Last update: Sept 14 2005, 11:00 pm

and Akhlesh Lakhtakia, Pennsylvania State University (Sect.5)

**Abstract**

A recent publication [1] contains a new view of electromagnetism according to the so-called
Evans field theory.
We show that the electromagnetic sector of this theory is seriously flawed for the
following reasons:
An ultraspecial version of the Î-tensor is used that applies
only for the case of constant metric determinant g (Sect.1).
The tetrad concept has been misapplied (Sect.2).
Finally, it leads to strange consequences (Sect.3),
has a serious internal contradiction (Sect. 4),
and is undermined by the negative experimental evidence of the B(3) field (Sect.5).

Introduction

The so-called Evans field theory, also known as the Generally Covariant Unified Field Theory, has been explicated in numerous papers published over a decade in this journal. Our focus here is solely on a recent publication [1] focusing on electromagnetism within the GCUFT. We show here that the electromagnetic sector of the GCUFT is theoretically flawed and counter-indicated by published experimental evidence.

Incidentally, Evans' article [1] is almost identical with Chap. 13 of his book manuscript [2], which has evidently not been published except on the web. The article being a copy of that book chapter 13 explains the 57 equation numbers misprinted as (13.xx) instead of (xx).

1. Evans' Î-tensor

On [1; p.261 below] we read

. . . the totally antisymmetric unit tensor in four dimensions,
Î^{μνρσ},
has been used as usual. This tensor is **the same for any non-Euclidean** spacetime
[Carroll**???**].

What is a *totally antisymmetric unit tensor*?
Evans refers to a work of S.M. Carroll [3] wherein the Î-tensor
was defined at p.16 as
the permutation symbol with the values

Î^{mnrs}
=
g^{−½}
π^{mnrs}

where π^{mnrs} is the "sign" of the permutation
(μνρσ), i.e. +1 for even permutations, −1 for odd permutations
and 0 otherwise.
Evans always assumes g = const for the Î-tensor.

By direct evaluation we obtain

dÙF
=
∂_{α}F_{βγ}
dx^{α}Ùdx^{β}Ùdx^{γ},

∂_{α}F_{βγ}
Î^{αβγ} = 0 ;

hence, as the Hodge dual *F (= F^{~} in Evans' notation) is defined by

F^{~μν}
=
½
Î^{μνρσ}
F_{ρσ} ,
(10)

due to the Leibniz rule, we get

∂_{μ}F^{~μν}
=
½(∂_{μ}ln g^{−½})
Î^{μνρσ}
F_{ρσ}
=
−¼(∂_{μ}ln g)
F^{~μν} ,

i.e. Evans' equation

∂_{μ}F^{~μν}
=
0
(11)

is **wrong for non-constant g**.

**Remark** If one would replace the variable coefficients
Î^{μνρσ}
in (10) with the constant coefficients π^{μνρσ}
(as Evans does), then (10) would not yield
a tensor.

2. Evans' tetrad concept

Let us discuss now the main basic tool of Evans' GCUFT:

. . . The second major advantage is that differential geometry
is developed in terms of the vector-valued tetrad one form q^{a}_{μ}
which is more fundamental than the metric tensor g_{μν} used by Einstein
because

g_{μν}
=
q^{a}_{μ}
q^{b}_{ν}
η_{ab}
(17)

In other words, the metric tensor is defined as the dot product of two (different???) tetrads, so the tetrad factorizes the metric tensor of the base manifold (non-Euclidean spacetime).

Here (η_{ab}) denotes the Minkowski diagonal matrix
diag(1,−1,−1,−1).

The statement "the metric tensor is defined as the dot product of two
tetrads" shows a misunderstanding of the tetrad concept that was
correctly introduced by S.M. Carroll [3; p.88]: Carroll defined the **tetrad to be a
local frame attached to the points of spacetime**, i.e. a set of 4 basis vectors
**e**_{a} (a= 0,1,2,3)
being orthonormal in a certain sense with the advantage of being not related to any
coordinate system.
That concept goes back to J.G. Darboux (around 1880) as the "method of moving frames"
on a surface and was later used by H. Cartan for his work on manifolds.
In Evans' understanding however, a tetrad is only a coefficient scheme
(q^{a}_{μ}), the tetrad index a (correctly the basis vector index)
giving rise to dubious further definitions (Sect.3).

Evidently, Evans intends to develop a new physical theory,
namely a generalization of General Relativity. Such a theory is a system of rules
to be obeyed by the physical observables, viz.
− the electric and magnetic field vectors **E** and **B**,
and the source densities **j** and ρ, in the case of electromagnetics.

Even in a very elegant formulation of the Maxwell equations, in the
equations dÙF = 0,
dÙ*F = J, we can identify these observables: We know
that the 2-form contains the field vectors **E** and **B** as coefficients
with a very clear experimental meaning:
The 2-form F is source-free; its integral over an arbitrary closed
surface, i.e. the flux of F through that surface, is always null-valued.
The 3-form J represents the observables **j** and ρ, with
the Poincaré Lemma yielding the "continuity law" dÙJ = 0.

Maybe, a new theory will modify these rules. However,
**we must retain rules of the physically measurable quantities, the observables,**
even if other quantities as gravitation, for example, should appear in addition.

3. The consequences

What about the observables in Evans' theory?

S.M. Carroll [3; p.89, (3.123)] describes accurately what happens to the
components V^{μ} of a vector
**V** = V^{μ}**∂**_{μ}
when being referred to the tetrad basis:

V^{a}
=
q^{a}_{μ} V^{μ}
.
[3; p.89, (3.123)]

Similarly the (scalar valued) 2-form
F = F_{μν}
dx^{μ}Ùdx^{ν}
transforms according to

F_{μν}
dx^{μ}Ùdx^{ν}
=
F_{ab}
θ^{a}Ùθ^{b},

θ^{a}
= q^{a}_{μ} dx^{μ}

of the tetrad.

Finally, the scalar valued 3-form
J = J_{μνρ}
dx^{μ}Ùdx^{ν}Ùdx^{ρ}
transforms to the scalar valued 3-form
J_{abc}
θ^{a}Ùθ^{b}Ùθ^{c}.

That's what should correctly happen to the involved forms when the reference frame is changed from the
(coordinate-dependent) basis {dx^{μ} | μ=0,1,2,3} to the tetrad 1-form basis
{θ^{a} | a=0,1,2,3}.

**What, however, happens to the observables in Evans' theory?**

Some questions arise immediately: What does the foregoing mean with respect to the observables contained in F? Do we have quadruples of the respective observables now? The answer is YES as can be seen by the Equ. (35-40) in [1; p.265] (the fourth components with index (0) are missing there). Which of the quadrupled observables is the correct one because it appears in experiments, and why not the other ones?

Evans claims quadrupling of observables especially for the case when gravitation is present. However, on Earth, we are living on the surface of a sphere under a virtually constant gravitational acceleration of 9.81 m/s². And all experimental physicists know that the usual Maxwell equations are a very good approximation to what they recognize in their experiments: They don't observe tripled or quadrupled electric or magnetic fields or quadrupled current densities.

The eqns. (35-40) at the bottom of [1; p.265] confirm Evans' misinterpretation. That (strange) quadrupling of observables will lead us to a serious contradiction in Evans' theory in the following section.

4. A severe internal contradiction

On p.265 we read:

When the electromagnetic and gravitational fields decouple:

DÙF = 0 , (31)

. . .

and in the MH limit

DÙF^{(a)}
→
dÙF
(33)

. . .

i.e. we have dÙF = 0, which is the coordinate independent formulation of the homogeneous Maxwell equations

Ñ**·B** = 0 ,
Ñ×**E**
+
∂**B**/∂t = **0** .

However, what we read in [1; p.265, (35-40)] is really the consequence of

DÙF^{(a)}
→
dÙF^{(a)}
(a= 1,2,3).
(33')

This modification (33') of Evans' Formula (33) agrees with the corrected formulation:

Ñ **· B**^{(a)} = 0
(a=1,2,3).
(35'-37')

Ñ × **E**^{(a)}
+
∂**B**^{(a)}/∂t
= 0
(a=1,2,3):
(38'-40')

However, Evans' text is evidently a garbled version of that (marked in red; compare the Eqns. (38-40) with our Eqns.(38'-40'), cf also [1a; Eqns.(38-40)]):

. . . In vector notation, Eq.(13.31) becomes the following six equations:

Ñ **· B**^{(1)} = 0 ,
(35)

Ñ **· B**^{(2)} = 0 ,
(36)

Ñ **· B**^{(3)} = 0 ,
(37)

Ñ **· E**^{(1)}
+
∂**B**^{(1)}/∂t
= 0 ,
(38)

Ñ **· E**^{(2)}
+
∂**B**^{(2)}/∂t
= 0 ,
(39)

**??? + **
∂**B**^{(3)}/∂t
= 0 .
(40)

The indices (1), (2), (3) refer to Evans' *complex circular basis* (cf. [1; p.264])
introduced in his book [4; p.7-14]. There, on p.7, we find the definitions

**e**^{(1)} = 2^{−½}(**i** − *i* **j**),
**e**^{(2)} = 2^{−½}(**i** + *i* **j**),
**e**^{(3)} = **k**
(1.1.1)

({**i**,**j**,**k**} = orthonormal basis of
**R**³, *i* = imaginary unit)

yielding

**A**
=
**A**^{(1)}
+
**A**^{(2)}
+
**A**^{(3)}
=
A^{(1)}**e**^{(1)}
+
A^{(2)}**e**^{(2)}
+
A^{(3)}**e**^{(3)} ,
(1.1.5)

where A^{(3)} = A_{z} is the z-coordinate of the arbitrary vector
**A** due to (1.1.6).

With these formulas of his the author Evans should check the following example, the (trivial) solution of the time independent Maxwell equations (hence Equ.(31) is fulfilled):

**E** = **0**,
**B** = γ ^{r}/_{|r|³}
where **r** = x**i**+y**j**+z**k** ≠ **0**

with some constant γ > 0.

The Eqns. (38-40) (and (38'-40') as well) are trivially satisfied, while the Eqns.
(35-37) are not:
We have
0 = Ñ**·B**
=
Ñ**·B**^{(1)}
+
Ñ**·B**^{(2)}
+
Ñ**·B**^{(3)}
and evidently
Ñ**·B**^{(3)}
=
∂_{z}^{z}/_{|r|³}
≠ 0, hence also
Ñ**·B**^{(1)}
+Ñ**·B**^{(2)}
≠ 0, which contradicts Evans' Eqns.(35-37).

The same would turn out for other simple solutions of the original Maxwell
equations with the exception of plane waves due to the special properties of
a plane transversal wave. However, already a superposition of two plane waves with
non-parallel directions of propagation will even cause a bigger problem:
The direction (3) is not well-defined.
Then the Eqns. (35-40) become *completely senseless*.

5. Negative experimental evidence

Evans proposed in 1992 that circularly polarized plane waves (and photons) in vacuum
are accompanied by an "elementary static magnetic field", but which "is not interpretable
as an ordinary uniform, magnetostatic field" [6]. This field was denoted by the symbol **B**_{Π}.
Various actual and possible effects were ascribed to this field, including the optical
Zeeman effect [7], the inverse Faraday effect [8], and the optical Faraday effect and
optical magnetic circular dichroism [9]. Such a field cannot be a solution of the Maxwell
equations. This was pointed out by Barron on the basis of charge conjugation symmetry [10],
but not accepted by Evans [11]. Lakhtakia showed that **B**_{Π} can be defined for elliptically
polarized plane waves as well, but it cannot be considered "fundamental" because it is
merely an analytical construct created by multiplying the product **E**×**E*** by a constant
with suitable units and the time-domain interpretation of **E**×**E*** is ambiguous [12].
Furthermore, as Grimes [13] and van Enk [14] also pointed out, any optical effects
in vacuum could be explained better by resorting to the well-established angular momentum.
This suggestion was rejected by Evans [15]. The conclusion that this field, by virtue of
Evans’ definition is independent of both space and time and is therefore unknowable [16]
was also rejected by Evans [17], who went on to write a comprehensive but dense, terse
and confusing reply to his theoretical critics [18].

By that time, Evans had replaced the symbol **B**_{Π} by **B**^{(3)} [19, 20] and invoked the so-called
B cyclic relations. Theoretical criticisms mounted but Evans defended against every criticism
[21] - [27], and inundated Foundations of Physics Letters with a plethora of publications
culminating in the paper under review here.

The inverse Faraday effect in actual materials is not in doubt, nor is the angular—momentum explanation for it, with the angular momentum sometimes interpreted analogously to an effective magnetic field. But that effective magnetic field is not an actual magnetic field, as van Enk [14] has carefully stated. Most importantly, that field does not exist in vacuum.

Spectral shifts observed by Warren et al. [28] in the NMR resonances of certain materials
exposed to circularly polarized light were proffered by Evans as “definitive evidence” of the
existence of **B**^{(3)} [20], and a competing “conventional calculation” by Harris and Tinoco
[29] for the same shifts was dismissed. Warren et al. [30] agreed that the effects were
small and could mostly be explained conventionally; most importantly, they emphatically
dismissed the idea that they had found any evidence of **B**^{(3)}. Additional consideration of the
spectral shifts by Buckingham and Parlett [31, 32], who also echoed the conclusion against
the effect of **B**^{(3)}, was dismissed by Evans [33].

Rikken actually set up an experiment to specifically measure optical Faraday effect purportedly due to the longitudinal magnetic field of a circularly polarized laser beam. He concluded that such an effect “does not exist in the form described by Evans” [34]. Evans’ commented that the conditions chosen by Rikken for his experiment were not sufficiently appropriate [35].

Raja et al. set up three different experiments to test the existence of **B**^{(3)} in vacuum via
photomagnetic induction, Faraday effect, and inverse Faraday effect [36].
The concluded that “all three experiments clearly disprove the claims of **B**_{Π}-theory” and
that "such fields are non-existent", but Evans continued to insist on the existence
of **B**^{(3)} [37]. Negative evidence against **B**^{(3)} in vacuum was also experimentally obtained by
Compton [38,39].

As **B**^{(3)} does not exist in vacuum, then a circularly polarized plane
wave cannot give rise to an Aharonov-Bohm effect.

Concluding Remarks

Thus we have shown that the electromagnetic sector of the GCUFT is not only theoretically flawed but is also counter-indicated by experimental evidence on the existence of B(3). The flaws are fatal, and the negative experimental evidence is overwhelming.

References

[1]
M.W. Evans, *The electromagnetic sector of the Evans Field Theory*,

Foundations of Physics Letters, Vol.18 No.3, p.259-273 (2005)

[1a]
M.W. Evans, *The electromagnetic sector of the Evans Field Theory*,

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

[2]
M.W. Evans, *Generally Covariant Unified Field Theory,
the geometrization of physics*,

http://www.aias.us/Comments/Evans-Book-Final.pdf

[3]
Sean M. Carroll, *Lecture Notes on General Relativity*,

http://arxiv.org/pdf/gr-qc/9712019

[4]
M.W. Evans, *The Enigmatic Photon Vol.5*,

1999 Kluwer Academic Publishers

[5]
K. Kleiner, *Most scientific papers are probably wrong*,

NewScientist.com news service,

http://www.newscientist.com/article.ns?id=dn7915

[6] M.W. Evans, *The elementary static magnetic field of the photon*,

Physica B 182 (1992) 227-236.

[7] M.W. Evans, *On the experimental measurement of the
photon's fundamental static magnetic field operator B _{Π}: the optical Zeeman effect*,

Physica B 182 (1992) 237-243.

[8] M.W. Evans, *The photon's magnetostatic flux density B _{Π},
the inverse Faraday effect revisited*,

Physica B 183 (1993) 103-107.

[9] M.W. Evans, *The optical Faraday effect and optical MCD*,

J. Molec. Liq. 55 (1993) 127-136.

[10] L.D. Barron, *Charge conjugation symmetry and the nonexistence of
the photon's static magnetic field*,

Physica B 190 (1993) 307-309.

[11] M.W. Evans, *Reply to Comment: "Charge conjugation symmetry and
the nonexistence of the photon's static magnetic field"*,

Physica B 190 (1993) 310-313.

[12] A. Lakhtakia, *Does the photon have an elementary magnetostatic
flux density? (I) Plane waves*,

Physica B 191 (1993) 362-366.

[13] D.M. Grimes, *Does the photon have an elementary magnetostatic
flux density? (II) A phasor description of photon fields*,

Physica B 191 (1993) 367-371.

[14] S.J. van Enk, *Is there a static magnetic field of the photon?*,

Found. Phys. Lett. 9 (1996) 183-190.

[15] M.W. Evans, *One photon and macroscopic B cyclic equations:
Reply to van Enk*,

Found. Phys. Lett. 9 (1996) 191-190.

[16] A. Lakhtakia, *Is Evans' longitudinal ghost field B ^{(3)}
unknowable?*,

Found. Phys. Lett. 8 (1995) 183-186.

[17] M.W. Evans, *Reply to A. Lakhtakia: Experimental measurement of
B ^{(3)}* Found. Phys. Lett. 8 (1995) 187-193.

[18] M.W. Evans, *Reply to the criticisms of the B ^{(3)} field*,

Found. Phys. Lett. 8 (1995) 563-573.

[19] M.W. Evans, *The photomagneton B ^{(3)} and longitudinal
ghost field B^{(3)} of electromagnetism*,

Found. Phys. Lett. 7 (1994) 67-74.

[20] M.W. Evans, *The photomagneton B ^{(3)} and
electrodynamic conservation laws*,

Found. Phys. Lett. 7 (1994) 209-217.

[21] E. Comay, *Comment on the longitudinal magnetic field of
circularly polarized electromagnetic waves*,

Chem. Phys. Lett. 261 (1996) 601-604.

[22] E. Comay, *Unphysical properties of the longitudinal-phase-free
magnetic field of circularly polarized electromagnetic waves*,

Physica A 242 (1997) 522-528.

[23] E. Comay, *Relativity versus the longitudinal magnetic field of
the photon*,

Found. Phys. Lett. 10 (1997) 245-254.

[24] M.W. Evans, *Reply to Comay's "Relativity versus the longitudinal
magnetic field of the photon*,

Found. Phys. Lett. 10 (1997) 255-271.

[25] V.V. Dvoeglazov, *Speculations on the Evans-Comay discussion*,

Apeiron 6 (1999) 227-232.

[26] E. Comay, *A reply to V.V. Dvoeglazov*,

Apeiron 6 (1999) 233-236.

[27] G. Hunter, *The nature of the B ^{(3)} field*,

Chem. Phys. 242 (1999) 331-339.

[28] W.S. Warren, D. Goswami, S. Mayr and A.P. West Jr,
*Laser-Enhanced NMR Spectroscopy*,

Science 255 (1992) 1683-1685.

[29] R.A. Harris and I. Tinoco, Jr., *Laser-enhanced NMR spectroscopy:
Theoretical considerations*,

Science 259 (1993) 835-836,

[30] W.S. Warren, D. Goswami, and S. Mayr, *Laser enhanced NMR
spectroscopy*,

revisited, Molec. Phys. 93 (1998) 371-375.

[31] A.D. Buckingham and L.C. Parlett, *High-Resolution Nuclear Magnetic Resonance Spectroscopy in a Circularly
Polarized Laser Beam, Science 264 (1994) 1748-1750
*

*[32] A.D. Buckingham and L.C. Parlett, The effect of circularly
polarized light on ESR spectra, Chem. Phys. Lett. 243 (1995) 15-21.
*

*[33] M.W. Evans, Optical NMR from the Dirac equation: A reply to
Buckingham and Parlett, Found. Phys. Lett. 9 (1996) 175-181.
*

*[34] G.L.J.A. Rikken, Nonexistence of the optical Faraday effect, Opt. Lett. 20 (1995) 846-847.
*

*[35] M.W. Evans, Proof of the Evans-Vigier Field from the Dirac
equation of the fermion in the classical field: Reply to Rikken,
Found. Phys. Lett. 9 (1996) 61-66.
*

*[36] M.Y.A.Raja, W.N. Sisk, M. Yousaf, and D. Allen,
In search of photon's elementary axial magnetostatic field, Appl. Phys. B 64 (1997) 79-84.
*

*[37] M.W. Evans, Rebuttal of M.Y.A. Raja et al., Apeiron 4 (1997) 94-95.
*

*[38] M.Y.A. Raja, D. Allen, and W. Sisk, Room-temperature inverse Faraday effect in terbium gallium garnet, Appl. Phys. Lett. 67 (1995) 2123-2125.
*

*[39] R.N. Compton (University of Tennessee), personal communication to A. Lakhtakia (Sept. 8, 2005).
*