Last update: Sept 14 2005, 11:00 pm


Commentary on Myron Evans' paper

"THE ELECTROMAGNETIC SECTOR OF THE EVANS FIELD THEORY"

by Gerhard W. Bruhn, Darmstadt University of Technology (Sect.1-4)
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, Îμνρσ, 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 +1 and 0. Note that Carroll restricted his definition expressly to flat spacetime while Evans is considering "non Euclidean spacetime". However, when Carroll introduced the Hodge duality [3; p.23, (1.87)] on an n-dimensional manifold he failed to repeat the restriction of his former definition of the Î-"tensor" to flat spacetime. (Carroll gives a correct defintion several pages later on p.52.) Evans took that over from Carroll to his book [2; p.21,(A3)] without regard to the restriction to flat spacetime. However, his definition does not yield a tensor under general coordinate transforms and hence is erroneous. A factor g−½ (where g := |det (gαβ)|) is required to obtain the correct transformation behavior of a tensor:

Îmnrs = g−½ πmnrs

where πmnrs 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

dF = ∂αFβγ dxαdxβdxγ,

to obtain from dF = 0 due to dxαdxβdxγ ~ Îαβγ

αFβγ Îαβγ = 0 ;

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

F~μν = ½ Îμνρσ Fρσ ,                                                                 (10)

due to the Leibniz rule, we get

μF~μν = ½(∂μln g−½) Îμνρσ Fρσ = −¼(∂μln g) F~μν ,

i.e. Evans' equation

μF~μν = 0                                                                 (11)

is wrong for non-constant g.

Remark If one would replace the variable coefficients Îμνρσ in (10) with the constant coefficients πμνρσ (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 qaμ which is more fundamental than the metric tensor gμν used by Einstein because

gμν = qaμ qbν ηab                                                                 (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 (ηab) 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 ea (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 (qaμ), 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 dF = 0, d*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 PoincarÚ Lemma yielding the "continuity law" dJ = 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 Vμ of a vector V = Vμμ when being referred to the tetrad basis:

Va = qaμ Vμ .                                                 [3; p.89, (3.123)]

Similarly the (scalar valued) 2-form F = Fμν dxμdxν transforms according to

Fμν dxμdxν = Fab θaθb,

i.e. F remains a scalar valued 2-form, only related now to the basis 1-forms

θa = qaμ dxμ

of the tetrad.

Finally, the scalar valued 3-form J = Jμνρ dxμdxνdxρ transforms to the scalar valued 3-form Jabc θaθbθc.

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

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

We learn at the top of p.264 that the scalar valued 2-form F has to be replaced by Fa, i.e. due to a=0,1,2,3 by the quadruple of the four scalar valued 2-forms F0, F1, F2, F3, or in terms of Evans, the scalar valued 2-form F has to be replaced by a vector valued 2-form, The analogue applies to the other forms too: All involved scalar valued forms are replaced by quadruples of those.

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:

DF = 0 ,                                                                 (31)

. . .

and in the MH limit

DF(a) → dF                                                                 (33)

. . .

i.e. we have dF = 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

DF(a) → dF(a)                 (a= 1,2,3).                                                 (33')

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

Đ · B(a) = 0                 (a=1,2,3).                                                (35'-37')

Đ × 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:

Đ · B(1) = 0 ,                                                                 (35)

Đ · B(2) = 0 ,                                                                 (36)

Đ · B(3) = 0 ,                                                                 (37)

Đ · E(1) + ∂B(1)/∂t = 0 ,                                                         (38)

Đ · 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(ii j), e(2) = 2−½(i + i j), e(3) = k                                   (1.1.1)

({i,j,k} = orthonormal basis of R³, 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) = Az 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 = xi+yj+zk0

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 = Đ·B = Đ·B(1) + Đ·B(2) + Đ·B(3) and evidently Đ·B(3) = ∂zz/|r|³ ≠ 0, hence also Đ·B(1) +Đ·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.

Thus, Evans' theory leads to contradictions even in very simple concrete applications.


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 BΠ. 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 BΠ 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 Evansĺ 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 BΠ 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 angularŚmomentum 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 ôdefinitive evidenceö of the existence of B(3) [20], and a competing ôconventional calculationö 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 ôdoes not exist in the form described by Evansö [34]. Evansĺ 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 ôall three experiments clearly disprove the claims of BΠ-theoryö 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

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

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[5]    K. Kleiner, Most scientific papers are probably wrong,
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[39]    R.N. Compton (University of Tennessee), personal communication to A. Lakhtakia (Sept. 8, 2005).