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.


Dedicated to Her Majesty the Queen's New Pensioner MWE:

Myron W. Evans' Most Spectacular Errors

Gerhard W. Bruhn, Darmstadt University of Technology



Summary Some fatal errors at the basis of M.W. Evans' Grand Unified Field Theory (GUFT) are listed and described here. The errors are not sophisticated but on elementary algebraic level.




INTRODUCTION

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

Dear Sir,

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


1. The "antisymmetric metric" in Minkowski spacetime

In Chapter 2, Duality and the Antisymmetric Metric [1], of the forthcoming book on his Grand Unified Field Theory (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                                                                
qρσ(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                                                                
qρσ(A) =                                 .                                                 (2.15)
1   1   0 1                                                                
1 1   1   0                                                                

The line element ds is then (correctly) given by

ω1 = ds2 = qρσ(S) dxσdxρ ,                                                 (2.16)

while the antisymmetric metric yields the line element ds by

ω2 = ds2 = 1/6 qρσ(A) dxσ dxρ ,                                                 (2.17)

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

(1.1)                                 ds2 = dx02 dx12 dx22 dx32

and from (2.17)

(1.2)         ds2 = 1/3 (dx0dx1 + dx0dx2 + dx0dx3 + dx1dx2 + dx2dx3 dx1dx3)

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

The metrics given by Evans' Eqns. (2.14-17) cannot agree.

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



2. The question of a general antisymmetric 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 (qρσ(A)) is able to produce a metric by means of Equ. (2.17) that is given by a symmetric matrix (qρσ(S)) using the well-known Equ. (2.16).

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

(2.1)                                 dxσdxρ = (dxσdxρ + dxρdxσ)

and likewise

(2.2)                                 dxσdxρ = (dxσdxρdxρdxσ) .

Inserting (2.1) into (2.16) we obtain

qρσ(S) dxσdxρ = (qρσ(S) dxρdxσ + qρσ(S) dxσdxρ),

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

qρσ(S) dxσdxρ = (qρσ(S) dxρdxσ + qσρ(S) dxρdxσ),

and finally, since qσρ(S) is symmetric,

(2.3)                                 qρσ(S) dxσdxρ = qρσ(S) dxρdxσ ,

i.e. the (symmetric) metric (2.16) can be expressed by the symmetric tensor qρσ(S) dxρdxσ. The result is therefore:

The expression qρσ(S) dxρdxσ defines a (symmetric) metric
if and only if the matrix (qρσ) is symmetric.

The analogous procedure can be applied to qρσ(A) dxσ dxρ:

qρσ(A) dxσ dxρ = (qρσ(A) dxρdxσqρσ(A) dxσdxρ),

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

qρσ(A) dxσdxρ = (qρσ(A) dxρdxσqσρ(A) dxρdxσ).

However, since the matrix (qσρ(A)) is skew symmetric,

(2.4)                                 qρσ(A) dxσdxρ = qρσ(A) dxρdxσ ,

i.e. the "antisymmetric metric" of (2.17) can be expressed by the skew symmetric tensor qρσ(A)dxσdxρ. This yields the result:

Equ. (2.17) with skew symmetric matrix (qσρ(A)) ≠ O cannot
represent a (symmetric) metric (2.16).

This result is fatal for M.W. Evans GUFT, which is essentially based
on the existence of such an "antisymmetric metric" (2.17).



3. A metric tensor for general spacetime?

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"

h0     h0h1     h0h2     h0h3                                                                
h1h0       h1     h1h2     h1h3                                                                
.                                                 (2.42)
h2h0     h2h1       h2     h2h3                                                                
h3h0     h3h1     h3h2       h3                                                                

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

hμhνξ μξ ν = (hμξ μ)2 > 0 .

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

ds < 0 for "space like" directions and ds > 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 [h0,h1,h2,h3]; hence it is not invertible, which is necessary for metric matrix.

The matrix (2.42) cannot be the metric tensor of spacetime.
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.


4. A Remark concerning ω1 in (2.16)

Evans calls ω1 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 dxσdxρ as a sum of tensor products of 1-forms, strictly speaking as a sum of 2-forms, is a 2-form again:

ω1 = (qρσ(S) dxρdxσ + qσρ(S) dxρdxσ) .


5. Metrics on Spacetime

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 (ηab) = 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 gμν := g(∂μ,∂ν). Equivalence of the metric g or the matrix (gμν) to (ηab) means that there exists an invertible matrix (qμa) such that

(5.1)                                 gμν = ηab qμa qνb.

Conclusion: The matrix (gμν), and hence the mapping g : T × T → R is symmetric.

Remark: There exist matrices (gμν) that cannot be equivalent to (ηab).
Example 1 The matrix (gμν) = diag(1,1,1,1) cannot be equivalent to the local Minkowski metric (ηab): Due to det(gμν) = 1 by taking the determinants on both sides of Equ.(5.1) we would obtain

1 = det(gμν) = det (ηab) det(qμa) det(qνb) = (−1) · det(qμa)2 = − det(qaμ)2 ,

which is a contradiction.

Example 2 The matrix (gμν) given by Evans' matrix (2.42) cannot be equivalent to the local Minkowski metric (ηab):
The determinant of matrix (2.42) vanishes, because the matrix has parallel line vectors: det(gμν) = 0 . Hence by taking the determinants on both sides of Equ.(5.1) we would obtain

0 = det(gμν) = det (ηab) det(qμa) det(qνb) = (−1) · det(qμa)2 = − det(qμa)2 ,

which is a contradiction to the invertibility of the matrix (qμa).

6. A fatal metric error

At the beginning of Chapter 2.3 Evans remarks for the first time that the quantity qμ is to be understood as "the inverse tetrad qaμ with unwritten index a". That suppression of an index will cause a fatal error as we shall see soon. We shall not follow this bad habit and rewrite Evans' equations with all indices and indicating the completion by an apostrophe at the equation number.

The first equation is

qμν(S) = qaμqbν ηab .                                                                 (2.73')

where (ηab) = (ηab) = diag(−1, 1, 1, 1) denotes the Minkowski matrix.

Taking the inverse matrices in (2.73') we obtain due to (ηab)−1 = (ηab)

qμν(S) = qμaqνb ηab .                                                                 (2.73'')

So Evans obtains the well-known relation

qμν(S)qμν(S) = 4 .                                                                 (2.137)

Then he concludes (in Evans' terms)

4 = qμν(S) qμν(S) = qμqνqμqν = (qμqμ)2 ,                                         (2.145)

i.e. written with all indices

4 = qμν(S) qμν(S) = qaμ qaν ηab qμa' qνb' ηa'b' = (? ? ?)2 ,                                         (2.145')

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

qaμqbνηab qμa'qνb'ηa'b' =                                                                                                
(−qμ0qν0 + qμ1qν1 + qμ2qν2 + qμ3qν3) (−q0μq0ν + q1μq1ν + q2μq2ν + q3μq3ν)

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 −qμ0qν0 + qμ1qν1. Thus, all the more generally, the substitute for qμqμ, the expression (? ? ?) cannot be defined. Hence Evans' further steps to prove his equ.(2.149) cannot be executed:

Equ. (2.149), qμqμ = −2, is erroneous. Consequently all further equations
referring to (2.149), to qμqμ = −2, are erroneous.

Remark 1
If Evans should be interested in the value of qμqμ in his short hand notation: We have

qaμqμb = δab ,

since the matrices (qaμ) and (qμb) are mutually inverses. Especially the choice a=b yields

qaμqμa = δaa = 4 .

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.



7. The calculation following (2.173)

The equation (2.173) multiplied by qν (= qcν) in Evans' dangerous short hand yields:

(Dρqμ) qν qν + qμ (Dρqν) qν = 0                                                 (2.173a)

The extended version is

(Dρqμa) ηab qνb qcν + qμa (Dρqνb) ηab qcν = 0                                 (2.173a')

Here we can simplify qνb qcν = δbc , while Evans, using the wrong equ. (2.149), obtains qν qν = −2 . We continue the correct calculation:

(Dρqμa) ηab δbc + qμa (Dρqνb) ηab qcν = 0

or

(Dρqμa) ηac + qμa (Dρqνb) ηab qcν = 0 .

Now Evans multiplies by qμqν , i.e. by qa'μ qνc' , to obtain

qa'μ qνc' (Dρqμa) ηac + qa'μ qμa (Dρqνb) ηab qcν qνc' = 0

or

qa'μ qνc' (Dρqμa) ηac + δa'a (Dρqνb) ηab δcc' = 0

or for c'=c

qa'μ qνc (Dρqμa) ηac + (Dρqνb) ηa'b = 0

This equation can be resolved for Dρqνb by multiplication by ηa'b'

ηa'b' qa'μ qνc (Dρqμa) ηac + (Dρqνb) ηa'bηa'b' = 0

or

ηa'b qa'μ qνc (Dρqμa) ηac + Dρqνb = 0

Now Evans sets μ=ν.

However, that is not allowed, since μ is a summation index.

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

Dρqνb = 0 .                                                                 (2.179)



8. Equation (2.182) and following

Evans' short hand equation

Dνqμ = ∂νqμ + Γλμν qλ = 0                                                 (2.182)

reads extended

Dνqaμ = ∂νqaμ + Γλμν qaλ = 0 .                                             (2.182')

However, the following compatibility condition

νqaμ + Γλμν qaλων ba qbμ = 0 .

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

ωνba qbμ = 0 ,

or, since the matrix (qbμ) is invertible, also

ωνba = 0

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

Equ. (2.182) cannot hold in general.

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

We consider Evans' short hand equation

qλ Dνqμ = qλνqμ + qλ qλ Γλμν = 0 ,                                                 (2.183)

i.e. in extended version

qλa Dνqaμ = qλaνqaμ + qλa qaλ Γλμν = 0 .                                             (2.183')

The product qλa qaλ yields the value 4, while Evans' short hand version due to the wrong equation (2.149) yields the wrong result qλ qλ = −2 .

Hence we obtain from (2.183')

Γλμν = − ¼ qλaνqaμ .                                                 (2.184')

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

Equ. (2.184) (or the correct equ.(2.184')) are unsuited for further use.



9. The Evans Lemma

Let Q denote the tetrad matrix (qνa), 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)                                                DμQ := (∂μqλa + ωaμb qλb − Γνμλ qνa)

defines a linear differential operator Dμ that anihilates the matrix Q = (qλa):

(9.2)                                                DμQ := (∂μqλa + ωaμb qλb − Γνμλ qνa) = O .

To make the way of operation of Dμ visible we temporarily suppress the index μ to obtain

(9.3)                                                DQ := (∂qλa + ωab qλb − Γνλ qνa)

Here we introduce the matrices Ω := (ωab) and Γ := (Γνλ) . Then we have

(9.4)                                                DQ = ∂Q + ΩQQΓ ,

or, with restored index μ and replaced with ν

(9.5)                                                 DνQ = ∂νQ + ΩνQQΓν ( = O) .

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

(9.6)                                 DνDμQ := ∂ν(DμQ) + Ων(DμQ) − (DμQ)Γν ,

hence by using the operators Dμ := gμνDν and ∂μ := gμνν also

(9.7)                                 DμDμQ = ∂μ(DμQ) + Ωμ(DμQ) − (DμQ)Γμ .

where Ωμ := gμνΩν and Γμ := gμνΓν .

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

(9.8)                               DμDμQ = ∂μ(DμQ) = ∂μ(∂μQ + ΩμQQΓμ)
                                                = ∂μμQ + gμνν(ΩμQQΓμ) ,
                                                = ∂μμQ + gμν [(∂νΩμ)Q + Ωμ(∂νQ) − (∂νQ)ΓμQ(∂νΓμ) ] ,

Here the derivations ∂νQ can be eliminated due to the vanishing of (9.5)

(9.9)                 DμDμQ = ∂μμQ + gμν [(∂νΩμ)Q + Ωμ(QΓνΩνQ) − (QΓνΩνQ) ΓμQ(∂νΓμ)] ,

or rewritten

            DμDμQ = ∂μμQ + gμν [(∂νΩμΩμΩν)Q + ΩμQΓν + ΩνQ ΓμQ(∂νΓμ + ΓνΓμ) ]
(9.10)                   = ∂μμQ + [(∂μΩμΩμΩμ)Q + 2ΩμQ ΓμQ(∂μΓμ + ΓμΓμ) ]
                         = o2Q + [(∂μμ − ∂μμ + ∂μΩμΩμΩμ)Q + 2ΩμQ ΓμQ(∂μΓμ + ΓμΓμ) ] ,

where o2Q is the d'Alembertian operator ∂μμ .

Remark
M.W. Evans uses two different d'Alembertian operators with the same notation: In [1; (2.188)] we read o := ∂μμ, while in [1; (9.42)] o := ∂μμ is used. For matter of distinction we index both versions:
                                o1 := ∂μμ         and         o2 := ∂μμ.
The use of o2 instead of o1 causes additional terms, which will be marked in red in the following.

Due to

                                (o2o1)Q = (∂μμ − ∂μμ)Q = ∂μ(gμννQ) − gμνμνQ
                                                                = (∂μgμν) ∂νQ = (∂νgμν) ∂μQ = (∂νgμν) (QΓμΩμQ)

we obtain from (9.10)

(9.11)        DμDμQ = o2Q + [(∂μΩμΩμ (Ωμ + ∂νgμν))Q + 2ΩμQ ΓμQ(∂μΓμ + Γμ (Γμ − ∂νgμν))] ;

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

(9.12)        o2Q + [(∂μΩμΩμ (Ωμ + ∂νgμν))Q + 2ΩμQ ΓμQ(∂μΓμ + Γμ (Γμ − ∂νgμν))] = O .

This is a linear partial differential equation for the tetrad matrix Q the principal part of which is the d'Alembertian o2Q, i.e. we have a "wave equation" for Q.

For comparison let's have a look at the "Evans Lemma" [1; Sect. 9.2]: After having introduced the "scalar curvatures" R1 and R2 by

R1 qλa := (Dμωaμb) qλb − (Dμ Γνμλ) qνa .                                 (9.44)

and

R2 qλa := − Γννμ ωμab qλb + Γννμ Γμνλ qνa ,                                 (9.48)

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

o2 qλa = R qλa ,                                                                 (9.49)

where

R = R1 + R2.                                                                 (9.50)

We remark that the "curvatures" R1 and R2depend 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 qλa. Especially there is no reason to consider equ. (9.49) as an eigenvalue problem of the dAlembertian operator o2Q 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 dAlembertian operator o, and the eigenfunction in this case is the tetrad. In Sec. 3 we will show that the eigenfunction can be any differential form, thus introducing a powerful class of wave equations to differential geometry and physics. The lemma is the subsidiary proposition leading to the Evans wave Eq. (9.2) through Eq. (9.3). The lemma is an identity of differential geometry, and so is comparable in generality and power to the well known Poincar lemma [14]. In other words, new theorems of topology can be developed from the Evans lemma in analogy with topological theorems [2,14] from the Poincar lemma. This can be the subject of future work in mathematics, work which may lead in turn to new findings in physics based on topology. The immediate importance of the lemma to physics is that it is the subsidiary proposition leading to the Evans wave equation, which is valid for all radiated and matter fields. Equation (9.49) can be solved for R given tetrad components, or vice versa, solved for tetrad components for a given R. The equation is non-linear in the spin and Christoffel connections, but for a given R it is a linear second order partial differential wave equation, or eigenequation. In this sense it is an equation of wave mechanics and therefore of quantum mechanics, and so unifies quantum mechanics, unified field theory and general relativity. Its power is therefore apparent and the wave equation (9.49) reduces to known equations of physics [3-7] in the appropriate limits. These include the four Newton equations, the Poisson equations of gravitation and electrostatics, the Schrodinger, Klein-Gordon and Dirac equations, and the equations of O(3) electrodynamics. Equation (9.49) produces the quark color triplet through a choice of eigenfunction (a three-spinor of the SU(3) representation), and so unifies the gravitational and strong fields. . . ."

Errors in Chap.13 and Chap.17

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

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





References

[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://www.mathematik.tu-darmstadt.de/~bruhn/Comment-Chap2.htm