(HTML version of http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.4433v1.pdf)

Friedrich W. Hehl, Institute for Theoretical Physics, University of Cologne, Germany,

and Department of Physics and Astronomy, University of Missouri-Columbia,

Arkadiusz Jadczyk, Château Saint Martin, 82100 Castelsarrasin, France, and

Center C.A.I.R.O.S, Laboratoire de Mathématiques Emile Picard,

Institut de Mathématiques de Toulouse, Université Paul Sabatier, France

We comment on a recent article of Evans [1]. We point out that the equations underlying Evans'
theory are highly problematic. Moreover, we demonstrate that the
so-called "spin connection resonance", predicted by Evans, cannot
be derived from the equation he used. We provide an exact solution
of Evans' corresponding equation and show that is has definitely
*no* resonance solutions.

PACS numbers: 03.50.Kk; 04.20.Jb; 04.50.+h

Keywords: Electrodynamics, gravitation, Einstein-Cartan theory, Evans' unified field theory

Over the last years, Evans' papers deal mainly with his so-called Einstein-Cartan-Evans (ECE) theory, which exists also under the former name "Generally covariant unified field theory" [2]. Evans aims at a fundamental unified field theory for physics. However, a long list of serious errors in his theory is well-known, see [3,4,5,6,7]. Evans never tried to take care of these errors and to improve his theory correspondingly. In fact, he believes that his theory is flawless.

In our opinion it is clear that Evans' theory has been disproved already and is untenable, both from a physical and a mathematical point of view. Nevertheless, he continues to publish papers and to predict new physical effects. In [1], Evans foresees a new "spin connection resonance" (SCR) effect. The aim of our article is to take a critical view on [1].

In Sec.2 we go through Evans' article [1] and point out numerous mistakes and inconsistencies in the set-up of his theory. Most of it is known from the literature [3,4,5,6,7]. In Sec.3 we turn to the new SCR effect, which Evans derives from a certain ordinary differential equation of second order. Even though the derivation of this equation is dubious, we start from exactly the same equation as Evans did and prove that this equation has no resonance type solutions as Evans claims. This shows that Evans' SCR effect is a hoax.

Evans' paper starts with what the author calls "the second Cartan structure equation",

R^{a}_{b} = D Ù
ω^{a}_{b} ,
((1))

and with the second Bianchi identity,

D Ù R^{a}_{b} := 0 .
((2))

The symbol DÙ stands, in Evans' notation, for the exterior
covariant derivative, ω and R are the connection and the
curvature forms, respectively. Eq.((1)) represents the *definition*
of the curvature form. The second structure equation,
which follows immediately from ((1)) and from the definition of D, is given as

R^{a}_{b} = d Ù
ω^{a}_{b} + ω^{a}_{c}
Ù ω^{c}_{b} .
((5))

The second Bianchi identity follows from ((5)) by exterior differentiation:

d Ù R^{a}_{b} +
ω^{a}_{c} Ù
R^{c}_{b} − R^{a}_{c}
Ù ω^{c}_{b} = 0.
((6))

Torsion is introduced according to

T^{a} = d Ù q^{a} +
ω^{a}_{b} Ù
q^{b} ,
((7))

with the tetrad one-forms q^{a}, which we interpret, according to the
context, as a local *orthonormal* coframe.

. . . Eq.((6)) can be rewritten as

d Ù R^{a}_{b} = j^{a}_{b} ,
((10))

d Ù R^{~ a}_{b} =
j^{~ a}_{b} ,
((11))

where

j^{a}_{b} = R^{a}_{c} Ù ω^{c}_{b}
− ω^{a}_{c} Ù
R^{c}_{b} ,
((12))

j^{~ a}_{b} = R^{~ a}_{c} Ù ω^{c}_{b}
− ω^{a}_{c} Ù
R^{~ c}_{b} .
((13))

The tilde denotes the Hodge dual [1-20] of the tensor valued two-form

R^{a}_{bμν} = − R^{a}_{bνμ} , . . .
((14))

While it is true that ((10)) and ((12)) are a rewriting of ((6)), this
is *false* for ((11)) and ((13)). Eqs.((11)) and ((13)) do not
follow from differential geometry. Especially the combination of
((11)) and ((13)), namely

d Ù R^{~ a}_{b} = R^{~ a}_{c}
Ù ω^{c}_{b} −
ω^{a}_{c} Ù R^{~ c}_{b} ,

cannot be derived from the second Bianchi identity ((6)) and does not
hold in general. Indeed, DÙR^{a}_{b}=0 does
*not* imply
DÙR^{~ a}_{b}=0 ,
since taking the Hodge dual doesn't commute with D.

Eqs.((17)) and ((18)) relate, according to Evans, a generalized
electromagnetic field strength F^{a} and a potential A^{a} to the
torsion and the tetrad, respectively,

F^{a} = *A*^{(o)} T^{a} ,
((17))

A^{a} = *A*^{(o)} q^{a} ,
((18))

where *A*^{(o)} is, presumably, a universal constant. Evans' next but one
equation is the first Bianchi identity,

d Ù T^{a} =
R^{a}_{b} Ù q^{b}
− ω^{a}_{b} Ù T^{b} .
((20))

Let us look at Evans' motivation for his choices ((17)) and ((18)).
Evans supposed an analogy of A^{a} and F^{a} with the Maxwellian
potential one-form A and the field strength two-form F according
to

A → A^{a} ,
F → F^{a} .
(1)

In Maxwell's theory, F = dÙA is then put in analogy to
Cartan's first structure equation (definition of the torsion) T^{a} =
DÙq^{a}.

One serious objection is based on the fact that Evans has not given
any information about the relations between the concrete
electromagnetic fields F=(**E**,**B**) in physics and his quadruple
of two-forms
F^{0}, F^{1}, F^{2}, F^{3}
and the associated quadruple of
one-forms
A^{0}, A^{1}, A^{2}, A^{3}.
Evans himself ignores that problem
of attaching a superscript 'a' to all electromagnetic field quantities
without giving a satisfying explanation of that index surplus.

Evans' attempts to interpret (1) appropriately doesn't
even work in the case of a simple circularly polarized plane (cpp)
wave. His considerations are contradictory and incomplete, and we see
no way to define
F^{0}, F^{1}, F^{2}, F^{3}
and
A^{0}, A^{1}, A^{2}, A^{3}
even for a bit more complicated field as, e.g., a superposition of
different cpp waves travelling in different directions. This is not a
mathematical error, but a physical gap, and we doubt that one can find
a general solution of that problem. Anyway, Evans never presented
such a solution.

Therefore, Evans' analogy F ↔ T^{a}, for a=0,1,2,3, causes a
*type mismatch* between the
* vector* valued torsion two-form T^{a} and the *scalar*
valued electromagnetic field strength two-form F. The analogous
holds for A ↔ q^{a} , for a=0,1,2,3 as well.

Evans' whole SCR paper is based on the dubious assumption that (1), and thus ((17)) and ((18)), make sense in physics. Without a concrete physical interpretation of (1), Evans' whole SCR paper is null and void, regardless whether there are other (mathematical) errors or not.

Moreover, as it was with the second Bianchi identity, so here, Evans' equations ((23)), ((16)), and ((17)), if combined, lead to

d Ù T^{~a} =
R^{~a}_{b} Ù q^{b}
− ω^{a}_{b} Ù T^{~b},.
(2)

Eq.(2), contrary to Evans' statement, is *not* a
consequence of the first Bianchi identity and does not hold in
Cartan's differential geometry. It represents an additional ad hoc
assumption.

Eq.((29)) is the field equation of Einstein's general relativity theory,

G^{μν} = k T^{μν} ,
((29))

after which Evans writes:

. . . Eq.(29) is well known, but much less transparent than the equivalent Cartan equation

D Ù ω^{a}_{b} =
k D Ù T^{a}_{b}
:= 0 . . .
((30))

Eq.((30)) is certainly *not* equivalent to ((29)), and it cannot
be a part of general relativity theory, be it tensorial or in
Cartan form. The reason is very simple: T^{a}_{b} in ((30)) has to
be a one-form. Therefore it should be integrated over a world-line
and not over a hypersurface of four-dimensional spacetime, as it is
done with the energy-momentum tensor. In other words, Eq.((30)) is
simply incorrect since the energy-momentum in exterior calculus is a
covector-valued three-form (or, if its Hodge dual is taken, a *
covector-valued* one-form).

Now Evans turns to the combined equation ((5)) and ((10)),

d Ù (d Ù
ω^{a}_{b} + ω^{a}_{c}
Ù ω^{c}_{b}) =
j^{a}_{b} ,
((31))

with his comment that *in vector notation* it gives, in
particular,

**Ñ ·
R**(orbital) = **J**^{0} ,
((32))

with

**R**(orbital) =
R^{0}_{1}^{01} **i**
+ R^{0}_{2}^{01} **j**
+ R^{0}_{3}^{01} **k**
((33))

It is evident that ((32)) is not equivalent to ((31)), if only for the simple reason that ((31)) involves a three-form, where all indices must be different from each other, while ((32)), with the divergence operator, involves summation over repeated indices. In ((37)), Evans evidently attempts to calculate the (0i) component of the curvature form:

**R**^{a}_{b} = − ^{1}/_{c}
^{∂ωab}/_{∂t}
− **Ñω**^{0a}_{b} −
ω^{0a}_{c} **ω**^{c}_{b} +
**ω**^{a}_{c} ω^{0c}_{b} .
((37))

This is again incorrect. In fact, starting from ((5)), the calculation
of the components (R_{0i})^{a}_{b} , for i=1,2,3 , yields

(R_{0i})^{a}_{b} =
∂_{b}(ω_{i})^{a}_{b} −
∂_{i}(ω_{0})^{a}_{b} +
(ω_{0})^{a}_{c} (ω_{i})^{c}_{b} −
(ω_{i})^{a}_{c} (ω_{0})^{c}_{b} .
(3)

Raising the index 0 of ω_{0} in the term
∂_{i}(ω_{0})^{a}_{b}, as Evans does,
is illegitimate,
because the metric component g^{00} of the Schwarzschild metric,
which Evans considers, is *not* a constant function of the
variables x^{i}. The sign in front of the time derivative in ((37)) is
also wrong.

Then in ((42)), when restricting to the static case, Evans 'forgets' one of the quadratic terms of his erroneous ((37)):

**R**^{a}_{b} = −
**Ñ**ω^{0a}_{b} +
**ω**^{a}_{c} ω^{0c}_{b} .
((42))

Again, this is wrong, since now **R**^{a}_{b} is not in the Lie
algebra of the Lorentz group. The same error applies to ((44)), where
ω^{0a}_{b} is substituted by Φ^{a}_{b} ,

**R**^{a}_{b} = −
**Ñ**Φ^{a}_{b} +
**ω**^{a}_{c} Φ^{c}_{b} .
((44))

Then Evans adds:

. . . It is convenient to use a negative sign for the vector part of the spin connection, so

**R**^{a}_{b} = − (**Ñ**
Φ^{a}_{b} + **ω**^{a}_{c} Φ^{c}_{b}) ...
((45))

This is another evident and grave error. Since the sign of the connection form is not a question of 'convenience' in the theory of gravity, where the curvature tensor contains both linear and quadratic terms in the connection. Changing the sign of the connection forms changes its curvature in an essential way.

Using incomprehensible and sometimes evidently wrong reasonings, such as skipping one term when going from ((37)) to ((42)), as we saw above, Evans postulates a potential equation ((63)) for an unidentified variable Φ for the case of the Schwarzschild geometry. We shall discuss the "electromagnetic analogue of Eq.(63)", namely Eq. ((65)), in the following section.

In the lines after ((31)), Evans writes:

"It is shown in this section that Eq.(31) produces an infinite number of resonance peaks of infinite amplitude in the gravitational potential [2-20]. To show this numerically, Eq.(31) is developed in vector notation . . . "

This is an unfounded claim followed by *no proof* and no numerical
results either. In addition the claim is erroneous as we shall see
below. At the very end of his article Evans at last arrives at the
topic `resonance' that is already announced in the title of his paper.
He reports:

. . . The electromagnetic analogue of Eq. (63) is

^{∂²φ}/_{∂r²} −
^{1}/_{r} ^{∂φ}/_{∂r} +
^{1}/_{r²} φ = −
^{ρ(0)}/_{εo} cos(κ r)
((65))

which has been solved recently using analytical and numerical methods [2-20]. These solutions for φ and Φ show the presence of an infinite number of resonance peaks, each of which become infinite in amplitude at resonance.

Evans' efforts (together with H. Eckardt) with respect to the resonance of ((65)) are available on his website. He attempts to find values of the parameter κ that yield resonances of the right hand side of ((65)) with the eigensolutions of this Euler type ordinary differential equation (ODE). However, the eigensolutions of the associated homogeneous ODE are well-known. The eigenspace is spanned by the special solutions

φ_{1} = r and
φ_{1} = r log r .
(4)

Resonance means that the driving term cos(κ r) belongs to the
eigenspace, i.e., is a linear combination, with constant coefficients,
of the functions φ_{1} and φ_{2} for any value of the parameter
κ. Obviously this is not the case.

Moreover, the general solution of ((65)) can be calculated. With the help of Mathematica, we obtain

φ(r) = c_{1} r + c_{2} r log r
− ^{ρ(0)}/_{εo}
^{r}/_{κ} Si(κr) ,
(5)

where Si( ) denotes the *sine integral* function defined by

Si(z) := ∫_{o}^{z sin t}/_{t} dt
(6)

for real z satisfying the estimate

| Si(z) | ≤ min(|z|,2) . (7)

The graph of Si(z) is displayed in Fig.1, Graph of the sine integral Si(z)

Thus, the κ dependent part of the solution (5) satisfies the estimate

| ^{ρ(0)}/_{εo}
^{r}/_{κ} Si(κr)|
≤ ^{ρ(0)}/_{εo} min (r²,
^{2r}/_{κ}) .
(8)

Consequently, the general solution of ((65)) is bounded for all real values of κ and r. For no value of κ, we will have a resonance of the right-hand-side of ((65)) with the eigensolutions (4).

However, Evans & Eckardt apply a lot of their specific 'new math': an
inadmissible rotation of the complex plane of eigenvalues by an angle
of 90° and multiplication by the imaginary unit *i*, among
other peculiarities, see [3] for details. Evans & Eckardt
succeed in detecting resonance peaks, unattainable to all who are
using standard mathematics only.

in Evans' theory previous to his equations ((63)) and ((65)).

[1] M.W. Evans, *Spin connection resonance in
gravitational general relativity*,

Acta Physica Polonica **B38** (2007) 2211-2220

[2] M.W. Evans, *Generally Covariant Unified Field
Theory: The Geometrization of Physics.*

Vols. 1 to 5, Abramis
Academic (2005,2006,2007); see also
http://www.aias.us/ and http://www.atomicprecision.com/

[3] G.W. Bruhn, *Remarks on Evans/Eckardt's Web-Note on Coulomb Resonance*,

http://www2.mathematik.tu-darmstadt.de/~bruhn/RemarkEvans61.html

[4] G.W. Bruhn, * web articles*,

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

[5] W.A. Rodrigues, Jr. and Q.A.~Gomes de Souza, *
An ambiguous statement called `tetrad postulate' and
the correct
field equations satisfied by the tetrad fields*,
Int.\ J.\ Mod.\
Phys.

http://arXiv.org/math-ph/0411085

[6] A. Jadczyk, * Remarks on Evans' "Covariant Derivatives"*,

http://www2.mathematik.tu-darmstadt.de/~bruhn/byArk260107.html

[7] F.W. Hehl, * An assessment of Evans' unified field theory I,*

F.W. Hehl and Yu.N. Obukhov, *An assessment of Evans' unified field theory II,*

both in Foundations of Physics, to be published (2007). See also

http://arXiv.org/abs/physics/0703116
and

http://arXiv.org/abs/physics/0703117