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

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                                                                
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

ω₁ = ds² = q^ρσ(S) dx_σdx_ρ ,                                                 (2.16)

while the antisymmetric metric yields the line element ds by

ω₂ = ds² = – ¹/₆ q^ρσ(A) dx_σ Ù dx_ρ ,                                                 (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 (dx₀Ùdx₁ + dx₀Ùdx₂ + dx₀Ùdx₃ + dx₁Ùdx₂ + dx₂Ùdx₃ – dx₁Ùdx₃)

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.

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 (q^ρσ(A)) is able to produce a metric by means of Equ. (2.17) that is given by a symmetric matrix (q^ρσ(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 Ù can both be expressed by means of the (non-commutative) tensor product Ä: We have

(2.1)                                 dx_σdx_ρ = ½ (dx_σÄdx_ρ + dx_ρÄdx_σ)

and likewise

(2.2)                                 dx_σÙdx_ρ = ½ (dx_σÄdx_ρ − dx_ρÄdx_σ) .

Inserting (2.1) into (2.16) we obtain

q^ρσ(S) dx_σdx_ρ = ½ (q^ρσ(S) dx_ρÄdx_σ + q^ρσ(S) dx_σÄdx_ρ),

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

q^ρσ(S) dx_σdx_ρ = ½ (q^ρσ(S) dx_ρÄdx_σ + q^σρ(S) dx_ρÄdx_σ),

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

(2.3)                                 q^ρσ(S) dx_σdx_ρ = q^ρσ(S) dx_ρÄdx_σ ,

i.e. the (symmetric) metric (2.16) can be expressed by the symmetric tensor q^ρσ(S) dx_ρÄdx_σ. The result is therefore:

The expression q^ρσ(S) dx_ρÄdx_σ defines a (symmetric) metric
if and only if the matrix (q^ρσ) is symmetric.

The analogous procedure can be applied to q^ρσ(A) dx_σ Ù dx_ρ:

q^ρσ(A) dx_σÙ dx_ρ = ½ (q^ρσ(A) dx_ρÄdx_σ − q^ρσ(A) dx_σÄdx_ρ),

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

q^ρσ(A) dx_σÙdx_ρ = ½ (q^ρσ(A) dx_ρÄdx_σ − q^σρ(A) dx_ρÄdx_σ).

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)dx_σÄdx_ρ. This yields the result:

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

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"

h₀²     h₀h₁     h₀h₂     h₀h₃                                                                
h₁h₀       h₁²     h₁h₂     h₁h₃                                                                
.                                                 (2.42)
h₂h₀     h₂h₁       h₂²     h₂h₃                                                                
h₃h₀     h₃h₁     h₃h₂       h₃²                                                                

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_μh_νξ ^μξ ^ν = (h_μξ ^μ)² > 0 .

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

ds² < 0 for "space like" directions and ds² > 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 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 dx_σdx_ρ as a sum of tensor products of 1-forms, strictly speaking as a sum of 2-forms, is a 2-form again:

ω₁ = ½ (q^ρσ(S) dx_ρÄdx_σ + q^σρ(S) dx_ρÄdx_σ) .

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 (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(∂_μ,∂_ν). Equivalence of the metric g or the matrix (g_μν) to (η_ab) means that there exists an invertible matrix (q_μ^a) such that

(5.1)                                 g_μν = η_ab q_μ^a q_ν^b.

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

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

1 = det(g_μν) = det (η_ab) det(q_μ^a) det(q_ν^b) = (−1) · det(q_μ^a)² = − det(q_a^μ)² ,

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(g_μν) = 0 . Hence by taking the determinants on both sides of Equ.(5.1) we would obtain

0 = det(g_μν) = det (η_ab) det(q_μ^a) det(q_ν^b) = (−1) · det(q_μ^a)² = − det(q_μ^a)² ,

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

6. A fatal metric error

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 q_a^μ 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

q^μν(S) = q_a^μq_b^ν η^ab .                                                                 (2.73')

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^(S) = q_μ^aq_ν^b η_ab .                                                                 (2.73'')

So Evans obtains the well-known relation

q^μν(S)q_μν^(S) = 4 .                                                                 (2.137)

Then he concludes (in Evans' terms)

4 = q^μν(S) q_μν^(S) = q^μq^νq_μq_ν = (q^μq_μ)² ,                                         (2.145)

i.e. written with all indices

4 = q^μν(S) q_μν^(S) = q_a^μ q_a^ν η^ab q_μ^a' q_ν^b' η_a'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

q_a^μq_b^νη^ab q_μ^a'q_ν^b'η_a'b' =                                                                                                
(−q_μ⁰q_ν⁰ + q_μ¹q_ν¹ + q_μ²q_ν² + q_μ³q_ν³) (−q₀^μq₀^ν + q₁^μq₁^ν + q₂^μq₂^ν + q₃^μq₃^ν)

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_μ⁰q_ν⁰ + q_μ¹q_ν¹. Thus, all the more generally, the substitute for q^μq_μ, the expression (? ? ?) cannot be defined. Hence Evans' further steps to prove his equ.(2.149) cannot be executed:

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

q_a^μq_μ^b = δ_a^b ,

since the matrices (q_a^μ) and (q_μ^b) are mutually inverses. Especially the choice a=b yields

q_a^μq_μ^a = δ_a^a = 4 .

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 q^ν (= q_c^ν) in Evans' dangerous short hand yields:

(D_ρq_μ) q_ν q^ν + q_μ (D_ρq_ν) q^ν = 0                                                 (2.173a)

The extended version is

(D_ρq_μ^a) η_ab q_ν^b q_c^ν + q_μ^a (D_ρq_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_ρq_μ^a) η_ab δ^b_c + q_μ^a (D_ρq_ν^b) η_ab q_c^ν = 0

or

(D_ρq_μ^a) η_ac + q_μ^a (D_ρq_ν^b) η_ab q_c^ν = 0 .

Now Evans multiplies by q^μq_ν , i.e. by q_a'^μ q_ν^c' , to obtain

q_a'^μ q_ν^c' (D_ρq_μ^a) η_ac + q_a'^μ q_μ^a (D_ρq_ν^b) η_ab q_c^ν q_ν^c' = 0

or

q_a'^μ q_ν^c' (D_ρq_μ^a) η_ac + δ_a'^a (D_ρq_ν^b) η_ab δ_c^c' = 0

or for c'=c

q_a'^μ q_ν^c (D_ρq_μ^a) η_ac + (D_ρq_ν^b) η_a'b = 0

This equation can be resolved for D_ρq_ν^b by multiplication by η^a'b'

η^a'b' q_a'^μ q_ν^c (D_ρq_μ^a) η_ac + (D_ρq_ν^b) η_a'bη^a'b' = 0

or

η^a'b q_a'^μ q_ν^c (D_ρq_μ^a) η_ac + D_ρq_ν^b = 0

Now Evans sets μ=ν.

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

D_ρq_ν^b = 0 .                                                                 (2.179)

8. Equation (2.182) and following

Evans' short hand equation

D_νq^μ = ∂_νq^μ + Γ_λ^μ_ν q^λ = 0                                                 (2.182)

reads extended

D_νq_a^μ = ∂_νq_a^μ + Γ_λ^μ_ν q_a^λ = 0 .                                             (2.182')

However, the following compatibility condition

∂_νq_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_νq^μ = q_λ ∂_νq^μ + q_λ q^λ Γ_λ^μ_ν = 0 ,                                                 (2.183)

i.e. in extended version

q_λ^a D_νq_a^μ = q_λ^a ∂_νq_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 ∂_νq_a^μ .                                                 (2.184')

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

9. The Evans Lemma

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_μQ := (∂_μq_λ^a + ω^a_μb q_λ^b − Γ^ν_μλ q_ν^a)

defines a linear differential operator D_μ that anihilates the matrix Q = (q_λ^a):

(9.2)                                                D_μQ := (∂_μq_λ^a + ω^a_μb q_λ^b − Γ^ν_μλ q_ν^a) = O .

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

(9.3)                                                DQ := (∂q_λ^a + ω^a_b q_λ^b − Γ^ν_λ q_ν^a)

Here we introduce the matrices Ω := (ω^a_b) and Γ := (Γ^ν_λ) . Then we have

(9.4)                                                DQ = ∂Q + ΩQ − QΓ ,

or, with restored index μ and replaced with ν

(9.5)                                                 D_νQ = ∂_νQ + Ω_νQ − QΓ_ν ( = O) .

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

(9.6)                                 D_νD_μQ := ∂_ν(D_μQ) + Ω_ν(D_μQ) − (D_μQ)Γ_ν ,

hence by using the operators D^μ := g^μνD_ν and ∂^μ := g^μν∂_ν also

(9.7)                                 D^μD_μQ = ∂^μ(D_μQ) + Ω^μ(D_μQ) − (D_μQ)Γ^μ .

where Ω^μ := g^μνΩ_ν and Γ^μ := g^μνΓ_ν .

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

(9.8)                               D^μD_μQ = ∂^μ(D_μQ) = ∂^μ(∂_μQ + Ω_μQ − QΓ_μ)
                                                = ∂^μ∂_μQ + g^μν ∂_ν(Ω_μQ − QΓ_μ) ,
                                                = ∂^μ∂_μQ + g^μν [(∂_νΩ_μ)Q + Ω_μ(∂_νQ) − (∂_νQ)Γ_μ − Q(∂_νΓ_μ) ] ,

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

(9.9)                 D^μD_μQ = ∂^μ∂_μQ + g^μν [(∂_νΩ_μ)Q + Ω_μ(QΓ_ν − Ω_νQ) − (QΓ_ν − Ω_νQ) Γ_μ − Q(∂_νΓ_μ)] ,

or rewritten

            D^μD_μQ = ∂^μ∂_μQ + g^μν [(∂_νΩ_μ − Ω_μΩ_ν)Q + Ω_μQΓ_ν + Ω_νQ Γ_μ − Q(∂_νΓ_μ + Γ_νΓ_μ) ]
(9.10)                   = ∂^μ∂_μQ + [(∂^μΩ_μ − Ω_μΩ^μ)Q + 2Ω_μQ Γ^μ − Q(∂^μΓ_μ + Γ^μΓ_μ) ]
                         = o₂Q + [(∂^μ∂_μ − ∂_μ∂^μ + ∂^μΩ_μ − Ω_μΩ^μ)Q + 2Ω_μQ Γ^μ − Q(∂^μΓ_μ + Γ^μΓ_μ) ] ,

where o₂Q is the d'Alembertian operator ∂_μ∂^μ .

Due to

we obtain from (9.10)

(9.11)        D^μD_μQ = o₂Q + [(∂^μΩ_μ − Ω_μ (Ω^μ + ∂_νg^μν))Q + 2Ω_μQ Γ^μ − Q(∂^μΓ_μ + Γ_μ (Γ^μ − ∂_νg^μν))] ;

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

(9.12)        o₂Q + [(∂^μΩ_μ − Ω_μ (Ω^μ + ∂_νg^μν))Q + 2Ω_μQ Γ^μ − Q(∂^μΓ_μ + Γ_μ (Γ^μ − ∂_νg^μν))] = 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 q_λ^a := (D^μω^a_μb) q_λ^b − (D^μ Γ^ν_μλ) q_ν^a .                                 (9.44)

and

− R₂ 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

o₂ q_λ^a = R q_λ^a ,                                                                 (9.49)

where

R = R₁ + R₂.                                                                 (9.50)

Errors in Chap.13 and Chap.17

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

Chap.17: http://www2.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://www2.mathematik.tu-darmstadt.de/~bruhn/Comment-Chap2.htm