Last update: 28.09.2004 10:00

htm copy from the original

http://www.aias.us/Comments/aphaselawpaper.pdf

with comments by G.W. Bruhn in

**Introduction**

**Remark 01** . . . Sign error in (20)

**Remark 02** . . . Dimension error for g in (19)

**Remark 03** . . . Dimension check for (24)

**Remark 04** . . . Dimension error for g in (80)

**Remark 05** . . . Writing error?

**Remark 06** . . . Sign error in (77)

**Remark 07** . . . Aharonov Bohm effect

**Remark 08** . . . Integration I

**Remark 09** . . . Integration II

**Remark 10** . . . Integration III

**Remark 11** . . . Aharonov Bohm effect and Poincaré Lemma

**Remark 12** . . . The Sagnac Effect

**Remark 13** . . . Calculation error in (73)

**Remark 14** . . . Topology of Nonsense

During the last weeks on the AIAS website a new paper of Myron W. Evans was announced with great advertising that should treat the application of Evans’ General Unified Field Theory to some well-known effects of experimental physics. So I decided to study this paper thoroughly in the version that was published as a preprint in the web under

**http://www.aias.us/Comments/aphaselawpaper.pdf**

Reading that paper I detected several points where I couldn’t agree with.

First of all some writing errors in the formulas, which nobody can avoid who writes a
paper and which can easily be removed:

**Remark 1**
and
**Remark 6**

Probably also
**Remark 5**
is of that type. Then there was a convention error,
physical dimensions, which would not agree on both sides of a formula

**Remark 3**
and
**Remark 4**

concerning the constant g that can readily be removed by correcting its definition.

There is another error, probably in equ. (82), which caused my
**Remark 7** :
(82) would yield curl **A**_{trans} = **0**, while the modified equation

**A**^{(1)} = **A**^{(2)}*
= A^{(0)}/sqr(2) (**i** - i **j**) e^{iφ} ,
where
φ = ωt - κz
(82')

would give the result probably intended by Evans.
However, the author and his reviewers had a lot of time to detect that error. They didn't.

Also
**Remark 13**
itself is of minor interest, but the corresponding error, a simple
calculation or writing error, has a certain importance in view of other errors in our list.

That list up to this point was not worth mentioning. But a conclusion can be derived from these errors:

**I doubt that any reviewer has ever read that article.**

I wrote these remarks on the basis of Evans' preprint in the web.
However, in the meantime, in August 2004 the article appeared in **Foundations of
Physics Letters**, Vol. 17, No. 4. and one can compare both versions. Even the obvious
and senseless writing errors have survived: No! No reviewer has seen that article.
Not a single word has changed on request of any reviewer.

But there are some points left being of *serious importance*:

The *main assertion* of the paper is that the physical effects under consideration
all can be described by means of the so-called phase integral appearing on the
left-hand side of the eqns. (21) and (22)

∫_{DS} K
and
∫_{dS} **κ·**d**r** respectively.

A basic tool for transforming such integrals along closed boundaries DS or dS into integrals over the "interior" S of the boundary is the Stokes-Cartan integral theorem [9; p. 54, (2.49)]. Here Evans makes a distinction between the «ordinary Stokes Theorem» and a so-called «generally covariant Stokes Theorem». His assertion is that only the latter would give correct evaluations of the phase integral while the «ordinary Stokes Theorem» would fail (cf. Section 3).

This assertion, if being true, would be very amazing, since the ordinary theorem is mathematically true, so why should the «generally covariant Stokes Theorem» - if correct - give different results?

A reason to have a look on the mathematics of the Stokes integral theorem: The received version transforms the above path integral

∫_{DS} K = ∫_{DS} **κ·**d**x**
= ∫_{DS} κ_{j} dx^{j}
(with Einstein's summation convention to be applied here and further more)

into a surface integral over the surface S spanned into the closed boundary curve dS:

∫_{DS} K = ∫_{S} d^K
= ∫_{DS} ∂_{j}κ_{k} dx^{j}^dx^{k}
(0 __<__ j , k __<__ n)

(1)

= ∫_{DS} (∂_{j}κ_{k}
− ∂_{k}κ_{j}) dx^{j}^dx^{k}
(0 __<__ j < k __<__ n),

valid for an (n+1)-dimensional space. What Evans calls «ordinary Stokes Theorem»
appears to be its spatial (three-dimensional) version merely (see the text that
belongs to the eqns. (35 – 38)), which, of course, does not hold for generic
surfaces in the four-dimensional spacetime. In that case additional terms would
appear, even if
K = κ_{1} dx^{1} + κ_{2} dx^{2} + κ_{3} dx^{3}
is only spatial. However, no doubt,
that the «ordinary Stokes theorem» in the general (4-dimensional) sense is valid.
Evans explains the different results of «ordinary Stokes Theorem» and his
«generally covariant Stokes Theorem» by the use of «covariant derivatives»
instead of usual partial derivatives.

Covariant derivatives are introduced in differential geometry since generally
partial derivatives of tensors are no tensors while the covariant derivation
of tensors yields tensors again. On a manifold with the metric tensor (g_{jk}),
e.g. the spacetime of general relativity,
the covariant derivative D_{j}κ_{k}
[9; p.58, (3.12)] is defined by

D_{j} κ_{k}
:= ∂_{j} κ_{k} − Γ^{s}_{jk} κ_{s}

where the so-called Christoffel symbols Γ^{s}_{jk}
are given [9; p.60, (3.21)] by

(2)
Γ^{s}_{jk}
= ½ g^{sr} (∂_{j} g_{rk} + ∂_{k} g_{rj} − ∂_{r} g_{jk}) .

Evidently the coefficients Γ^{s}_{jk}
are symmetric with respect to j,k. Due to the antisymmetry of dx^{j}^dx^{k}
that yields
Γ^{s}_{jk} dx^{j}^dx^{k}
= 0 and thus from (1)

(1')
∫_{DS} K = ∫_{S} D^K
:=
∫_{DS} (D_{j} κ_{k} – D_{k} κ_{j}) dx^{j}^dx^{k}
(0 __<__ j < k __<__ n).

Therefore we can use the covariant derivatives instead of the partial derivatives
without any additions. That was the case of a *torsion-free* connection.

More generally one can use connections *with torsion*. The quantity

(3)
T^{s}_{jk} := Γ^{s}_{jk} − Γ^{s}_{kj}

is called torsion tensor. In that case the torsion tensor appears in the «generally covariant Stokes theorem». Then we have the integral theorem

(1")
∫_{DS} K = ∫_{S} (D_{j} κ_{k}
– D_{k} κ_{j}
+ T^{s}_{jk} κ_{s}) dx^{j}^dx^{k}
(0 __<__ j < k __<__ n).

To say it clearly:

I. The «generally covariant Stokes theorem» without
these torsion terms T^{s}_{jk} κ_{s} dx^{j}^dx^{k}
would be wrong.

II. The correct «generally covariant Stokes theorem» (1") yields *no other result* than what could already be attained by using the «ordinary Stokes theorem»: We have

∫_{S} (∂_{j} κ_{k} – ∂_{k} κ_{j}) dx^{j}^dx^{k}
= ∫_{S} (D_{j} κ_{k} – D_{k} κ_{j}
+ T^{s}_{jk} κ_{s}) dx^{j}^dx^{k}

However, Evans asserts a discrepancy existing between his «generally covariant Stokes theorem» and the «ordinary Stokes theorem», which we could not confirm after having checked that above by reviewing the corresponding theory. In addition, Evans’ concrete examples of applications of his integral theorem contain calculation errors and, after removal of these errors, confirm the result attainable by the «ordinary Stokes theorem».

**The «ordinary Stokes theorem» and the «generally covariant Stokes theorem»,
if applied correctly, cannot disagree.**

This is the background on which we have to see the examples of integrations in Evans’ paper.

The correctness of Evans’ «generally covariant Stokes Theorem» must be doubted again due to its failing in the application to the Sagnac effect where the area integral on the right-hand side of (33) according to Evans is given by

∫_{S} **B**^{(3)}**·k** dAr
= π r² B_{z}^{(3)}
= π r² B^{(0)}
(51)

while the left-hand side path integral due to Evans’ corrected calculation
(two errors; see
**Remark 10**) is zero.

The situation in the case of the Aharonov Bohm effect is not better: From (82-83) we obtain

^{i}/_{g} A^{(0)} **B**^{(3)}
= **A**^{(1)} × **A**^{(2)} / A^{(0)} = − i A^{(0)} **k**

and from Evans’ cyclic equations {The Enigmatic Photon Vol. 5, p. 14, (1.1.24)} we know

**A**^{(3)}
= − i **A**^{(1)} ×**A**^{(2)} / A^{(0)}
= − A^{(0)} **k** ,

i.e. ∫_{dS} **A**^{(3)}**·**d**r** = 0 for arbitrary closed pathes dS.
But then (76) yields

∫_{S} **B**^{(3)}**·k** dAr = ∫_{dS} **A**^{(3)}**·**d**r** = 0,

which due to **B**^{(3)}**·k** = − g A^{(0)}² implies A^{(0)} = 0
and **B**^{(3)} = **0**.
**That is a contradiction.**

So in summary it may be said that Evans has not achieved the aim of his article: He failed in showing that the phase factor (32) would give the key for the description of important effects of physical optics on the basis of his «generally covariant field theory». Additionally, he has shown by claiming a discrepancy between the «ordinary Stokes theorem» and his «generally covariant Stokes theorem» that something in that «generally covariant Stokes theorem» is not well-understood.

by

Myron W. Evans,

Alpha-Institute for Advanced Study,

**ABSTRACT**

A generally covariant and gauge invariant description of physical optics, the Sagnac effect and the Aharonov Bohm effect is developed using the appropriate phase factor for electrodynamics. The latter is a generally covariant development of the Dirac Wu Yang phase factor based on the generally covariant Stokes Theorem. The Maxwell Heaviside field theory fails to describe physical optics, interferometry, and topological phase effects in general because the phase factor in that theory is under-determined. A random number from a U(1) gauge transformation can be added to the Maxwell Heaviside phase factor. The generally covariant phase factor of the Evans unified field theory is gauge invariant and has the correct property under parity inversion to produce observables such as reflection, interferograms, the Sagnac and Aharonov Bohm effects, the Tomita Chao effect and topologieal phase effects in general. These effects are described simply and self consistently with the generally covariant Stokes Theorem in which the ordinary exterior derivative is replaced by the covariant exterior derivative of differential geometry.

Keywords: Evans unified field theory, **B**^{(3)}
field, physical optics, reflection, interferometry, Sagnac effect, Aharonov
Bohm effect, Tomita Chao effect, topological phase effects.

In generally
covariant electrodynamics {1-8} the
potential field is the tetrad within a C negative coefficient, and the gauge
invariant electromagnetic field is a wedge product of tetrads. The covariant
exterior derivative is used to describe the electromagnetic field using Maurer
Cartan structure relation of differential geometry {9}. The generally covariant
field theory leads to O(3) electrodynamics and the Evans Vigier field
**B**^{(3)}. The latter is observed in several experiments, including the inverse Faraday
effect {10}. The experimental evidence for the **B**^{(3) }field,
and thus for generally covariant electrodynamics, is reviewed in the literature
{1}.

In this paper a generally covariant Dirac Wu Yang phase is defined within Evans unified field theory by using a generally covariant exterior derivative to develop the Stokes Theorem of differential geometry {12}. The electromagnetic phase factor is described by this generally covariant Stokes Theorem. This procedure produces the correct parity inversion symmetry needed to describe fundamental physical optical phenomena such as reflection and interferometry. The Maxwell Heaviside theory of electrodynamics is not generally covariant, i.e is a theory of special relativity, and for this reason is unable to describe physical optics and interferometry, and unable to describe the Sagnac and Aharonov Bohm effects. In Section 2, these shortcomings of the Maxwell Heaviside theory are summarized. In Section 3 the Evans unified field theory is applied to physical optics, which is correctly described with a phase factor constructed from the appropriate contour and area integrals of the generally covariant Stokes Theorem. This procedure gives the correct parity inversion symmetry of the electromagnetic phase factor, and so is able to correctly describe physical optical phenomena such as reflection, interferometry, the Sagnac effect and the Aharonov Bohm effect. In each case the generally covariant theory is simpler than the Maxwell Heaviside theory and at the same time is the first correct theory of physical optics as a theory of electrodynamics.

In the Maxwell-Heaviside field theory the phase factor is defined by:

Φ_{MH}
= exp(i(ω t − κZ))
(1)

where ω is the angular frequency of radiation at instant t, and κ is the wave-number of radiation at point Z in the propagation axis. Under a U(1) gauge transformation {13} the phase factor (1) becomes

Φ_{MH}
→ exp(iα) Φ_{MH}
= exp(i(ωt − κZ + α))
(2)

where α is arbitrary. In consequence the phase factor is not gauge invariant, a major failure of Maxwell Heaviside field theory. In other words the phase factor (2) is under-determined theoretically, because any number, α, can be added to it without affecting the description of experimental data. In Michelson interferometry {14}, for example, an interferogram is formed by displacing a mirror in one arm of the interferometer, thus changing Z in eqn. (1) for constant ω , t , and κ. The path of a beam of light from the beam-splitter to the mirror and back to the beam-splitter is increased by 2Z. The conventional description of Michelson interferometry incorrectly asserts that this increase in the path of the light beam, 2Z, results in a change of phase factor:

Δ Φ_{MH}
= exp (2iκZ)
(3)

producing the observed interferogram, cos(2κZ), for a monochromatic light beam in Michelson interferometry.

This basic error is the conventional theory has been reviewed in detail in ref. {11}. The main purpose of this paper is to show that the error is corrected when the theory of electrodynamics is developed into a generally covariant unified field theory.

The origin of the
error is found when parity inversion symmetry is examined. Consider the parity
inversion symmetry of the product **κ·r** of wave-vector
**κ** and position vector **r**:

P^ (**κ·r**) = **κ·r**
(4)

because:

P^ (**κ**) = − **κ** ,
(5)

P^ (**r**) =
− **r** .
(6)

Parity inversion is equivalent to reflection, and so κZ does not change under reflection in the Maxwell Heaviside phase factor (1). This means that the reflected phase factor is the same as the phase factor of the beam before reflection. In terms of the Fermat Principle of least time in optics {15} this means:

Φ_{2}
= e^{0} Φ_{1}
(7)

where Φ_{1} is the phase factor
of the wave before reflection and Φ_{2} after reflection.

The experimentally observed interferogram in the Michelson interferometer {14} and in reflection in general must be described, however, by:

Φ_{2}
= e^{2iκZ} Φ_{1}
(8)

and so the Maxwell Heaviside field theory of electrodynamics fails at a fundamental level to describe physical optics. Various hand-waving arguments have been used over the years to address this problem, but there has never been a solution. The reason is that there cannot be a solution in special relativistic gauge field theory because as we have seen, the phase factor is not gauge invariant, a random number α can be added to it {13}. This number is random in the same sense as χ is random in the usual gauge transform of the Maxwell Heaviside potential, i.e.:

**A** → **A** + Ñ χ
(9)

The U(1) gauge
transformation is a rotation of the phase factor (2), a rotation of a gauge
field brought about by multiplication by the rotation generator of U(1) gauge
field theory, the exponential e^{iα} . It is always argued in
U(1) gauge transformation that χ is unphysical, so for
self-consistency, α must be unphysical. However, an unphysical factor
α cannot appear in the phase factor {13} of a physical theory, i.e. a
theory of physics aimed at describing data. So the Maxwell Heaviside field
theory fails at a fundamental level to describe physical optics. In other words
Maxwell Heaviside gauge field theory fails to describe reflection and
interferometry.

The Sagnac interferometer {13} with platform at rest produces an observable interferogram both in electromagnetic waves and matter waves (interfering electron beams {11}):

γ = cos
(2 ^{ω2}/_{c2} Ar)
(10)

where Ar is the area enclosed by the interfering beams (electromagnetic or electron beams). When the platform is rotated at an angular frequency Ω a shift is observed in the interferogram:

Δ γ = cos
(4 ^{ωΩ}/_{c2} Ar)
(11)

The interferograms (10) and (11) are independent of the shape of the area Ar, and are the same to an observer on and off the rotating platform. There have been many attempts to explain the Sagnac effect since it was first observed, all have their shortcomings, as reviewed by Barrett {13}. In particular the Maxwell Heaviside field theory (a U(1) gauge field theory) is wholly incapable of explaining the effect {13}, either with platform at rest or in motion. In Section 3 it is shown that the Sagnac effect is described straightforwardly in the Evans unified field theory as a change in the Cartan tetrad {1-8}. It is well known that the tetrad plays the role of metric in differential geometry. This gets to the core of the problem with Maxwell Heaviside theory - it is a theory of flat space-time, and so is metric invariant {13}. The phase factor (1) of the Maxwell Heaviside field theory is a number, and so is invariant under both parity and motion reversal symmetry. The phase factor (1) has no sense of parity, of handedness or of chirality (of being clockwise or anticlockwise) and does not change under frame rotation or platform rotation. The Maxwell Heaviside theory cannot describe any feature of the Sagnac effect, and again fails at a fundamental level.

There has been a forty year controversy over the Aharonov Bohm effect{12}. The only way to describe it with Maxwell Heaviside field theory is to shift the problem to the nature of space-time itself. It is asserted conventionally {12} that the latter is multiply connected and that this feature (rather magically) produces an Aharonov Bohm effect by U(1) gauge transformation into the vacuum. It is straightforward to show as follows that this assertion is incorrect because it violates the Poincaré Lemma. In Section 3 we show that the Evans unified field theory explains the Aharonov Bohm effect with the correct generally covariant phase factor. This is the same in mathematical structure as for the Sagnac effect and physical optics, only its interpretation is different. The Evans unified field theory is therefore much simpler, as well as much more powerful, than the earlier Maxwell Heaviside field theory. To prove that the Maxwell Heaviside theory of the Aharonov Bohm effect is incorrect consider the Stokes Theorem in the well known {9} notation of differential geometry:

Why should the Poincaré Lemma be violated by the usual treatment of the
Aharonov Bohm effect? And why should spacetime be multiply connected in the
usual treatment of the Aharonov Bohm effect?

We consider a coil given by
x² + y² = R²
inside of which there is a constant magnetic field **B** = B_{o} **k** while
outside the coil we have **B** = **0**.
In this specified case we have the well-known potential

**A** = ^{1}/_{2} B_{o} (−y **i** + x **j**)
for
r²
= x² + y² __<__ R²
and
**A** = ^{1}/_{2} B_{o} (−y **i** + x **j**)
^{R²}/_{r²}
for
r²
= x² + y² __>__ R².

Therefore we obtain

**B** = curl **A** = B_{o} **k**
inside the coil
and
**B** = curl **A** = **0**
outside the coil.

The potential **A** given above is a special one that yields the prescribed magnetic field
**B**. If **A'** is another possible potential, then the difference
**A'**−**A** fulfils the condition curl (**A'**−**A**) = **0**
on the whole space. As is well-known this is equivalent to the existence of a function
χ defined on the whole space such that **A'**−**A** = grad χ. This
function χ must fulfil ∫_{Γ} grad χ**·dr** = 0 for
every closed path Γ, since otherwise χ could not be univalent.
Thus, no correct treatment of the Aharonov Bohm effect can rely on Evans' equ. (17).
Hence Evans must have misunderstood something.

see also **Remark 7**

exp (ig ∫_{dS} A) = exp (ig ∫_{dS} d^A)
(12)

where A is the potential one-form and d^A the exterior derivative of A, a two-form. The factor g in the Aharonov Bohm effect is e/h¯ where e is the charge on the electron and h¯ is the Dirac constant (h/(2π)). Under the U(1) gauge transformation

A → A + dχ (13)

the two-form becomes:

d^A → d^A + d^d χ (14)

However, the Poincaré Lemma asserts that for any topology:

d ^ d = 0 (15)

Therefore the two-form d^A is unchanged under the U(1) gauge transformation {13}. The Stokes Theorem (12) then shows that:

∫_{dS} dχ = 0
(16)

However, the conventional description of the Aharonov Bohm effect {12} relies on the incorrect assertion:

∫_{dS}dχ ≠ 0
(17)

c.f.

This is incorrect because it violates the Poincaré Lemma (15). The Lemma is true for multiply connected as well as simply connected regions. In ordinary vector notation the Lemma states that for any function χ

Ñ×Ñ χ = 0 (18)

and this is true for a periodic function because it is true for any function. A standard freshman textbook {16} will show that the Stokes Theorem and the Green Theorem are both true for multiply connected regions as well as simply connected regions.

There is therefore no correct explanation of the Aharonov Bohm effect in Maxwell Heaviside theory and special relativity. Similar arguments show that Maxwell Heaviside theory cannot be used to describe a topological phase effect such as the Tomita Chao effect, a shift in phase brought about by rotating a beam of light around a helical optical fibre. In Section 3 we shown that the Tomita Chao effect is a Sagnac effect with several loops, and is a shift in the Cartan tetrad of the Evans unified field theory. Similarly, the Berry phase of matter wave theory is a shift in the tetrad of the Evans unified field theory. Many more examples could be given of the advantages of the Evans unified field theory over the Maxwell Heaviside theory.

The first correct description of the phase factor in electrodynamics was given in ref. {11}. Since then, a unified field theory has been developed {1-8} which in turn gives further insight to the phase factor of electrodynamics and physical optics. In order to construct a phase factor which has the correct symmetry under parity inversion, and which is generally covariant and valid for all topologies, and which is furthermore, correctly gauge invariant, we use the generally covariant Stokes Theorem in differential geometry within the exponent of the electromagnetic phase factor. The phase factor is therefore an application of Eq. (12) for the propagating electromagnetic field, which is considered to be part of the generally covariant unified field theory {1-8}. Therefore the phase factor of electrodynamics and physical optics is:

Φ = exp
(ig ∫_{DS} A)
= exp (ig ∫_{S} D^A)
(19)

The product gA must have the physical dimension 1. See

under all conditions (free or radiated field and field matter interaction). We shall show in this section that the phase factor (19) produces fundamental phenomena of physical optics such as reflection and interferometry through a difference in contour integrals of Eq. (19):

ΔΦ = exp (i(∫_{0}^{Z}
**κ·**d**r
−
**∫

It is also shown that the phase factor (19) can be written as:

Φ = exp (i
∫_{dS} **κ·**d**r **)
= exp (i ∫_{dS} **κ**² dAr )
(21)

and automatically gives the Sagnac effect from the right hand side of eqn. (21). Eqn. (21) follows from the generally covariant-Stokes formula applied to the one-form K of differential geometry, the one-form that represents the wave-number. In Eq. (19 ) D^A is the covariant exterior derivative {1-9} necessary to make the theory correctly covariant in general relativity. Eq. (19) is the generally covariant Stokes Theorem, i.e. the Stokes Theorem defined for the non-Euclidean geometry of the base manifold and therefore in general relativity {11}, and it follows from differential geometry that D^A is a two-form. The electromagnetic phase can be written in general as:

Φ = exp (ig ∫_{DS} K)
= exp (ig ∫_{S }D^K)
(22)

and Eq. (19) and (22) are equivalent definitions provided that we recognise the following duality between the electromagnetic field as a potential one-form, denoted A in differential geometry {9} and the wave-number one-form K:

K = g A (23)

Here g is a constant of proportionality between the A and K one-forms. From gauge theory {11} g is given by the wave particle dualism represented by:

p = h¯ K = e A . (24)

Dimension check for (24):
Let [X] denote the physical dimension of the quantity X.

We have

[h¯ κ]
= m² s^{–1} g · m^{–1}
= m s^{–1} g
and
[e A] = m^{½} s^{–1} g^{½} · m^{3/2} s^{–1} g^{½}
= m² s^{–2} g,

since the potential form A yields [A]
= [vec-pot] · m = m^{3/2} s^{–1} g^{½} .

Thus, a velocity, e.g. c, is missing in the product h κ :
[h¯ κ c]
= m² s^{–2} g = [e A].

** Equ. (24) cannot be true.**
As a consequence the equs. (25) and (80) are wrong also.

where p is a momentum one-form. The duality (24) asserts that the photon is a particle with momentum h¯ K and also a field with momentum eA, where −e is the charge on the electron. The meaning of the duality (24) has been explained extensively in the literature on O(3) electrodynamics {11}. The charge e is defined as the ratio of magnitudes:

e = |p|/|A| (25)

and so eqn. (23) in physical optics and electrodynamics is multiplied by the potential and divided by the magnitude of the potential to inter-relate the field as particle with momentum h¯ and the field as wave, with momentum A. This is the de Broglie wave particle duality applied the electromagnetic field.

The electromagnetic phase in unified field theory and general relativity is therefore:

Φ = exp (i
∫_{DS} K) = exp (^{i}/_{h¯} ∫_{DS}
p) = exp (ig ∫_{DS} A)
(26)

which is the generally covariant Dirac phase or Wilson loop for electrodynamics as part of a unified field theory {1-8}. The covariant derivative appearing in Eq. (19) is defined in the notation of differential geometry by:

D^A = d^A + g A^A (27)

The symbol A is
convenient shorthand notation for the tetrad A^{a}_{μ} {1-9}, so Eq. (27) for example denotes:

(D^A^{c})_{μν}
= (d^A^{c})_{μν} + g A^{a}_{μ} ^ A^{b}_{ν}
(28)

For O(3) electrodynamics with orthonormal space index:

a = (1), (2), (3) (29)

the complex circular basis, then:

D^A^{(1)*} =
d^A^{(1)*} − ig A^{(2)}^A^{(3)}
et cyclicum.
(30)

In eqn. (30) the
indices μ and ν of the base manifold have been suppressed,
as is the custom in differential
geometry {9} because Eq. (27) to (30) are equations of differential geometry
valid for all geometries of the base manifold. So the indices μ and ν
are always the same
on both sides and can be suppressed for convenience of notation {9}. The
shorthand notation of Eq. (27) is therefore a summary of the essential features
of eqns. (28) to (30). The B^{(3)} form of differential geometry

B^{(3)} =
D^A^{(3)}
= − ig A^{(1)}^A^{(2)}
(31)

is the expression
of the fundamental Evans Vigier field {11} in differential geometry. The B^{(3)}
form is a component of a torsion tensor, (a vector valued two-form or
antisymmetric tensor) and is the spin Casimir invariant of the Einstein group.
It will be demonstrated in this section that it is responsible for and
therefore observed in all physical optical effects through the generally
covariant and gauge invariant phase factor of electrodynamics:

Φ = exp (ig ∫_{DS}
A^{(3)})
= exp (ig ∫_{S} B^{(3)})
(32)

Eqn. (32) is the
result of a generally covariant theory of electrodynamics and it is necessary
to be precise about the meaning of the contour integral on the left hand side
and the area integral on the right hand side of Eq. (32).
In particular, A^{(3)} is an irrotational function, whose curl
vanishes. It follows that eqn. (32) cannot be the result of the ordinary Stokes
Theorem:

∫_{δS} A = ∫_{S} B = ∫_{S} d^A
(33)

Probably the subscript

in which d^ is the ordinary exterior derivative {11} of differential geometry. In vector notation Eq. (33) is:

∫_{dS}
**A·**d**r** = ∫_{S} **B·k **dAr
= ∫_{S} (Ñ×**A**)** ·k **dAr
(34)

and for an
irrotational function A both sides of Eq. (34) vanish. Therefore in Eq. (32),
∫ A^{(3)} is not the result of an integration around a closed
loop as defined in the ordinary Stokes Theorem {16}.

Another way of seeing this is that if we attempt to integrate the ordinary plane wave:

**A** = ^{A(0)}/_{sqr(2)}
(**i** − i**j**) e^{iφ}
(35)

around a closed loop (such as a circle) using the ordinary Stokes Theorem then we obtain the result:

∫_{dS}
**A·**d**r**
= ∫_{S} (Ñ×**A**)** ·k** dAr = 0
(36)

The right hand
side is true because **k** is perpendicular to **i** and **j** and the
left hand side is proven by parameterizing the circle as:

dx = − x_{o} sin θ
dθ , dy = y_{o}
cos θ dθ
(37)

and using:

∫_{0}^{2π} sin θ dθ
= ∫_{0}^{2π} cos θ dθ = 0
(38)

**At first:** The denotation **A** in the equs. (35 - 36) is not correct.
The quantity given by (35) is **A**^{(1)}.

**A** itself (more precise: its tranversal component) is given by

½ (**A**^{(1)}+**A**^{(2)})
= ½ (**A**^{(1)}+**A**^{(1)}*)
= Re **A**^{(1)}.

Therefore for the following we have to replace **A** in the equs. (35 - 36)
with **A**^{(1)}:

**A**^{(1)} = ^{A(0)}/_{sqr(2)}
(**i** − i**j**) e^{iφ}
(35')

∫_{dS}
**A**^{(1)}**·**d**r**
= ∫_{S} (Ñ×**A**^{(1)})** ·k** dAr = 0
(36')

The analogues of (36') and (37') hold for **A**^{(2)} = **A**^{(1)}*.

**Secondly: **As one can see from equ. (1) or from literature
the quantity φ in equ. (35) depends on z **and** t:

φ = φ(z,t) = ωt − κz

But apparently Evans has forgotten that fact since there is no explicit declaration
of φ in his whole paper.

The integration described by (36 - 38) is apparently meant at constant time t.
Then **A**^{(1)} is a gradient:

**A**^{(1)} = grad (^{i}/_{κ} e^{iφ})
where φ = ωt−κz,

hence the line integral ∫_{dS}
**A**^{(1)}**·**d**r** vanishes for every *closed* path of
integration (at fixed time t) in accordance with Evans result (36).

But for more complicated cases we have to observe that the quantity φ = ωt−κz
of the integrand **A**^{(1)} depends on
z and on t as well. Hence the definition of a path must contain a prescription
of the time t at each point z of the path. For the calculation above we therefore
added the prescription «apparently meant at constant time t».
But below Evans will consider another (more complicated) example *without giving
any prescription* of the time along the path of integration, which is a
helix in **that case**.

This means that the ordinary Stokes Theorem of special relativity cannot be used to represent the phase factor of electrodynamics. The essential reason for this is that electrodynamics must be a theory of general relativity in which the exterior derivative is replaced by the covariant exterior derivative {1 - 9} under all conditions (free field and field matter interaction). Only in this way can a unification of gravitation and electromagnetism be achieved {1-8} within general relativity.

This type of field unification is the result of the Principle of General Relativity, i.e. that all theories of physics must be theories of general relativity. (The Maxwell Heaviside theory is the archetypical theory of special relativity, and is not generally covariant. This leads to the fundamental problems summarized in Section 2.)

The method of interpreting Eq. (32) is found by considering integration around the transverse part of a helix. The transverse part of a helix is a position vector which coils around the Z axis as follows:

**r** = (X**i** + Y**j**) e^{iφ}
(39)

and defines a non-Euclidean (curling) base-line in the non-Euclidean base manifold of general relativity. The product:

**r**² = **r·r**
= X² + Y²
(40)

for the helix is the same as for a circle of radius r. Therefore integration around the transverse part of a helix is the same as integration around a circle of circumference 2πr and area πr² provided that the arc length of the helix is also 2πr. The arc length is the distance along the helix from A to B in the following diagram:

Fig. 1

The diagram summarizes the process of taking a circle of diameter 2r and drawing it out into a helix coiled along the Z axis. The distance from A to B around the circumference of the circle is the same as the distance from A to B along the transverse part of the helix (its arc length {16}). The difference between the helix and the circle that there is a longitudinal component of the helix, the distance along the Z axis from A to B. The arc length of the helix is along its transverse component from A to B, therefore the distance along Z from A to B is defined by:

Z __<__
2πr
(41)

When this distance is equal to 2πr the helix becomes a straight line of length 2πr along the Z axis.

The difference between the helix and the circle illustrates the difference between the generally covariant Stokes Theorem and the ordinary Stokes Theorem, and is also the essential difference between generally covariant electrodynamics and Maxwell Heaviside electrodynamics in which radiation is the plane wave (35), with no longitudinal component. In generally covariant electrodynamics there is a longitudinal component, the Z component of the helix. The latter can be parameterized {16} by:

x = x_{o} cos θ , y = y_{o}
sin θ , z = z_{o} θ ,
(42)

and so contains both longitudinal and transverse components. The circle can be parameterized {16} by:

x = x_{o} cos θ ,
y = y_{o} sin θ
(43)

and contains only components which are perpendicular, or transverse, to the Z axis. The latter is evidently undefined in the circle, but well defined in the helix.

It is well
accepted that electromagnetic radiation propagates and so contains both
rotational and translational components of motion. These must be described by
the helix, and not by the plane wave of the Maxwell Heaviside theory in special
relativity. There must be longitudinal components of the free electromagnetic
field, represented by **A**^{(3)} and **B**^{(3)}.

The ordinary Stokes Theorem cannot be applied to the helix, because the latter does not define a closed curve, i.e. the path from A to B of the helix is not closed. In the circle the path around the circumference from A to B brings us back to the starting point, and defines the area of the circle. In order to close the path in the helix the path from A to B along its transverse part must be followed by the path back from B to A along its Z axis, as in the following diagram:

Fig. 2

The closed path on the right hand side of this diagram defines the contour integral of the generally covariant Stokes Theorem appearing on the left hand side of Eq. (32). Furthermore, as we have seen, integration around the transverse part of the helix is equivalent to integration around a circle whose circumference is equal to the arc length of the helix. Therefore the plane wave contributes nothing to the generally covariant Stokes Theorem because the integration of the plane wave around a circle is zero, as shown in Eq. (34) to (38).

It is concluded
that the electromagnetic phase in generally covariant electrodynamics and
physical optics is completely described by Eq. (31). The contour integral on
the left hand side of this equation is along the Z axis of the helix whose arc
length is the same as the circumference of a circle of radius r. The area
integral on the right hand side of Eq. (32) is an integral around the area
πr² of this circle. The contour integral is an integral over
the irrotational and longitudinal potential field **A**^{(3)}, and
the area integral is an integral over the Evans Vigier field **B**^{(3)},
the spin invariant of the Einstein group. Therefore all of physical optics and
electrodynamics are described by a phase factor which is completely defined by **A**^{(3)}
and **B**^{(3)}.

As we have already remarked above the quantity φ in equ. (35) depends
on z **and** t:

φ(z,t) = ωt − κz

where κ = ω / c .

But here, for the helix, a precise description of the path of integration and
parametrization of t along that path are totally missing in Evans' paper.
In that situation no «generally covariant Stokes»
can help:

**The integrand along the path of integration, and hence the line
integral along that path,
is simply not defined without that parametrization.**

If we would assume φ to be varying along the helix then the "transversal part" of the line integral (36) would in general give a nonvanishing contribution since one could not refer to the eqns. (38). Therefore, to simplify matters, we assume φ to be constant along the helix, which implies ωt - κz = 0 and hence t(z) = z/c. But then due to

But even if we assume

∫

due to the special form of

∫

In the following sections, applications of this law are given for reflection, interferometry, the Sagnac effect and the Aharonov Bohm effect. The catastrophic failings of the Maxwell Heaviside field theory of special relativity (summarized in Section 2) are corrected in each case. The easiest way to see that the Maxwell Heaviside theory is not generally covariant is to note that the plane wave defines a finite scalar curvature:

R = |A^{(1)}|^{−1}
^{∂²A(1)}/_{∂Z²}
= κ²
(44)

However, in special relativity, the scalar curvature R = 0 by definition because the space-time of a theory of special relativity is Euclidean (or "flat"). In the generally covariant description of electrodynamics, the scalar curvature of the helix, κ², is consistent with the fact that we are considering a baseline that coils around the Z axis in the base manifold.

The generally covariant phase law (32) can be exemplified by integration around a circle of radius r. The area on the right hand side of equ. (32) in this case is πr², so we obtain

∫_{S} **B**^{(3)}**·k **dAr
= π r² B_{z}^{(3)}
= π r² B^{(0)}
(51)

The phase factor observed in the Sagnac effect therefore originates in the phase law (32) as follows:

Φ = exp (igB^{(0)}π r²)
= exp (iκ² Ar)
= exp (i^{ω²}/_{c²} Ar)
(52)

where we have used

B^{(0)} = κ A^{(0)}

g = κ/A^{(0)}
(53)

Ar = π r²

The Sagnac effect with platform at rest is an interferogram formed from interference
of clockwise and counter-clockwise waves, one with phase e^{iκ²Ar}
and one with phase factor e^{−iκ²Ar} giving the difference
in phase factor e^{2iκ²Ar} and interferogram:

observed experimentally {13} with great precision . . .

It's a rather miraculous result that Evans presents here referring to T. W. Barrett and D. M. Grimes from a book that appeared at WORLD SCIENTIFIC (1996). Obviously the two waves return to their starting point "in phase", i.e. with phase difference Δλ = 0 due to symmetry reasons. For further information about the Sagnac effect we recommend the following web source

For

Δλ =

and especially Δλ = 0 for Ω = 0.

Evans, after some not understandable calculations, arrives at a similar result, remarkably without any restrictions for the rotation frequency Ω. He shows -

ΔΔγ = cos (^{4 ωΩ} / _{c²} Ar).
(56)

In equs. (10) and (11) Evans had already announced these results with one Δ less per equation. But even this Δ left in (11) is not justified since Δγ is no difference; it doesn't tend to 0 for Ω → 0, as our Δλ .

. . .

We can cross check this result for the Sagnac effect by integrating the
wave number vector **κ** around the helix parameterized {16} by:

X = X_{o} cos θ , Y = Y_{o}
sin θ , Z = Z_{o} θ ,
(67)

^{dX}/_{dθ}
= –X_{o} sin θ ,
^{dY}/_{dθ}
= Y_{o} cos θ ,
^{dZ}/_{dθ}
= Z_{o} .
(68)

The wave number vector is:

**κ** = κ_{x} **i**
+ κ_{y} **j**
+ κ_{z} **k**
(69)

and the contour integral on the left hand side of the phase law (32) is:

∫_{Γ} **κ****·**d**r**
= – κ_{x}X_{o}
∫_{Γ} sin θ dθ
+ κ_{y}Y_{o}
∫_{Γ} cos θ dθ
+ κ_{z}Z_{o}^{}
∫_{Γ} **θ** dθ .
(70)

Where is the factor

∫

That causes changes in (72) at the marked positions:

∫

If (73) would be correct the first eq. in (74) would have to be corrected to: Z = 2π Z

We stop here since (73) is not correct as we shall see in

The transverse part of the helix gives no contribution to this contour integral because:

∫_{0}^{2π} sin θ dθ
= ∫_{0}^{2π} cos θ dθ = 0 .
(71)

The longitudinal component along the Z axis from B to A in fig. (2) gives the only non-zero contribution to the contour integral, a contribution originating in:

∫_{0}^{2π}
**θ**
dθ
= 2π**²**
.
(72)

So:

∫_{Γ} **κ****·**d**r**
= 2 π**²**
κ_{z} Z_{o}
(73)

This result is Eq. (58) after identifying:

Z = 2 π**²**
κ_{z} Z_{o}, κ Z = g A^{(0)} Z
(74)

The whole of Eq. (73) comes from the contribution along the Z axis in Fig. (2) through the axis of the helix. This contribution is not present in special relativity and the conventional Stokes Theorem. Our generally covariant unified field theory {1-8} gives the observed phase factor of the platform at rest from:. . .

Φ = exp (iκZ) = exp (^{iω²}/_{c²} Ar)
(75)

and also with the platform in motion as argued already.

According to your

OK. What about the constants X

X

And what's with Z

OK. You are the boss.

Now to the longitudinal component:

Stop, Myron, we aren't in a hurry. First of all notice that eq. (70) is not correct, which has consequences for eq. (72) and (73): We shall need the corrected result here:

∫

But apart from that we are not satisfied by your answer!

r(

to its starting point A (see

R(

which can be reached by a vertical straight line (parallel to the Z axis) where the parameter z is running from z = Z

∫

and balances with the other longitudinal integral.

And that's no miracle! You have chosen a

∫

for arbitrary

This result is in accordance with our calculation in

But remember, due to your theory the result of the path integration should agree with (51), with π r² B(0), and that is not fulfilled.

The topological fundament of this result is that a circle can be shrunk continuously to a point and is simply connected, but a helix shrinks continuously to a line, and in this sense is not simply connected. This is the essential difference between special and general relativity in optics and electrodynamics.

If this difference should really mark the difference between special and general relativity
then the author has done a great work: Circle and (closed) helix (see
**Fig. 2**).

are toplogically
equivalent since both are doubly connected and can be shrunk continuously to a point.
Hence for Myron W. Evans
any difference between special and general relativity has disappeared.

The Aharonov Bohm (AB) effect in generally covariant electrodynamics is also described straightforwardly by the phase law (32):

exp(ig ∫ **B**^{(3)}**·k** dAr)
= exp(ig ∫ **A**^{(3)}**·dr**)
= exp(g² ∫
**A**^{(1)}×**A**^{(2)}
**· k** dAr)
(76)

The interpretation of eqn. (76) for the AB effect is as follows.

a) The magnetic
field **B**^{(3)} is the static magnetic field
(iron whisker or
solenoid) placed between the two openings of a Young interferometer which
measures the interferogram formed by two electron beams.

b) Any magnetic field (including a static magnetic field) in generally covariant electrodynamics is the two-form {1-9}:

B = D^A **+** g A^A
(77)

Due to (27) the equ. (77) contains a sign error. It should read

B = D^A **−** g A^A .

which is the torsion form, or wedge product of tetrads:

B^{c} =
− i g A^{a}^A^{b}
(78)

Eqn. (78) follows from the fact that a magnetic field must be a component of a torsion two-form (or antisymmetric second rank tensor) that is the signature of spin in general relativity. The concept of spin is missing completely from Einstein's general covariant theory of gravitation {9}. In the Evans unified field theory spin (or torsion) gives rise to the generally covariant electromagnetic field. Curvature of spacetime gives rise to the gravitational field, and the latter is the symmetric product of tetrads {1-8}. In the complex circular basis {11}

c = (3) , a = (1) , b = (2) . (79)

c) The factor g in eqn. (76) for the AB effect is:

g = ^{e}/_{h¯}
(80)

and is the ratio of the modulus of the charge on the electron (situated in the electron beams) to the reduced Planck constant.

For that g given by (80) the exponents in (81) won't be dimensionless.

d) The potentials **A**^{(1)}
= **A**^{(2)}* are defined by the cyclic relation (76) and extend
outside the area of the solenoid because they are components of a tetrad
multiplied by the scalar A^{(0) }. The tetrad is the metric tensor in
differential geometry, so **A**^{(1)} = **A**^{(2)}* are
properties of non-Euclidean (i.e. spinning) spacetime. This spacetime,
evidently, is not restricted to the solenoid.

e) The area Ar is the area enclosed {12} by the electron beams of the Young interferometer. The AB effect is therefore

ΔΦ
= exp(i ^{e}/_{h¯}
∫ **B**^{(3)} **·k**
dAr)
= exp(i ^{e}/_{h¯} f)
(81)

where f is
the magnetic flux produced by the solenoid of magnetic flux density **B**^{(3)}. The magnetic flux (weber = volts s^{−1})
is **B**^{(3)} (tesla) multiplied by the area Ar enclosed by the
electron beams.

The effect (81) is observed experimentally {12} as a shift in the interferogram of the Young interferometer, a shift caused by the iron whisker. It is seen that the AB effect is closely similar to the Sagnac effect and to Michelson interferometry and reflection in physical optics. All effects are described straightforwardly by the same phase law (76), which is therefore verified with great precision in these experiments. Recall that the Maxwell Heaviside, or U(1), phase factor (1) fails qualitatively in all four experiments (Section 2).

When we use the correctly covariant phase law of general relativity, eqn. (76), the AB effect becomes a direct interaction of the following potentials with the electron beam:

**A**^{(1)}
= **A**^{(2)}*
= ^{A(0)}/_{sqr(2)}
(**i** − i **j**) e^{iωt} .
(82)

These potentials define the static magnetic field of the iron whisker or solenoid as follows:

**B**^{(3)} = − ig **A**^{(1)} ×
**A**^{(2)}
(83)

>From (82-83) we obtain

^{i}/_{g A(0)} **B**^{(3)}
= **A**^{(1)}×**A**^{(2)}/ A^{(0)}
= − i A^{(0)} **k**

and from Evans' cyclic equations {The Enigmatic Photon Vol. 5, p. 14, (1.1.24)}
we know

**A**^{(3)}
= − i **A**^{(1)}×**A**^{(2)}/A^{(0)}
= − A^{(0)} **k** ,

i.e. ∫_{dS} **A**^{(3)} **·** d**r** = 0
for arbitrary closed pathes dS. But then (76) yields
∫_{S} **B**^{(3)}**·k** dAr
= ∫_{dS} **A**^{(3)}**·**d**r** = 0,
which due to **B**^{(3)}**·k**
= − g A^{(0)}^{2}
implies A^{(0)} = 0 and **B**^{(3)} = 0.

**That is a contradiction.** Thus, Evans method has failed,
while

In the specified case we have the well-known potential

**A** = ^{1}/_{2} B_{o} (−y **i** + x **j**)
for
r² = x² + y² __<__ R²
and
**A** = ^{1}/_{2} B_{o} (−y **i** + x **j**)
^{R²}/_{r²}
for
r² = x² + y² __>__ R²

up to an additive gradient. Therefore we obtain

**B** = curl **A** = B_{o} **k** inside the coil
and
**B** = curl **A** = **0** outside the coil.

∫

Of course, evaluation of the area integral yields the same result in accordance with the Stokes integral theorem

∫

where Ω denotes a surface the border of which is the path of integration ∂Ω.

where ω is an angular frequency defining the rate at which the potentials spin around the Z axis of the solenoid. The potential

**A**^{(1)} = **A**^{(2)}*
(84)

is defined
completely by these scalar magnitude A^{(0)},
the frequency, ω and the complex unit vectors:

**e**^{(1)} = **e**^{(2)}* .
(85)

The magnitude of the static magnetic field is:

B^{(0)} = g A^{(0)}²
(86)

and

**B**^{(3)} = B^{(0)} **e**^{(3)}
(87)

where

**e**^{(1)} × **e**^{(2)}
= i **e**^{(3)}*
= i **k**
et cyclicum
(88)

Therefore the
essence of the AB effect in general relativity is that it is an effect of
spinning spacetime itself, the spinning potential **A**^{(1)}
= A^{(2)}* extends outside the solenoid
(to whose Z axis is confined) and **A**^{(1)} = A^{(2)}*
interacts directly with the electron beam, giving the observed phase shift
(81). The spin of the spacetime is produced by the iron whisker, or static
magnetic field. Conversely, a static magnetie field is spacetime spin as
measured by the torsion form of the Evans theory {1-8} of generally covariant electrodynamics. This theory
completes the earlier Einstein theory of gravitation {9}.

This explanation has the advantage of simplicity (Okham's Razor) and there is no need for the obscure, and mathematically incorrect, assumptions used in the received view of the AB effect (Section 2). The explanation shows that the AB effect originates in general relativity with spin (the Evans unified field theory {l-8}). In analogy the AB effect is a whirlpool effect, the whirlpool is created by a stirring rod (the iron whisker) and the effect at the edges of the whirlpool is evidently measurable even though the stirring rod is not present there. The water of the whirlpool is the analogy to spinning spacetime. In the same type of analogy the Sagnac effect is a rotational effect caused by a physical rotation of the platform with respect to a fixed reference frame, whereas in the AB effect the magnetic field is a rotation or spin of spacetime (the reference frame itself). The Evans unified field theory gives a quantitative explanation of these whirlpool effects with a precision of up to 1 part on 10 (contemporary ring laser gyro {11, 13}.

Finally in this paper a brief discussion is given of the gauge invariance of the generally covariant phase Law (32). The Evans theory (1-8) is a theory of general relativity, in which the base manifold is non-Euclidean in nature, therefore covariant derivatives are always used from the outset instead of ordinary derivatives {9}. In both gravitation and electrodynamics the use of a covariant derivative implies that the Evans theory is intrinsically gauge invariant. The reason is that local gauge invariance in relativity theory is defined as the replacement of the ordinary derivative by a covariant derivative {12, 13). In the Maxwell Heaviside (or U(l) electrodynamics, the covariant derivative is the RESULT of a local gauge transformation of the generic gauge field ψ, a transformation under which the Lagrangian is invariant {12}. This local (or type two) gauge transformation is defined only in special relativity (flat spacetime) in Maxwell Heaviside theory and introduces the covariant derivative by changing the ordinary derivative to the U(1) covariant derivative:

∂_{μ} →
∂_{μ} − ig
A_{μ}
(89)

The introduction
of the U(1) Potential A in this way is equivalent to the well known minimal
prescription in the received view {12}, and so is equivalent to the interaction
of field with matter (i.e. of A_{μ} with e). The
shortcomings of this point of view, i.e. of the Maxwell Heaviside theory, are
reviewed briefly in Seetion 2, and elsewhere {1-8,11}. One of several problems
with the received view is that A_{μ} itself is not
gauge invariant (i.e. assumed not to be a physical quantity), yet is used in
the minimal prescription to represent the physical electromagnetic field. The
same type of problem appears in the U(1) phase through the introduction of the
random factor α (Section 2). In consequence there has been a long,
confusing, and misleading debate within U(1) electrodynamics as to whether or
not A_{μ} is physical {13}. This type of debate has
also bedevilled the understanding of the AB effect.

In the Evans
unified field theory the debate is resolved using general relativity by
recognising that A^{a}_{μ}
is a tetrad and therefore an element of a
metric tensor in a spinning spacetime. The gauge invariance of the Evans theory
is evident through the fact that it always uses covariant derivatives. In the
language of differential geometry this is the covariant exterior derivative D^
{9}, without which the geometry of spacetime is incorrectly defined. A magnetic
field for example is always (as we have seen) the covariant derivative of a
Potential form, which is a vector valued one-form. The correctly and generally
covariant magnetic field is therefore a vector valued two-form. The latter is
the torsion form of spacetime within a C negative scalar. This concept is
missing both from the Einstein theory of gravitation and the Maxwell Heaviside
theory of electrodynamics. It is the concept needed for a unified field theory
of gravitation and electromagnetism in
terms of general relativity and differential geometry.

In this paper we have developed the unified field theory into a novel phase law (32) that is gauge invariant and generally covariant. The phase law gives the first correct description of physical optics, interferometry, and related effects such as the Sagnac and Aharonov Bohm effects. If we use many loops of the Sagnac effect (11) we obtain the Tomita Chao effect, and applying the phase law (32) to matter waves, we obtain the Berry phase effects. This will be the subject of future communications.

ACKNOWLEDGEMENTS

Craddock Inc. and the Ted Annis Foundation are acknowledged for funding, and the staff of AIAS and others for many interesting discussions.

As argued in the text the Sagnac effect is a change in frequency caused by rotating the platform of the interferometer:

ω
→ ω __+__ Ω
(A1)

It is a shift in wavenumber:

κ
→ κ __+__ ^{Ω}/_{c}
(A2)

and thus in the following component of the potential:

A^{(3)}_{z}
→ A^{(3)}_{z} __+__
^{Ω}/_{ω} A^{(3)}_{z}
(A3)

In the unified field theory:

A^{a}_{μ
}= A^{(0)} v^{a}_{μ}
(A4)

and is the Sagnac effect is a shift in a tetrad component:

v^{(3)}_{z}
→ (1 __+__ ^{Ω}/_{ω})_{
}v^{(3)}_{z
}
(A5)

brought about by rotating the platform. The Sagnac effect is therefore one of general relativity applied to physical optics, and so cannot be explained by Maxwell Heaviside field theory, which is a theory of special relativity and so metric invariant {13}. Recall that the tetrad is the equivalent of the metric matrix in differential geometry. The components of the tetrad in the Evans unified field theory are clearly not those of special relativity.

{l) M. W. Evans, A
Unified Field Equation for Gravitation and Electromagnetism, Found. Phys.
Lett., 16, 367 (2003)

{2} M. W. Evans, A
Generally Covariant Wave Equation for Grand Unified Field Theory, Found. Phys.
Lett, Dec. 2003, preprint on
www.aias.us.

{3} M. W. Evans,
The Equations of Grand Unified Field Theory in terms of the Maurer Cartan
Structure Relations of Differential Geometry, Found. Phys. Lett., Feb., 2004,
preprint on www.aias.us

{4} M. W. Evans,
Derivation of Dirac's equation from the
Evans Wave Equation, Found. Phys. 1,ett., (2004, in press), preprint on www.aias.us.

{5} M. W.- Evans,
Unification of the Gravitational and Strong Nuclear Fields, Found. Phys. Lett.
2004, in press), preprint on www.aias.us.

{6} M. W.- Evans, The Evans Lemma of
Differential geometry (submitted for publication), preprint on www.aias.us.

{7} M. W. Evans:
Derivation of the Evans Wave Equation from the Lagrangian and Action. Origin of
the Planck Constant in General Relativity, Found. Phys. Lett., (submitted for
publication), preprint on www.aias-us

{8} M. W. Evans, Development of the Evans Equation in the Weak Field Limit, the Electrogravitic Equation, Found. Phys. Lett., submitted for publication.

{9) S. M. Carroll,
Lecture Notes in General Relativity, a graduate course in Univ. California, Santa
Barbara, arXiv:gr-qe/9712019 vl 3 Dec 1997.

{10} M. Tatarikis
et al., Measurements of the Inverse Faraday Effect in High Intensity Laser
Produced Plasmas, (CFL Annual Report, 1998/1999, EPSRC, Rutherford Appleton
Laboratories, m.tatarikis@ic.ac.uk).

{11} M. W. Evans, 0(3) Electrodynamics, in M. W.
Evans (ed.), Modern Nonlinear Optics, series eds. I. Prigogine and S. A. Rice, Advances in Chemical Physics (Wiley
Interscience, New York, 2001, 2^{nd} ed.), vol. 119(2), (e book cross
link on www.aias); L. Felker (ed.), The Evans Equations (World Seientific, in
press, 2004); M. W. Evans, J.-P. Vigier et al., The Enigmatic Photon (Kluwer,
Dordrecht, 1994 to 2002, hardback and paperback) ten volumes; M. W. Evans and
L. B. Crowell, Classical and Quantum Electrodynamics and the B^{(3)}
Field (World Scientific, Singapore,
2001); M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field
Theory (World Scientifie, Singapore, 1994).

{12} L. H. Ryder,
Quantum Field Theory (Cambridge Univ. Press, 1996, 2^{nd} ed.).

{13} T. W.
Barrett, in T. W. Barrett and D. M Grimes, Advanced Electrodynamics (World
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