Last Update: Jan 17 2007

The Electromagnetic and Gravitational Fields In The Evans Unified Field Theory"

Abstract

Some serious objections to Evans' GCUFT are found in Chap.17 of the book [4] and
presented here:

- The misunderstanding of the tetrad concept. (**Sect.1**)

- The failing of Evans' outer product of tetrads to construct an antisymmetric metric. (**Sect.2**)

- The severe inconsistencies of Evans' two "Tetrad Postulates" and

the "Metric Compatibility Condition", which turns out to be invalid in general. (**Sect.3**)

- The non Lorentz invariance of the associated tetrad to a plane wave. (**Sect.4**)

- The non-reality of the metric associated to Evans' Complex Circular Base. ((**Sect.5**))

Introduction

The paper [1] that is to appear in Springer's FPL is copy of Chap.17 of just that book [4]
that Springer's wise referees declined a few months ago. With exception of the equation numbering
we could not find any differences between book Chap.17 [4] and paper [1].
Even some evident misprints survived the FPL refereeing, see **Appendix 2**.
−
Evans denotes that chapter as a "cornerstone" of his theory.
Nevertheless we shall point out several serious flaws in the following sections
with consequences for the whole GCUFT.

For a later discovered error in the Introduction of Evans' paper the reader is
kindly requested to click at

A Remark on Evans' 2nd Bianchi Identity

All equation numbers without reference [...] refer to Evans web-published book manuscript [4]. For convenience we have quoted those equations here. Sometimes slight modification of an Evans equation were necessary: In that case an apostrophe ' is added to the equation number: (xxx) is the original equation and (xxx') our modification.

Original quotations from Evans' book manuscript appear in **black**
, our comments are colored.

1. Misunderstanding the Tetrad Concept

Quote from M.W. Evans [4; p.302f.]:

The type of Riemann geometry almost always used by Einstein
and others [2] for generally covariant gravitational field theory is a
special case of the more general Cartan differential geometry [2] in
which the connection is no longer symmetric and which *the metric is in
general the outer or tensor product of two more fundamental tetrads*
q^{a}_{μ}.

g_{μν}
= q^{a}_{μ} q^{b}_{ν}^{} η_{ab} ,
(17.10)

η_{ab} = diag (−1, 1, 1, 1) .
(17.11)

The sentence "*the metric is in
general the outer or tensor product of two more fundamental tetrads*
q^{a}_{μ}"
shows at first a strange understanding of the tetrad concept that was criticized already
elsewhere [3]: A tetrad is a coordinate independent frame, i.e. set of four tangent vectors
**e**_{a} (a=0,1,2,3) of the spacetime manifold defined for the diverse points
of spacetime.
The relation to the coordinate dependent tangent vectors vectors
**e**_{μ} = ∂_{μ}
(μ=0,1,2,3)

**e**_{μ}
=
q^{a}_{μ}
**e**_{a}

2. The Antisymmetric Metric

Quote from the break before equ.(9)/(17.10):
. . . in which the connection is no longer symmetric (ok.)
and which the metric is in general the outer or tensor product
of two (???) more fundamental tetrads q^{a}_{μ}.
Thus, in differential geometry the metric tensor is in general an
asymmetric matrix. Any asymmetric matrix is always the sum of a
symmetric matrix and an antisymmetric matrix [3], so it is possible to
construct an antisymmetric metric tensor. The symmetric metric used
by Einstein to describe gravitation is therefore the special case defined
by the inner or dot product of two tetrads [2]:

The outer product
q^{a}_{μ}Ä q^{b}_{ν}
as proposed by M.W. Evans *depends on the choice of the tetrad field* attached to the manifold, while a metric should
be *independent of the choice of the tetrads*. Therefore the tetrad indices a,b must be removed.
The only possibility to do that is to multiply the outer product by the corresponding tetrad
vectors, namely by the scalar products
e_{a}**·** e_{b} = η_{ab}. This way we obtain
the tetrad independent tensor
η_{ab} q^{a}_{μ}Ä q^{b}_{ν}
the symmetric part of which is the well-known *symmetric* metric tensor

g_{μν} =
η_{ab} q^{a}_{μ}q^{b}_{ν}
=
η_{ab} ½
(q^{a}_{μ}Ä q^{b}_{ν} +
q^{a}_{ν}Ä q^{b}_{μ})

η_{ab} q^{a}_{μ}Ù q^{b}_{ν}
=
η_{ab} ½
(q^{a}_{μ}Ä q^{b}_{ν} −
q^{a}_{ν}Ä q^{b}_{μ})

That is confirmed by the fact that Equ.(17.10) yields a *symmetric* metric tensor g_{μν}
= g_{νμ}.

3. The Tetrad Postulate and the Metric Compatibility

We come to another quote from p.306:

In the Einstein theory the tetrad postulate [2] of differential geometry:

D_{ν} q^{a}_{μ}
= 0
(17.32)

is specialized to the metric compatibility condition [2] for a symmetric metric:

D_{ν} g^{μρ}
=
D_{ν} g_{μρ} = 0.
(17.33)

The attached is a simple and powerful demonstration that the tetrad postulate is a fundamental property of Cartan differential geometry, a property that IMPLIES the less general metric compatibility condition of Riemann geometry used by Einstein and Hilbert in their famous theory of general relativity (1915 to 1916). The tetrad postulate leads directly to metric compatibility and the former is more fundamental, because it relies on no assumptions concerning the connection.

And in [4; p.65]:

The tetrad postulate: D_{ν}e^{a}_{μ}
= 0, ... ,
where D_{ν} denotes the covariant derivative [2], is **true for any connection,
whether or not it is metric compatible or torsion free.**

However, Evans' notation is *not unique*: The quantity e^{a}_{μ}
has later in the book converted to q^{a}_{μ}
and denotes tetrad coefficients. And more important:
**Also the derivative D _{ν}q^{μ}_{a} is not uniquely defined:**
In the first part of his book manuscript [4] (Chap.2 - 7) he uses the version (1.59)
(after tacit correction of an index exchange error at
Γ here named (1.59'))

D_{ρ}q^{μ}_{a}
=
∂_{ρ}q^{μ}_{a}
+
Γ^{μ}_{ρσ}q^{σ}_{a}
(1.59')

which, as we shall see, restricts the spacetime manifold to an uninteresting special case: Using the well-known tetrad identity

(I)
∂_{ν}q^{μ}_{a}
− ω^{b}_{νa}
q^{μ}_{b}
+ Γ^{μ}_{νλ}q^{λ}_{a}
= 0

we obtain from Evans' *first* "tetrad postulate"

D_{ν}q^{μ}
= ∂_{ν}q^{μ}
+ Γ^{μ}_{νλ} q^{λ}
= 0
(2.182)

the result

ω^{a}_{μb}
= 0 for all index combinations.

Thus, the "tetrad postulate" D_{ρ}q^{μ} = 0 is valid if and only if
the manifold ** M** is equipped with a tetrad field such that
ω

**The second version** of D_{ν} appears first at [4; p.149]:

The Evans wave equation and lemma are derived [1-3] from the tetrad postulate of differential geometry [9,10]:

D_{μ} q^{a}_{ν}
= ∂_{μ}q^{a}_{ν}
+ ω^{a}_{μb}
q^{b}_{ν}
− Γ^{λ}_{μν}q^{a}_{λ}
= 0
(8.13)

without any further word of introduction or explanation of the contrast of (8.13) to (1.59').

An **important disadvantage of that second covariant derivative (8.13)** is that
now Evans' proof of the metric compatibility (2.170) (cf. [6])

Applying the Leibniz rule to
g_{μν}
=
η_{ab} q^{a}_{μ} q^{b}_{ν}
yields

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

According to S.M. Carroll [2; p.91] one has

D_{ρ}q^{a}_{μ}
= ∂_{ρ}q^{a}_{μ}
+ ω^{a}_{ρb}
q^{b}_{μ}
−
Γ^{σ}_{ρμ}
q^{a}_{σ}
= 0 ,

which due to the preceding equation yields the metric compatibility
D_{ρ} g_{μν} = 0 , q.e.d..

becomes erroneous
since in that case D_{μ}η_{ab} does not vanish,
[6; (7)] is wrong. So nothing has been proved:

Wherever M.W. Evans refers to "metric compatibility" in his GCUFT
(a search in the GCUFT book manuscript yields about 20 spots) it should be remembered
that this is an *important additional condition on the "γ" connection of the
spacetime manifold that is not fulfilled in general*.

The author M.W. Evans should feel challenged to check the consequences of that important
restriction. Since D_{ρ}q^{a}_{μ} has an ambiguous meaning
in his book manuscript it is necessary to reconsider the
consequences for the diverse spots of GCUFT under concern.
Clear, however, is that Evans' first "tetrad postulate" (2.182),
D_{ν} q^{a}_{μ} = 0, and the metric compatibility condition
(17.33),
D_{ρ} g_{μν} = 0, **are not valid in general**.

Therefore *all nice conclusions from these equations in GCUFT (at about 20 spots)
have lost their justification now and have become dubious.*

4. Evans' Complex Circular Base

Another Quote from M.W. Evans [1; p.307]:

The tetrad is defined for the electromagnetic field by:

V^{a} = q^{a}_{μ} V^{μ}
(17.35)

where V^{μ} is a vector in the base manifold, and where V^{a} is a vector in the
tangent bundle. The tetrad is therefore the four by four invertible transformation
matrix [2] between base manifold and tangent bundle.

The wrong tetrad definition has already been criticized above and in [3]. In addition
this means that Evans believes in having two *different* vectors under consideration.
Thus, this quote shows Dr Evans' complete misunderstanding of the vector concept: Both
(V^{μ}) and (V^{a}) are descriptions of the *same* vector **V**
in the tangent space at some point P of the manifold under consideration,
related to *two different* frames in the tangent space at P:
(V^{μ}) are the coefficients of some vector **V** relative to the coordinates,
i.e. to the coordinate related tangent basis
**e**_{μ} = ∂_{μ}
:= ∂/∂x_{μ} (μ=0,1,2,3) while (V^{a}) is the same relative
to the coordinate independent (freely definable) basis of given tetrad vectors **e**_{a}
(a=0,1,2,3) (the *Greek* index μ indicating that the coordinate related basis
**e**_{μ} = ∂_{μ}
(μ=0,1,2,3) is referred to, and the *Latin* index a refers to the tetrad basis
{**e**_{a} | a=0,1,2,3}. Equ.(17.35) is nothing but the coordinate transform
between the two different frames in the tangent space at some point P of the manifold.

In the sequel Evans considers the vector potential **A** of a circularly polarized
plane wave related to his Complex Circular Basis using the "electromagnetic phase"
Φ = ω(t − z/c) defined in Equ.(1.37):

Circular polarization, discovered experimentally by Arago in 1811, is described geometrically
by elements of A^{a}_{μ} from Eq. (17.12), i.e. by the following complex valued tetrad elements:

A^{(1)}_{x}
= (A^{(0)}/2^{½}) e^{iΦ},
(17.36)

A^{(1)}_{y}
= −*i* (A^{(0)}/2^{½}) e^{iΦ},
(17.37)

where Φ is the electromagnetic phase. The complex conjugates of these elements are:

A^{(2)}_{x}
= (A^{(0)}/2^{½}) e^{−iΦ},
(17.38)

A^{(2)}_{y}
= *i* (A^{(0)}/2^{½}) e^{−iΦ}.
(17.39)

Therefore these tetrad elements are individual components of the following vectors:

**A**^{(1)} = (A^{(0)}/2^{½})
(**i** − *i***j**) e^{iΦ},
(17.40)

**A**^{(2)} = (A^{(0)}/2^{½})
(**i** + *i***j**) e^{−iΦ},
(17.41)

representing a spinning and forward moving frame. This frame is multiplied
by A^{(0)} to give the generally covariant electromagnetic potential field. In 1992
it was inferred by Evans [22] that these vectors define the Evans spin field,
**B**^{(3)}, of electromagnetism:

**B**^{(3) *} = − *i*g **A**^{(1)} × **A**^{(2)},
(17.42)

whose unit vectors are related to the orthonormal Cartesian unit vectors **i**, **j**, **k**
of the tangent space by

**e**^{(1)} =
2^{−½}
(**i** − *i***j**),
**e**^{(2)} =
2^{−½}
(**i** + *i***j**), **e**^{(3)} = **k**.
(5.4)

. . .

Let the orthonormal tangent space rotate and translate with respect to the space i = 1, 2, 3 by introducing the phase Φ = ωt − κZ of the wave equation [5-10]. Here ω is an angular frequency at instant t and κ is a wave-vector at point Z. The orthonormal tangent space is thereby defined by the metric vectors

**q**^{(1)} = **e**^{(1)}e^{iΦ},
**q**^{(2)} = **e**^{(2)}e^{−iΦ},
**q**^{(3)} = **e**^{(3)}
(5.7)

whose magnitudes are . . . follows an erroneous line of formulas (5.8).

We insert (5.4) into (5.7) to obtain

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

Evans' expositions show his idea of the special tetrad frame to be used, a *twisted*
Complex Circular Basis, now renamed with **e** instead of **q**:

**e**^{(1)} := 2^{−½}
(**i** − *i***j**) e^{iΦ},
**e**^{(2)} := 2^{−½}
(**i** + *i***j**) e^{−iΦ},
**e**^{(3)} := **k**
(17.40'-41')

which is no tetrad but an orthonormal spatial triad, that must be completed to a tetrad by introducing
a unit vector **e**_{o} in time direction
having vanishing scalar products with the other basis vectors
**e**^{(a)} (a=1,2,3).

5. The metric and Evans complex Circular Basis

A further objection:

According to Evans the Eqns.(17.36-39) define the "*complex valued tetrad elements*",
i.e. the coefficients q^{a}_{μ} (cf. (17.35) and the
**Appendix 1**).
Therefore we may conclude from (17.36-39)

q^{1}_{1}
= 2^{−½} e^{iΦ},
q^{1}_{2}
= −*i* 2^{−½} e^{iΦ},
(17.36'-37')

q^{2}_{1}
= 2^{−½} e^{−iΦ},
q^{2}_{2}
= *i* 2^{−½} e^{−iΦ},
(17.38'-39')

due to the invariance of **e**^{(0)} and **e**^{(3)} to be
completed with

q^{0}_{μ} = δ^{0}_{μ} ,
q^{3}_{μ} = δ^{3}_{μ} .

Then due to Evans' equation

g_{μν}
= q^{a}_{μ}q^{b}_{ν} η_{ab}
(17.10)

we should obtain a *real* metric tensor (g_{μν}), however,
the *complex* Eqns. (17.36-39) yield a *non-real* result for (17.10):

The reason of that error is that Evans violates the rules of real manifolds as spacetime
is considered in General Relativity. The tetrad vectors **e**_{a} (a=0,1,2,3)
have to be defined as *real* linear combinations of the coordinate related tangent
vectors ∂_{μ}, i.e. the coefficients q^{a}_{μ} of the
inverse representation
∂_{μ}
=
q^{a}_{μ}
**e**_{a}
must be real also [2; p.31 ff.]. **Non-real linear combinations as the basis vectors of
Evans' Complex Circular Basis are inadmissible.**

Conclusion

That is a *serious error*, which should make the author reconsider the concept
of his Complex Circular Basis which is of severe influence for the whole GCUFT book.

**Remark 1** The first way out of the dilemma could appear to be replacing Equ.(17.10)
with

g_{μν}
= q^{a}_{μ} q^{b}_{ν}^{*} η_{ab} ,
(17.10*)

however, that would yield a Hermitean metric, which would in general be non-real likewise.

**Remark 2** Let Q be the matrix (q^{a}_{μ}), Q^{T} its
transposed and M = diag(1,−1,−1,−1) the Minkowski matrix
(see **Appendix 1**). Then,
as can easily be seen from Eqns. (17.36'-39') above, we have Q^{T} M Q* = M.
That means that Evans' introduction of his twisted Complex Circular Basis by (17.40'-41'),
(**e**_{o}, **i**, **j**, **k**)
→
(**e**^{(0)}, **e**^{(1)}, **e**^{(2)}, **e**^{(3)}),
is nothing but a (complex) Lorentz transform and hence the spacetime under consideration is *flat*.
Of course, the modified metric (17.10*) has to be used here. Instead of all that trouble
with Evans' twisted Complex Circular Basis (17.36-41) a *real* twisted basis
(**e**_{o}, **i** cos Φ + **j** sin Φ,
− **i** sin Φ + **j** cos Φ, **k**)
could be introduced with the same (useless) result and using a *real* metric
(17.10) only.

**Remark 3** Another problem would be the definition of the well suited tetrad vectors
**e**^{(a)} (a=0,1,2,3) in more general cases than plane waves, say e.g.
for the case of the superposition of two plane waves with different directions of
propagation.

References

[1]
M.W. Evans, *The Spinning and Curving of Spacetime:
The Electromagnetic and Gravitational Fields In
The Evans Unified Field Theory*

Foundations of Physics Letters Vol. 18, No. 5, p.431-454

The Electromagnetic and Gravitational Fields In The Evans Unified Field Theory

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

[2]
Sean M. Carroll, *Lecture Notes on General Relativity*,

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

[3]
G.W. Bruhn and A. Lakhtakia, *Commentary on Myron W. Evans' paper
"The Electromagnetic Sector ..." *,

http://www.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html

[4]
M.W. Evans, *Generally Covariant Unified Field Theory,
the geometrization of physics*,

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

[5]
M.W. Evans, *METRIC COMPATIBILITY FROM TETRAD POSTULATE*,

http://www.aias.us/Comments/comments01172005b.html

[6]
M.W. Evans, *METRIC COMPATIBILITY FROM TETRAD POSTULATE*,

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

Appendix 1

Equ. (17.40'-41') in extended matrix notation

æ **e**^{(0)} ö
æ 1 0 0 0 ö
æ **e**_{o} ö

| **e**^{(1) } | |
0 2^{−½} e^{iΦ}
−*i* 2^{−½} e^{iΦ}
0 | | **i** |

| | = ^{ } | | | |

| **e**^{(2) } | | 0
2^{−½} e^{−iΦ}
*i* 2^{−½} e^{−iΦ}
^{ }0 | | **j** |

è **e**^{(3)} ø
è 0 0 0 0 ø
è ^{ }**k** ^{ }ø

i.e.
(**e**^{(a)}) =
Q
(**e**_{μ})

defines the *complex* tetrad coefficient matrix Q = (q^{a}_{μ}).
The reader is kindly asked to convince himself of the validity of the equation (17.10*),

(g_{μν})
=
(q^{a}_{μ} η_{ab} q^{b}_{ν}*)
= Q^{T} M Q* = M
= (η_{μν})

and that Evans' Equ.(17.10) gives a *non-real* result.

The analogous effect occurs for the *real* twisted tetrad system given by

æ **e**^{(0)} ö
æ 1 0 0 0 ö
æ **e**_{o} ö

| **e**^{(1) } | |
0 cos Φ
sin Φ
0 | | **i** |

| | = ^{ } | | | |

| **e**^{(2) } | | 0
− sin Φ
cos Φ
^{ }0 | | **j** |

è **e**^{(3)} ø
è 0 0 0 0 ø
è ^{ }**k** ^{ }ø

with the difference that we need not modify the real metric equation (17.10).
However, in both cases it turns out that the underlying metric (g_{μν}) is Minkowski,
so for Evans' purpose of introducing a metric with torsion we can speak with Shakespeare:

Appendix 2

Identical Misprints in book [4] and article [1]

"F^{a} = ... ≠ 0" should be "F^{a} = ... = 0" in Equ.(17.65).

Wrong equation reference on p.311: (17.68) should be (17.69).

Equ.(17.91) should probably run as

R = κκ*
= (T^{2}_{32})² + (T^{1}_{32})² .

Appendix 3

We consider a plane linearly polarized em-wave with the phase function
Φ = ω(t − **x·n**/c). Φ
must be invariant under Lorentz transforms. That yields the transformation rules
for two inertial systems S and S', S' travelling against S with velocity v in x direction:

ω = ω' γ (1 + β n_{x}')

ω n_{x} = ω' γ (β + n_{x}')

ω n_{y} = ω' n_{y}'

ω n_{z} = ω' n_{z}'

where β = ^{v}*/*_{c}, γ
= ^{1}*/*_{(1 − β²)½} .

For a linearly polarized plane em-wave travelling in z-direction of S, we obtain
n_{x} = n_{y} = 0 ,
n_{y}' = 0 and n_{x}' = − β,
hence ω = ω'/γ .
The propagation direction **n** = **k** is transformed to
**n'** = − β **i**' + ^{1}*/*_{γ} **k**' .

We assume the **E** field || **i**+**j**
and the **H** field || **i**−**j**,
hence E_{z} = H_{z} = 0 while E := E_{x} = E_{y} and
H := H_{x} = − H_{y} where due to Maxwell E = H.

The transformation of the fields **E** and **H** yields

E_{x}' = E_{x} = E,
H_{x}' = H_{x} = H

E_{y}' = γ (E_{y} − βH_{z})
= γ E_{y}
= γ E ,
H_{y}' = γ (H_{y} + βE_{z})
= γ H_{y} = − γ H

E_{z}' = γ (E_{z} + βH_{y})
= βγ H_{y}
= − βγ H,
H_{y}' = γ (H_{z} − βE_{y})
= − βγ E_{y} = − βγ E

Therewith we obtain

**E' · H'** = (1 − γ² − β²γ²) EH
= 0 EH = 0

Thus, the S' vectors **n'**, **E'**, **H'** are pair wise orthogonal
like the S vectors **n**, **E**, **H**.

Consider the orthonormal "triad"
**e**^{(1)}, **e**^{(2)}, **e**^{(3)}
associated with the wave vectors:
In S we have

**E** || **e**^{(1)} := (**i**+**j**)/2^{½} ,

**H** || **e**^{(2)} := (**i**−**j**)/2^{½} ,

**n** || **e**^{(3)} := **k** .

However the Lorentz transform L: S → S' **does neither preserve the orthonormality
of the triad nor its association to the plane wave**: We obtain

L **e**^{(1)} := 2^{−½} (γ^{−1}**i'**+**j'**)
not || **E'**,

L **e**^{(2)} := 2^{−½} (γ^{−1}**i'**−**j'**)
not || **H'**,

L **e**^{(3)} = **k'**
not || **n'**

where the orthonormality of the vector L **e**^{(1)} and L **e**^{(2)}
and the parallelity to the wave vectors is evidently lost.