Last update: 11.09.2005, 21:00 CEST

08.03.2005: Remark added, see Sect. 4

11.03.2005: Sect.5 added

18.03.2005: INTRODUCTION added.

19.03.2005: Sect.6 added.

20.03.2005: Sect.7 added.

21.03.2005: Sect.8 added.

27.03.2005: Evans' equ. (2.149)

inserted in Section 6.

01.04.2005: Sect.9 added.

11.09.2005: Links to error descriptions in Chap.13 and Chap.17 added.

Subject: Recipient of highest physics award in Great Britain

Date: Tue, 8 Mar 2005 20:16:49 +0100

From: **"John B. Hart" jhart@cinci.rr.com**
To: pt@aip.org

You may already know that Myron W. Evans was given the highest award in Great Britain for a physicist. I believe that he is also a Fellow of the AIP. Below is a summary of previous people given the award called the Civil List Pension. Following this is the press release about the award. I believe that Evans is the new supernova of physics for the next century. Take a look at his www.aias.us. Scroll down a coupld of screens and read the quote by Prof. Dr. Bo Lehnert, a member of the Royal Swedish Academy of Sciences and, presumably, a member of the nominating committee of the Nobel Prize in Physics and Chemistry. My quote follows his. A Civil List Pension is indeed rare: one was awarded to the Scottish botanist Robert Brown (1773-1858) of the Brownian motion. Einstein explained the Brownian motion in 1905 in terms of molecular motion. Dr Evans worked with Prof. William Coffey at Trinity College Dublin on the Brownian motion applied to the far infra red and computer simulation using memory functions and non-Markovian statistical dynamics. So this is pleasing to find. One of the very few other Civil List Pensions in science was awarded to Sir William Huggins KCB, OM, PRS for his work on spectroscopy applied to astronomy. He was PRS 1900-1905. Alfred Russel Wallace was awarded one on the advice of Charles Darwin, and also Mary Fairfax Somerville (1780 - 1872). Probably the most well known recipients in science are Faraday, Joule and Brown.

PRESS RELEASE

AWARD OF CIVIL LIST PENSION TO PROF. MYRON W. EVANS

Following a nomination by the Royal Society of Chemistry and on the advice of the Prime Minister, Queen Elizabeth II has awarded a Civil List Pension to Prof. Myron W. Evans, Director of the Alpha Foundation for Advanced Study (AIAS) in recognition of his contributions to science. This is a high honour granted to few British scientists. Only two or three other leading British scientists have been awarded a Civil List Pension: these include the father of classical electrodynamics, Michael Faraday, (1797 - 1867), who was awarded a Civil List Pension in 1836, and James Prescott Joule, (1818 - 1889), the father of thermodynamics, awarded a Civil List Pension in 1878. In the arts, notable recipients of a Civil List pension include Lord Byron, Lord Tennyson and William Wordsworth.

Prof. Evans was born in 1950 and was educated at Pontardawe Grammar School and University of Wales Aberystwyth, and is the author of some seven hundred papers and monographs in chemistry and physics. He is Harrison Memorial Prizewinner and Meldola Medallist of the Royal Society of Chemistry, sometime Junior Research Fellow of Wolfson College Oxford, University of Wales Fellow and SERC Advanced Fellow, formerly an IBM Research Professor at IBM Kingston New York and a visiting scientist at Cornell Theory Center.

He has made many contributions to both chemistry and physics, and has recently completed Einstein's work of 1925 to 1955 in the development of a unified field theory sought after by physicists for four hundred years. This achievement has been universally recognised worldwide in the past two years. He becomes the only British scientist currently on the Civil List.

Respectfully,

John B. Hart

Professor Emeritus of Physics

Xavier University Cincinnati, Ohio

In Chapter 2, Duality and the Antisymmetric Metric [1], of the forthcoming book
on his **G**rand **U**nified **F**ield **T**heory (GUFT) Evans claims the
existence of an "antisymmetric metric"; more, with the Eqns. (2.14-17) he specifies his
claim for the Minkowski spacetime. We reproduce the equations here.

The matrix of the symmetric metric tensor is (correctly) given by

1 0 0 0

0 –1 0 0

*q ^{ρσ}*(S) =
.
(2.14)

0 0 –1 0

0 0 0 –1

The superscript "(S)" means "symmetric". The antisymmetric metric (superscript "(A)") is according to Evans given by

0 –1 –1 –1

1 0 –1 1

*q ^{ρσ}*(A) =
.
(2.15)

1 1 0 –1

1 –1 1 0

The line element *ds* is then (correctly) given by

*ω*_{1}
=
*ds*^{2}
=
*q ^{ρσ}*(S)

while the antisymmetric metric yields the line element *ds* by

*ω*_{2}
=
*ds*^{2}
=
– ^{1}/_{6} *q ^{ρσ}*(A)

Since all data are defined we can readily evaluate both (2.16) and (2.17) to obtain from (2.16)

(1.1)
*ds*^{2}
=
*dx*_{0}^{2}
– *dx*_{1}^{2}
– *dx*_{2}^{2}
– *dx*_{3}^{2}

and from (2.17)

(1.2)
*ds*^{2}
=
^{1}/_{3}
(*dx*_{0}Ù*dx*_{1}
+
*dx*_{0}Ù*dx*_{2}
+
*dx*_{0}Ù*dx*_{3}
+
*dx*_{1}Ù*dx*_{2}
+
*dx*_{2}Ù*dx*_{3}
–
*dx*_{1}Ù*dx*_{3})

The reader, even if not being familiar with differential forms, will see the disagreement of both results.

In the next section we shall give a formal proof that even in more general
cases the Equ. (2.17) with any antisymmetric matrix
(*q ^{ρσ}*(A)) cannot give a symmetric metric.

In a further part of Chapter 2 of his forthcoming book on the GUFT Evans gives an antisymmetric matrix (2.49) that should define a metric eqivalent that one defined by the symmetric matrix (2.42).

We shall show here that no antisymmetric matrix
(*q ^{ρσ}*(A))
is able to produce a metric by means of Equ. (2.17) that is given by a symmetric matrix
(

First of all we remark that the commutative product of differentials in Equ. (2.16) and the anticommutative wedge product Ù can both be expressed by means of the (non-commutative) tensor product Ä: We have

(2.1)
*dx _{σ}dx_{ρ}*
= ½ (

and likewise

(2.2)
*dx _{σ}*Ù

Inserting (2.1) into (2.16) we obtain

*q ^{ρσ}*(S)

and by swapping the characters *σ* and *ρ* in the second summand

*q ^{ρσ}*(S)

and finally, since *q ^{σρ}*(S)
is symmetric,

(2.3)
*q ^{ρσ}*(S)

i.e. the (symmetric) metric (2.16) can be expressed by the *symmetric* tensor
*q ^{ρσ}*(S)

if and only if the matrix (

The analogous procedure can be applied to
*q ^{ρσ}*(A)

*q ^{ρσ}*(A)

and by swapping the characters *σ* and *ρ* in the second term

*q ^{ρσ}*(A)

However, since the matrix (*q ^{σρ}*(A)) is skew symmetric,

i.e. the "antisymmetric metric" of (2.17) can be expressed by the *skew symmetric* tensor
*q ^{ρσ}*(A)

represent a (symmetric) metric (2.16).

on the existence of such an "antisymmetric metric" (2.17).

Though the GUFT is erroneous due to Sect.1 and 2, we consider Equ. (2.42) that was "derived" by MWE as "the following 4 × 4 symmetric metric tensor in spacetime"

*h*_{0}²
*h*_{0}*h*_{1}
*h*_{0}*h*_{2}
*h*_{0}*h*_{3}

*h*_{1}*h*_{0}
*h*_{1}²
*h*_{1}*h*_{2}
*h*_{1}*h*_{3}

.
(2.42)

*h*_{2}*h*_{0}
*h*_{2}*h*_{1}
*h*_{2}²
*h*_{2}*h*_{3}

*h*_{3}*h*_{0}
*h*_{3}*h*_{1}
*h*_{3}*h*_{2}
*h*_{3}²

Evans uses that matrix in the sequel for further conclusions.

However, matrices of that type are well-known and readily proven to be positive semi-definite: The corresponding quadratic form is

*h _{μ}h_{ν}ξ^{ μ}ξ^{ ν}*
=
(

However, the metric of spacetime given by the quadratic form (2.14+16), is indefinite:

*ds*² < 0 for "space like" directions and *ds*² > 0 for "time like" directions.

In addition the matrix (2.42) has a vanishing determinant, because all of its lines are parallel to
the vector [*h*_{0},*h*_{1},*h*_{2},*h*_{3}];
hence it is not invertible, which is necessary for metric matrix.

That is fatal for Evans' further considerations.

The reason for that contradiction is mainly that Evans has suppressed or forgotten that each matrix element has a cofactor, a scalar product of basis vectors which, of course, would change the situation, see [2; remarks to (2.42)]. And even the scalar product of vectors in spacetime needs a definition.

Evans calls *ω*_{1} in (2.16) a "zero-form".
**That is wrong** and may be the reason of further fallacies:
Due to the representation (2.1) of the symmetric product
*dx _{σ}*

*ω*_{1}
=
½ (*q ^{ρσ}*(S)

Let *M* be a manifold and T be its tangent space at an arbitrary point P of *M*.
A *spacetime metric* is a bilinear mapping g : T × T → **R**,
which is equivalent to the local Minkowski metric given by
(*η*_{ab}) = diag(1, −1, −1, −1):

Let {∂_{μ}} be the basis vectors of T given by local coordinates in a neighborhood
of the point P. We define
*g _{μν}* := g(∂

(5.1)
*g _{μν}*
=

**Conclusion**: The matrix (*g _{μν}*), and hence the
mapping g : T × T →

**Remark**: There exist matrices (*g _{μν}*)
that cannot be equivalent to (

1
=
det(*g _{μν}*)
=
det (

which is a contradiction.

__Example 2__
**The matrix
( g_{μν}) given by Evans' matrix (2.42)
cannot be equivalent to the local Minkowski metric
(η_{ab})**:

The determinant of matrix (2.42) vanishes, because the matrix has parallel line vectors: det(

0
=
det(*g _{μν}*)
=
det (

which is a contradiction to the invertibility of the matrix
(*q*_{μ}^{a}).

At the beginning of Chapter 2.3 Evans remarks for the first time
that the quantity *q ^{μ}* is to be understood as "the inverse tetrad

The first equation is

*q ^{μν}*

where (*η*^{ab}) = (*η*_{ab}) = diag(−1, 1, 1, 1)
denotes the Minkowski matrix.

Taking the inverse matrices in (2.73') we obtain due to
(*η*^{ab})^{−1} = (*η*^{ab})

*q _{μν}*

So Evans obtains the well-known relation

*q ^{μν}*(S)

Then he concludes (in Evans' terms)

4 =
*q ^{μν}*

i.e. written with all indices

4 =
*q ^{μν}*

where the indices a, a', b, b' are *independent*.
The term (? ? ?) on the right hand side shows that there is a problem in determining the
contents of the brackets. Evans claims with (2.145) that
the expression in the middle of equ. (2.145') can be rewritten
as the quadrat of a sum containing only one
summation index (*μ*). But that is wrong, since

*q*_{a}^{μ}*q*_{b}^{ν}*η*^{ab}
*q _{μ}*

(−

cannot be factorized in that way: That would require that each of the two brackets above
could be decomposed into two factors depending solely on *μ* or on *ν*
respectively. The reader doubting that should try to decompose the simpler expression
−*q _{μ}*

referring to (2.149), to

**Remark 1**

If Evans should be interested in the value of
*q ^{μ}*

*q*_{a}^{μ}*q _{μ}*

since the matrices (*q*_{a}* ^{μ}*) and
(

*q*_{a}^{μ}*q _{μ}*

**
Both results are different from Evans' result (2.149),
which is therefore unsuitable for further use. This affects e.g. the equations
(2.175 - 179).**

**Remark 2**

The reason for Evans' mischief was clearly his suppression of indices and of the
Minkowski matrix.

The equation (2.173) multiplied by *q*^{ν}
(= *q*_{c}^{ν})
in Evans' dangerous short hand yields:

(*D*_{ρ}*q*_{μ})
*q*_{ν}
*q*^{ν}
+
*q*_{μ}
(*D*_{ρ}*q*_{ν})
*q*^{ν}
= 0
(2.173a)

The extended version is

(*D*_{ρ}*q*_{μ}^{a})
*η*_{ab}
*q*_{ν}^{b}
*q*_{c}^{ν}
+
*q*_{μ}^{a}
(*D*_{ρ}*q*_{ν}^{b})
*η*_{ab}
*q*_{c}^{ν}
= 0
(2.173a')

Here we can simplify
*q*_{ν}^{b}
*q*_{c}^{ν}
=
δ^{b}_{c} ,
while Evans, using the *wrong* equ. (2.149), obtains
*q*_{ν}
*q*^{ν}
=
−2 . We continue the correct calculation:

(*D*_{ρ}*q*_{μ}^{a})
*η*_{ab}
δ^{b}_{c}
+
*q*_{μ}^{a}
(*D*_{ρ}*q*_{ν}^{b})
*η*_{ab}
*q*_{c}^{ν}
= 0

or

(*D*_{ρ}*q*_{μ}^{a})
*η*_{ac}
+
*q*_{μ}^{a}
(*D*_{ρ}*q*_{ν}^{b})
*η*_{ab}
*q*_{c}^{ν}
= 0 .

Now Evans multiplies by *q*^{μ}*q*_{ν} , i.e. by
*q*_{a'}^{μ} *q*_{ν}^{c'} , to obtain

*q*_{a'}^{μ} *q*_{ν}^{c'}
(*D*_{ρ}*q*_{μ}^{a})
*η*_{ac}
+
*q*_{a'}^{μ}
*q*_{μ}^{a}
(*D*_{ρ}*q*_{ν}^{b})
*η*_{ab}
*q*_{c}^{ν}
*q*_{ν}^{c'}
= 0

or

*q*_{a'}^{μ} *q*_{ν}^{c'}
(*D*_{ρ}*q*_{μ}^{a})
*η*_{ac}
+
δ_{a'}^{a}
(*D*_{ρ}*q*_{ν}^{b})
*η*_{ab}
δ_{c}^{c'}
= 0

or for c'=c

*q*_{a'}^{μ} *q*_{ν}^{c}
(*D*_{ρ}*q*_{μ}^{a})
*η*_{ac}
+
(*D*_{ρ}*q*_{ν}^{b})
*η*_{a'b}
= 0

This equation can be resolved for *D*_{ρ}*q*_{ν}^{b}
by multiplication by *η*^{a'b'}

*η*^{a'b'}
*q*_{a'}^{μ} *q*_{ν}^{c}
(*D*_{ρ}*q*_{μ}^{a})
*η*_{ac}
+
(*D*_{ρ}*q*_{ν}^{b})
*η*_{a'b}*η*^{a'b'}
= 0

or

*η*^{a'b}
*q*_{a'}^{μ} *q*_{ν}^{c}
(*D*_{ρ}*q*_{μ}^{a})
*η*_{ac}
+
*D*_{ρ}*q*_{ν}^{b}
= 0

Now Evans sets * μ*=

So we have the next serious error, and Evans calculation stops here *without obtaining
the desired result*

*D*_{ρ}*q*_{ν}^{b} = 0 .
(2.179)

Evans' short hand equation

*D*_{ν}*q*^{μ}
=
∂_{ν}*q*^{μ}
+
Γ_{λ}^{μ}_{ν}
*q*^{λ}
= 0
(2.182)

reads extended

*D*_{ν}*q*_{a}^{μ}
=
∂_{ν}*q*_{a}^{μ}
+
Γ_{λ}^{μ}_{ν}
*q*_{a}^{λ}
= 0 .
(2.182')

However, the following compatibility condition

∂_{ν}*q*_{a}^{μ}
+
Γ_{λ}^{μ}_{ν}
*q*_{a}^{λ}
−
*ω*_{ν}
^{b}_{a}
*q*_{b}^{μ}
= 0 .

is holding always (see [3;(3.9-10)] or Evans e.a.]. Thus, comparison of both equations yields

*ω*_{ν}^{b}_{a}
*q*_{b}^{μ}
= 0 ,

or, since the matrix (*q*_{b}^{μ})
is invertible, also

*ω*_{ν}^{b}_{a}
= 0

for all index combinations. This is an *essential restriction for
the spacetime manifold under consideration*.

This means that all conclusions from equ. (2.182) can only be proven only under the same restrictions.

We consider Evans' short hand equation

*q*_{λ}
*D*_{ν}*q*^{μ}
=
*q*_{λ}
∂_{ν}*q*^{μ}
+
*q*_{λ}
*q*^{λ}
Γ_{λ}^{μ}_{ν}
= 0 ,
(2.183)

i.e. in extended version

*q*_{λ}^{a}
*D*_{ν}*q*_{a}^{μ}
=
*q*_{λ}^{a}
∂_{ν}*q*_{a}^{μ}
+
*q*_{λ}^{a}
*q*_{a}^{λ}
Γ_{λ}^{μ}_{ν}
= 0 .
(2.183')

The product
*q*_{λ}^{a}
*q*_{a}^{λ}
yields the value 4,
**while Evans' short hand version due to the wrong equation (2.149)
yields the wrong result
q_{λ}
q^{λ}
= −2**
.

Hence we obtain from (2.183')

Γ_{λ}^{μ}_{ν}
= − ¼ *q*_{λ}^{a}
∂_{ν}*q*_{a}^{μ}
.
(2.184')

instead of Evans' wrong equation (2.184). However, due to the mentioned
restrictions, that is a *rather useless result*. Thus,

Let **Q** denote the tetrad matrix (*q*_{ν}^{a}),
where the upper index is the line index and the lower index
is the column index.
Then the compatibility relation of frames means that the equation

(9.1)
*D _{μ}*

defines a linear differential operator *D _{μ}* that anihilates the matrix

(9.2)
*D _{μ}*

To make the way of operation of *D _{μ}* visible
we temporarily suppress the index

(9.3)
*D***Q**
:=
(∂*q _{λ}*

Here we introduce the matrices
**Ω**
:=
(*ω*^{a}_{b})
and
**Γ**
:=
(Γ* ^{ν}_{λ}*) .
Then we have

(9.4)
*D***Q**
=
∂**Q**
+
**Ω****Q**
−
**Q****Γ** ,

or, with restored index *μ* and replaced with *ν*

(9.5)
*D*_{ν}**Q**
=
∂_{ν}**Q**
+
**Ω**_{ν}**Q**
−
**Q****Γ**_{ν} ( = **O**) .

Equ.(9.5) holds for arbitrary 4×4 matrices **Q**. Therefore the matrix **Q**
can be replaced with the matrix *D*_{μ}**Q**
(with fixed value *μ*) to obtain

(9.6)
*D*_{ν}*D*_{μ}**Q**
:=
∂_{ν}(*D*_{μ}**Q**)
+
**Ω**_{ν}(*D*_{μ}**Q**)
−
(*D*_{μ}**Q**)**Γ**_{ν} ,

hence by using the operators
*D*^{μ}
:=
*g ^{μν}D_{ν}*
and
∂

(9.7)
*D*^{μ}*D*_{μ}**Q**
=
∂^{μ}(*D*_{μ}**Q**)
+
**Ω**^{μ}(*D*_{μ}**Q**)
−
(*D*_{μ}**Q**)**Γ**^{μ} .

where
**Ω**^{μ}
:=
*g ^{μν}*

The last two terms vanish due to equ.(9.2), and so we have the simplification

(9.8)
*D*^{μ}*D*_{μ}**Q**
=
∂^{μ}(*D*_{μ}**Q**)
=
∂^{μ}(∂_{μ}**Q**
+
**Ω**_{μ}**Q**
−
**Q****Γ**_{μ})

=
∂^{μ}∂_{μ}**Q** +
*g ^{μν}*
∂

= ∂

Here the derivations ∂_{ν}**Q** can be eliminated due to the
vanishing of (9.5)

(9.9)
*D*^{μ}*D*_{μ}**Q**
=
∂^{μ}∂_{μ}**Q** +
*g ^{μν}*
[(∂

or rewritten

*D*^{μ}*D*_{μ}**Q**
=
∂^{μ}∂_{μ}**Q** +
*g ^{μν}*
[(∂

(9.10) = ∂

= o

where o_{2}**Q** is the d'Alembertian operator
∂_{μ}∂^{μ} .

**Remark**

M.W. Evans uses two *different* d'Alembertian operators *with the same notation*:
In [1; (2.188)] we read
o
:= ∂^{μ}∂_{μ},
while in [1; (9.42)]
o
:= ∂_{μ}∂^{μ}
is used. For matter of distinction we index both versions:

o_{1}
:= ∂^{μ}∂_{μ} and
o_{2}
:= ∂_{μ}∂^{μ}.

The use of o_{2} instead of
o_{1} causes additional terms,
which will be marked in **red** in the following.

Due to

(o_{2}
−
o_{1})**Q**
=
(∂_{μ}∂^{μ}
−
∂^{μ}∂_{μ})**Q**
=
∂_{μ}(*g ^{μν}*∂

= (∂

we obtain from (9.10)

(9.11)
*D*^{μ}*D*_{μ}**Q**
=
o_{2}**Q**
+
[(∂^{μ}**Ω**_{μ}
−
**Ω**_{μ}
(**Ω**^{μ}
+
∂_{ν}*g ^{μν}*))

hence, due to equ.(2)
we obtain the "wave equation" for **Q**:

(9.12)
o_{2}**Q**
+
[(∂^{μ}**Ω**_{μ}
−
**Ω**_{μ}
(**Ω**^{μ}
+
∂_{ν}*g ^{μν}*))

This is a * linear partial differential equation*
for the tetrad matrix

For comparison let's have a look at the "Evans Lemma" [1; Sect. 9.2]:
After having introduced
the "scalar curvatures" *R*_{1} and *R*_{2} by

− *R*_{1}
*q*_{λ}^{a}
:=
(*D*^{μ}*ω*^{a}_{μb})
*q _{λ}*

and

− *R*_{2}
*q*_{λ}^{a}
:=
−
Γ^{ν}_{νμ}
*ω*^{μa}_{b}
*q*_{λ}^{b}
+
Γ^{ν}_{νμ}
Γ^{μν}_{λ}
*q*_{ν}^{a}
,
(9.48)

then the main equation, the "Evans Lemma", is

o_{2}
*q*_{λ}^{a}
= *R*
*q*_{λ}^{a}
,
(9.49)

where

*R* = *R*_{1} + *R*_{2}.
(9.50)

We remark that the "curvatures" *R*_{1} and
*R*_{2}depend on the tetrad coefficients, while
the main equation (9.49) pretends to be a single linear equation for
each single tetrad coefficients. Actually, however, equ. (9.49)
represents a *nonlinear system of partial differential equations* for
the 16 tetrad coefficients *q*_{λ}^{a}.
Especially there is no reason to consider equ. (9.49) as an eigenvalue problem
of the d’Alembertian operator
o_{2}**Q**
since the "eigenvalues" *R*
are non constant and solution dependent. Therefore all conclusions contained
in the following quotation from Evans' book are erroneous:

*
"Given the tetrad postulate, the lemma shows that scalar curvature R
is always an eigenvalue of the wave equation (9.49) for all spacetimes, in other
words, R is quantized. The eigenoperator is the d’Alembertian operator
*o

Chap.13: http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html

Chap.17: http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansChap17.html

[1]
M.W. Evans: GENERALLY COVARIANT UNIFIED FIELD THEORY:

THE GEOMETRIZATION OF PHYSICS;
Web-Preprint,

http://www.aias.us/book01/GCUFT-Book-10.pdf

[2]
G.W. Bruhn: Comments on M.W.Evans’ preprint Chapter 2:

Duality and the Antisymmetric Metric;

http://www2.mathematik.tu-darmstadt.de/~bruhn/Comment-Chap2.htm