Last update: Febr 20, 2006

Remarks on an Evans Paper [1]
concerning the "Evans Lemma"

Gerhard W. Bruhn, Darmstadt University of Technology

1. Evans' paper

The Evans paper is a slightly modified copy of Chap. 4 of his book manuscript [2].

Quote from M.W. Evans [1; Section 2]:

2 Derivation Of The Generally Covariant Wave Equation

The wave equation is based on the following expression for the covariant d'Alembertian operator:

D^ρD_ρ = o + D^μ Γ_μ^ρ_ρ , (7)

where

D^μ Γ_μ^ρ_ρ = ∂^μ Γ_μ^ρ_ρ + Γ^μρ_λ Γ_μ^λ_ρ (8)

is the covariant derivative of the index contracted Christoffel symbol Γ_μ^ρ_ρ.

The operators D^μ and D_μ of Equ.(7) operate context-sensitive, e.g. the result of an application of the covariant derivative D_μ (or D^μ) depends on the index type of the quantity where it is applied to. We have

D_μF = ∂_μF

for a (0,0)-tensor F, a funktion; for a (1,0) tensor F^ρ we obtain

D_μ F^ν = ∂_μF^ν + Γ_μ^ν_σ F^σ

while a (0,1) tensor yields

D_μF_ν = ∂_μF_ν − Γ_μ^σ_ν F_σ .

Tetrad related quantities F_a and F^a give

D_μF_a = ∂_μF_a − ω_μ^s_ν F_s .

and

D_μF^a = ∂_μF^a + ω_μ^a_s F^s .

Therefore it appears very unlikely that Equ.(7) should be true without regard of the operand where it is applied to. And evidently the author Evans has the same doubts as can be seen by comparing Equ.(7) with his equation

D^ρD_ρ = o + D^μ Γ^ρ_μρ + 2 Γ^λ_μρ D_λ, (14)

In Equ.(16) applies the operator D^μD_μ of Equ.(14) to his object of interest, the tetrad coefficient q_μ^a :

D^ρ(D_ρq_μ^a) := D^ρD_ρ e_μ^a = (o + D^μ Γ^ρ_μρ + 2 Γ^λ_μρ D_λ) e_μ^a
(16)
= (o + D^μ Γ^ρ_μρ) e_μ^a = 0 .

Here the index μ appears in double meaning, which could cause misunderstandings. In addition the quantity e is to be read as q. From the context it is clear that the author's claim is

D^ρ(D_ρq_ν^a) := D^ρD_ρ q_ν^a = (o + D^μ Γ^ρ_μρ + 2 Γ^λ_μρ D_λ) q_ν^a
(16')
= (o + D^μ Γ^ρ_μρ) q_ν^a = 0 .

Finally we can insert here Evans' Equ.

D^μΓ_μ^ρ_ρ = ∂^μ Γ^ρ_μρ + Γ^μρ_λ Γ^λ_μρ , (13)

to obtain

(o + ∂^μ Γ^ρ_μρ + Γ^μρ_λ Γ^λ_μρ) q_ν^a = 0 . (16")

2. An independent calculation

The author's reasoning and calculation for his result (16') is rudimentary and not convincing. Therefore we are going to check (16") now by calculating D^ρD_ρ q_ν^a independently.

The derivative D^ρ is g^ρλD_λ. In addition we have in accordance with the above rules for the covariant derivative

(2.1) D_λ q_ν^a = ∂_λ q_ν^a − Γ_λ^σ_ν q_σ^a + ω_λ^a_s .

A well-known frame relation gives

(2.2) ∂_λ q_ν^a − Γ_λ^σ_ν q_σ^a + ω_λ^a_s = 0 .

Therefore we have

(2.3) D_λ q_ν^a = 0

By virtue of the last equations we obtain

(2.4) 0 = D^ρ (D_ρq_ν^a) = ∂^ρ(D_ρq_ν^a) = o q_ν^a − ∂^ρ(Γ_ρ^σ_ν q_σ^a) + ∂^ρ(ω_ρ^a_b q_ν^b)

We apply the Leibniz rule to obtain

(2.5) o q_ν^a − (∂^ρΓ_ρ^σ_ν) q_σ^a − Γ_ρ^σ_ν ∂^ρq_σ^a + (∂^ρω_ρ^a_b) q_ν^b + ω_ρ^a_b ∂^ρq_ν^b = 0 .

Here the partials of q can be removed by means of the frame identity (2.3):

o q_ν^a − (∂^ρΓ_ρ^σ_ν) q_σ^a + (∂^ρω_ρ^a_b) q_ν^b
(2.6)
− Γ_ρ^τ_ν (Γ^ρσ_τ q_σ^a − ω^ρa_b q_τ^b) + ω_ρ^a_c (Γ^ρτ_ν q_τ^c − ω^ρc_b q_ν^b) = 0 .

or, after rearranging

o q_ν^a + (∂^ρω_ρ^a_b − ω_ρ^a_c ω^ρc_b) q_ν^b
(2.7)
− (∂^ρΓ_ρ^σ_ν + Γ_ρ^τ_ν Γ^ρσ_τ) q_σ^a + 2 ω^ρa_b Γ_ρ^τ_ν q_τ^b = 0 .

This is a system of linear second order partial differential equations all of them having the same principal part o q_ν^a and no first order derivatives. However, the equations cannot be separated, all of them are linked by the derivative-free terms.

The spin-free case

Then all ωs vanish and we obtain

(2.8) o q_ν^a − (∂^ρΓ_ρ^σ_ν) q_σ^a − Γ_ρ^σ_ν Γ^ρτ_σ q_τ^a = 0 .

3. Comparison with Evans' result

While our general equation (2.7) contains spin coefficients ω they are remarkably missing in Evans equation (16). So Evans equation does not depend on whether there is spin or not. That the spin-coefficients ω do not appear in Evans' equation is amazing. However, one glance at the rudiments of his calculation, at the equations (9)-(14), shows that the ωs are absent everywhere. So why not in his final equation.

However, in the calculation of Equ.(9) the ωs must have appeared at least firstly.

There is another spot where spin-coefficients must appear: The reasoning of Equ.(13) by means of Equ.(12), which is correct for a (1,0) tensor. Evans applies this rule to the contraction Γ_ν^ρ_ρ of the Christoffels Γ_ν^ρ_λ as if Γ_ν^ρ_ρ were a (1,0) tensor. However, it is well-known that the Christoffels Γ_ν^ρ_λ are no (1,2) tensor, so why should the contraction Γ_ν^ρ_ρ of non-tensors be a tensor in general ,i.e. for non-Riemannian connection? *)

There is a third objection against the separated form of Evans' differential equations: Due to Evans each tetrad coefficient is solution of the same linear differential equation independent from the other coefficients. This is very unlikely even in the spin-free case: Comparing the Eqns. (16") and (2.8) we obtain

(3.1) (∂^μ Γ^ρ_μρ + Γ^μρ_λ Γ^λ_μρ) q_ν^a = − (∂^ρΓ_ρ^σ_ν) q_σ^a − Γ_ρ^σ_ν Γ^ρτ_σ q_τ^a

which would be a set of 16 linear homogeneous restrictions to be fulfilled by the 16 tetrad coefficients q.

In the rest of the paper Evans tries to transform his bad equation (16) to a form where the scalar curvature R appears. A remarkably mis-formed equation in that context is contained in:

Quote with errors marked in red:

R is found by a contraction of indices in the Riemann tensor, which has several well-known symmetry properties [2], for example � it is anti-symmetric in its last two indices. Using the following choice of index contraction:

R := R^μ_νμν = ∂_μΓ_ν^μ_ν − ∂_νΓ_μ^μ_ν + Γ_μ^μ_ν Γ_ν^ν_ν − Γ_ν^μ_ν Γ_μ^ν_ν , (22)

the author himself should bring to formal correctness (if possible).

These dark points show that Evans result does not at all deserve confidence. With respect of our different result attained by a correct calculation I think that

Evans result is wrong.

That means that his paper "Generally Covariant Wave Equation for Grand Unified Field Theory" is completely questionable. However, that is a central point of M.W. Evans "Einstein Cartan Evans Theory".

References

[1] Myron W. Evans, A Generally Covariant Wave Equation for Grand Unified Field Theory
submitted to arXiv as a slightly modified copy of Chap. 4 of his book manuscript [2].

[2] M.W. Evans, GENERALLY COVARIANT UNIFIED FIELD THEORY:
THE GEOMETRIZATION OF PHYSICS; Web-Preprint,
http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[3] G.W. Bruhn, Remarks on the "Evans Lemma" ;
http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansLemma.html

[4] W. A. Rodrigues Jr. and Q.A.G. de Souza, An ambigous statement called
the "tetrad postulate",
Int. J. Mod. Phys. D 12, 2095-2150 (2005)

*) For a Riemanian torsion-free and metric compatibel connection we have

Γ_ρ^σ_ν = ½ g^σρ (∂_μg_νρ + ∂_νg_μρ − ∂_ρg_μν) ,

which yields Γ_μ^ν_ν = ½ g^νρ ∂_μg_νρ . That expression can be shown not to transform as a tensor under coordinate transforms. Thus, even in that well-known case a tensor property of the contraction Γ_μ^ν_ν cannot be deduced.

Remarks on an Evans Paper [1] concerning the "Evans Lemma"