurs: ω^a_b = κ Î^a_bc q^c "showing that the spin connection is dual to the tetrad". Further occurences at [1, vol.1, p.138 (7.6); p.183 (9.76); p.331 (18.6); p.438 (26.24); p.450 (27.29); p.488 (F.10)]. Especially at [1, vol.1, p.449 f.] we read:

                ω^a_b = − κ Î^a_bc q^c                 (27.28)

where Î^a_bc is the Levi-Civita tensor in the flat tangent bundle spacetime. Being a flat spacetime, Latin indices can be raised and lowered in contravariant covariant notation and so we may rewrite Eq.(27.28) as:

                ω_ab = κ Î_abc q^c.                 (27.29)

Eqn.(27.29) states that the spin connection is an antisymmetric tensor dual to the axial vector within a scalar valued factor with the dimensions of inverse metres. Thus Eq.(27.29) defines the wave-number magnitude, κ, in the unified field theory. It follows from Eqn.(27.29) that the covariant derivative defining the torsion form in the first Maurer-Cartan structure equation (27.25) can be written as:

                T^a = d Ù q^a + ω^a_b Ù q^b                 (27.30)

                    = d Ù q^a + κ q^b Ù q^c                 (27.31)

This is nonsense since a 3-index Î-tensor does not exist in 4D-tensor calculus [4,5]. (Notice e.g. the not well-formed eq. (27.31).) But it shows what the authors are meaning by the term ω^a_b × q^b: the spatial part of ω^a_b Ù q^b after application of the "Evans' duality" ω^a_b = − κ Î^a_bc q^c, i.e. the term ω^a_b Ù q^b = − κ Î^a_bc q^cÙ q^b, the spatial part of which is given with direction depending wave numbers κ_a

for a=1: −κ₁ q³×q² = κ₁ q²×q³ , hence: Ñ × q¹ = κ₁ q²×q³ ,
for a=2: −κ₂ q¹×q³ = κ₂ q³×q¹ , hence: Ñ × q² = κ₂ q³×q¹ ,
for a=3: −κ₃ q²×q¹ = κ₃ q¹×q² , hence: Ñ × q³ = κ₃ q¹×q² ,

Compare with the eqs. (18-20). Whether the form of the eqs. (18-20) can be attained due to special properties of Evans' circular basis may remain open here.

For plane wave solutions [1–12]:

q⁽¹⁾ = q⁽²⁾* = ¹/_2^½ (i − ij) exp(i(ωt − κZ)),         (21)

q⁽³⁾ = q⁽³⁾* = k         (22)

and the condition (16) becomes a Beltrami condition [1–12]:

Ñ × q⁽¹⁾ = −κq⁽¹⁾,         (23)

Ñ × q⁽²⁾ = −κq⁽²⁾,         (24)

Ñ × q⁽³⁾ = −0q⁽³⁾         (25)

with eigenvalues −κ, κ and O, indicating O(3) symmetry. These may be regarded as being generated from the boson eigenvalues −1, 0 and 1. The fermion eigenvalues are −½ and ½.

The Beltrami condition is in turn a Helmholtz wave equation. This is shown for example as follows:

Ñ × (Ñ × q⁽¹⁾) = −κ²Ñ × q⁽¹⁾ = κ² × q⁽¹⁾.         (26)

Two typos: The correct equation is
Ñ × (Ñ × q⁽¹⁾) = −κÑ × q⁽¹⁾ = κ² q⁽¹⁾.         (26')

Using the vector identity:

Ñ × (Ñ × q⁽¹⁾) = Ñ(Ñ · q⁽¹⁾) − Ѳq⁽¹⁾         (27)

and using the property of the plane wave:

Ñ × q⁽¹⁾ = 0,         (28)

Wrong! see eq. (23). Probably a typo. Meant is:
Ñ · q⁽¹⁾ = 0,         (28')

we obtain three Helmholtz wave equations:

(Ѳ + κ²)q⁽¹⁾ = 0,         (29)

Each of the following eqs. (30-33) contains a typo a missing exponent ².

(Ѳ + κ²)q⁽²⁾ = 0,         (30)

(Ѳ + 0)q⁽³⁾ = 0.         (31)

Note carefully that these are wave equations of the space–time itself, when there is no Cartan torsion and where a rotational symmetry has been assumed for the space part of the space–time. This is equivalent to assuming an O(3) or isotropic symmetry. Therefore the Helmholtz equations define the balance condition under which there is no torque between the magnetic dipole moment of the assembly and the space–time surrounding it. This is the condition of, for example, a non-rotating bar magnet resting on a laboratory bench. The magnet’s own magnetic field cannot rotate it, and there is no space–time torque to rotate it.

3. Rotating the magnetic assembly

In order to rotate the magnetic assembly as observed experimentally [14] it is assumed that the Helmholtz equation of space-time, of general form:

(Ѳ + κ²)q = 0         (32)

Due to the gap between the eqs. (16) and (18)ff. there is no connection of this result with Evans' general theory.

What Evans has indeed shown here is nothing but the plane wave of eq. (21) fulfilling the wave equation, or, after separating the time-factor, fulfilling the Helmholtz equation.

The rest is silence - i.e. is Evans' and Eckardt's wishful thinking.

is changed into an undamped resonator Eq. (17) by the addition of a right-hand driving term as follows:

(Ѳ + κ²)q = R cos(κ_o·r),         (33)

where R has the units of inverse square meters, the units of curvature. In the Z-axis:

(Ѳ + κ²)q_Z = R_Z cos(κ_oZ).         (34)

The solution of Eq. (34) is [1–12,17]

q_Z = ^{R_Z cos(κ_oZ)}/_{κo² − κ²}         (35)

and at the resonance condition

κ_o = κ,         (36)

A deep insight, really! The inhomogenious Helmholtz equation (33) gives resonance at κ_o = κ, that's true. But what has that to do with Evans' theory???

q_Z goes to infinity. This means that the potential [1–12]:

A_Z = A^(o) q_Z         (37)

goes to infinity and the torque:

Tq = m_Z × B_Z         (38)

between m and space-time goes to infinity. In practical terms enough torque is generated so that the magnetic motor starts to rotate indefinitely as observed [14]. This is a qualitative explanation, which shows that the rotation of the magnetic assembly is due to a torque between its magnetic dipole moment and space-time. It is also shown that resonant amplification is needed, and it is observed experimentally [14] that the magnetic assembly starts to rotate only when a critical design is completed, for example, by the addition of a small component. Without the right design, rotation does not occur. The key design of the assembly is described by the resonance condition

κ_o = κ,         (39)

of Eq. (36). Here κ_o is a characteristic wave-number (inverse distance) of the assembly and K of space-time. The small driving term on the right-hand side of Eq. (34) is amplified at resonance, and it is assumed that the driving term is a property of the magnetic assembly. When the latter is such that Eq. (39) holds, the torque is amplified enough to produce rotation. It is observed [14] that the rotation occurs only when the magnetic design is correct and when the rotation starts, it is continuous. This means that the driving term must be periodic, the simplest type being the cosine of Eq. (34).

It is well known that the compass needle rotates to magnetic north and stops. In ECE theory this is due to the Cartan torsion of space–time indicating the presence of the Earth’s static magnetic field B. The torque on the compass needle is

Tq = m × B,         (40)

where m is its magnetic dipole moment. When m is parallel to B the torque vanishes and the needle points to magnetic north and stops rotating. In order to make a magnetic assembly rotate continuously without stopping, as observed [14], a spinning torque is needed as in Eq. (34), a static torque is not sufficient.

This is a qualitative and essentially simple explanation of magnetic motors in ECE theory. In MH theory there is no explanation of magnetic motors, because there is no mechanism in MH theory through which m can form a torque with space-time.

References