**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(*i*ω*t*)-factor - the Helmholtz equation is fulfilled.

Hence the aimed explanation for the alleged magnetic motor effects has failed.

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

*B*^{a} = **Ñ × A**

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 x^{1}, x^{2}, x^{3} and x^{o}= ct
we obtain:

F^{a}_{μν}
dx^{μ}Ùdx^{ν}
=
(∂_{μ}A^{a}_{ν}
−
ω^{a}_{b}_{μ}
A^{b}_{ν})
dx^{μ}Ùdx^{ν}
=
½[(∂_{μ}A^{a}_{ν}
−
∂_{ν}A^{a}_{μ})
−
(ω^{a}_{b}_{μ}A^{b}_{ν}
−
ω^{a}_{b}_{ν}A^{b}_{μ})]
dx^{μ}Ùdx^{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].

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

*B*^{a} =
(^{μoe}/_{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} π*r*³
(5)

is the volume for an assumed spherical symmetry.

In the standard model:

** B** =

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**

*T q*^{a} = ** m** ×

*En*^{a} = − *m* · *B*^{a}.
(9)

In ECE theory the torque is

*T q*^{a} = ^{1}/_{3}
(^{μoe}/_{mV}) ** m** ×

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}** =

In this expression ** m** and

**Ñ** × ** q^{a}** =

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**

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**

**Ñ × q**

**Ñ × q**

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^{c}Ù q^{b},
the spatial part of which is given with direction depending wave numbers κ_{a}

for a=1:
−κ_{1} *q*^{3}×*q*^{2} =
κ_{1} *q*^{2}×*q*^{3} ,
hence:
**Ñ** × *q*^{1} =
κ_{1} *q*^{2}×*q*^{3} ,

for a=2:
−κ_{2} *q*^{1}×*q*^{3} =
κ_{2} *q*^{3}×*q*^{1} ,
hence:
**Ñ** × *q*^{2} =
κ_{2} *q*^{3}×*q*^{1} ,

for a=3:
−κ_{3} *q*^{2}×*q*^{1} =
κ_{3} *q*^{1}×*q*^{2} ,
hence:
**Ñ** × *q*^{3} =
κ_{3} *q*^{1}×*q*^{2} ,

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}/_{2½}
(** i** −

*q*^{(3)} = *q*^{(3)}* = ** k**
(22)

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

**Ñ × q**

**Ñ × q**

**Ñ × q**

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**

Two typos: The correct equation is

**Ñ** ×
(**Ñ × q**

Using the vector identity:

**Ñ** ×
(**Ñ × q**

and using the property of the plane wave:

**Ñ
× q**

Wrong! see eq. (23). Probably a typo. Meant is:

**Ñ · q**

we obtain three Helmholtz wave equations:

(**Ñ**² + κ²)*q*^{(1)} = 0,
(29)

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

(**Ñ****²** + κ²)*q*^{(2)} = 0,
(30)

(**Ñ****²** + 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.

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

(**Ñ****²** + 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.

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:

(**Ñ****²** + κ²)** q** =

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

(**Ñ****²** + κ²)*q*_{Z} =
*R _{Z}* cos(κ

The solution of Eq. (34) is [1–12,17]

* q_{Z}* =

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}* =

goes to infinity and the torque:

* Tq* =

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* =

where * m* is its magnetic dipole moment. When

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.

[1] M.W. Evans, *Generally Covariant Unified Field Theory,
the Geometrization of Physics*,

Abramis, Vols. 1-3, 2005-2006.

[2] M.W. Evans, H. Eckardt, *Spin connection resonance in magnetic motors*,

Physica B 400 (2007) 175-179.
PHYSB302065.pdf

[3] G.W. Bruhn, F.W. Hehl, A. Jadczyk, *Comments on "Spin Connection Resonance in
Gravitational General Relativity"*,

Acta Physica Polonia B **39** (2008) 1001-1008.

[4] G.W. Bruhn, *Evans' "3-index, totally antisymmetric unit tensor"*,

http://www2.mathematik.tu-darmstadt.de/~bruhn/Evans3indEtensor.html

[5] G.W. Bruhn, *Comments on Evans' Duality*,

http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansDuality.html

Coordinates

x^{μ} : x^{o} = ct, x^{1} = x, x^{2} = y, x^{3} = z .

Coordinate related basis vectors

∂_{o} = **h**, ∂_{1} = **i**, ∂_{2} = **j**, ∂_{3} = **k**,

Minkowski metric

(η_{μν}) := diag(−1, 1, 1, 1)

g(∂_{μ},∂_{ν}) = η_{μν}
where g(.,.) is bilinear.

**e**_{o} = ∂_{o} = **h**,

**e**_{1} =
2^{−½} e^{iΦ}
(∂_{1} − *i* ∂_{2}) =
2^{−½} e^{iΦ}
(**i** − *i* **j**) = **e**_{2}*

**e**_{2} =
2^{−½} e^{−iΦ}
(∂_{1} + *i* ∂_{2}) =
2^{−½} e^{−iΦ}
(**i** + *i* **j**) = **e**_{1}*

**e**_{3} = ∂_{3} = **k**,

g(**e**_{a},**e**_{b}) = η_{ab}

**e**_{a} = q_{a}^{μ} ∂_{μ}
yields the coefficient matrix, i.e. the tetrad frame vectors related to the coordinate frame:

q_{o} = (q_{o}^{μ}) = (1 , 0 , 0 , 0)

q_{1} = (q_{1}^{μ}) = (0, 2^{−½} e^{iΦ} ,
−*i*2^{−½} e^{iΦ}, 0)

q_{2} = (q_{2}^{μ}) = (0, 2^{−½} e^{−iΦ},
*i*2^{−½} e^{−iΦ}, 0)

q_{3} = (q_{3}^{μ}) = (0 , 0 , 0 , 1)

The corresponding coframe is

q^{o} = dx^{o}

q^{1} = 2^{−½}
(e^{−iΦ}dx^{1} + e^{iΦ}dx^{2})

q^{2} = 2^{−½}*i*
(e^{−iΦ}dx^{1} − e^{iΦ}dx^{2})

q^{3} = dx^{3}