Last update: Febr 22, 2006

Remarks on the "Evans Wave Equation"

Gerhard W. Bruhn, Darmstadt University of Technology

1. Evans' book [1]

Quote from M.W. Evans [1; Chap. 4.2]:

4.2 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 , (4.7)

D^μ Γ_μ^ρ_ρ = ∂^μ Γ_μ^ρ_ρ + Γ^μρ_λ Γ_μ^λ_ρ (4.8)

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

Here o is ∂^μ∂_μ as can be seen from the context.

The operators D^μ and D_μ of Equ.(4.7) 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

(1.1) D_μF = ∂_μF

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

(1.2) D_μ F^ν = ∂_μF^ν + Γ_μ^ν_σ F^σ

while a (0,1) tensor yields

(1.3) D_μF_ν = ∂_μF_ν − Γ_μ^σ_ν F_σ .

Tetrad related quantities F_a and F^a give

(1.4) D_μF_a = ∂_μF_a − ω_μ^s_ν F_s .

and

(1.5) D_μF^a = ∂_μF^a + ω_μ^a_s F^s .

Remark
One should notice that the covariant derivatives D^μ and D_μ cannot be applied to quantities that are no tensor components in some sense. E.g. quantities like D_μ Γ_ρ^ν_σ or D_μ ω_ρ^s_ν are not defined since the coefficients Γ_ρ^ν_σ and ω_ρ^s_ν are no tensor components. Therefore greater parts of Evans calculations are senseless. Especially the Eqns. (4.7-8) contain the not defined quantity D^μ Γ_μ^ρ_ρ. Here we can eliminate the undefined quantity to obtain

D^ρD_ρ = o + ∂^μ Γ_μ^ρ_ρ + Γ^μρ_λ Γ_μ^λ_ρ . (4.7')

However, even that equation must be handled with caution since the operator D^ρD_ρ is context-sensitive and only applicable to tensor components. ¨

Therefore it appears very unlikely that Equ.(4.7') 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.(4.7) with his equation

D^ρD_ρ = o + D^μ Γ^ρ_μρ + 2 Γ^λ_μρ D_λ, (4.14)

In Equ.(4.16) applies the operator D^μD_μ of Equ.(4.14) 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
(4.16)
= (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
(4.16')
= (o + D^μ Γ^ρ_μρ) q_ν^a = 0 .

Finally we can eliminate here the undefined expression D^μ Γ^ρ_μρ by means of Evans' Equ.(4.8) to obtain

(o + ∂^μ Γ^ρ_μρ + Γ^μρ_λ Γ^λ_μρ) q_ν^a = 0 . (4.16")

2. An independent calculation

The author's reasoning and calculation for his result (4.16') is rudimentary and not convincing. Therefore we are going to check (4.16") 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 q_ν^s .

A well-known frame relation gives

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

where o is the d'Alembertian ∂^ρ∂_ρ. 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.

Remark

The same result (2.7) was already attained in [2; (2.6)] by a different calculation that in contrast to (2.4) started with the direct evaluation of the derivative o q_ν^a = ∂^μ∂_μ q_ν^a avoiding double covariant differentiation. ¨

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 (4.16). 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 (4.9)-(4.14), shows that the ωs are absent everywhere. So why not in his final equation.

However, in the calculation of Equ.(4.9) the ωs must have appeared at least firstly.

There is another spot where spin-coefficients must appear: The reasoning of Equ.(4.13) by means of Equ.(4.12), identical with our Equ.(1.2), 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. (4.16") 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 (4.16) 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 , (4.22)

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

Remark

According to Carroll [4; (3.67)] we have

R^ρ_σμν = ∂_μ Γ_ν^ρ_σ − ∂_ν Γ_μ^ρ_σ + Γ_μ^ρ_λ Γ_ν^λ_σ − Γ_ν^ρ_λ Γ_μ^λ_σ (C 3.67)

hence

R^μ_σμν = ∂_μ Γ_ν^μ_σ − ∂_ν Γ_μ^μ_σ + Γ_μ^μ_λ Γ_ν^λ_σ − Γ_ν^μ_λ Γ_μ^λ_σ (C 3.67')

and thus

R = g^σν R^μ_σμν = g^σν (∂_μ Γ_ν^μ_σ + Γ_μ^μ_λ Γ_ν^λ_σ) − g^σν (∂_ν Γ_μ^μ_σ + Γ_ν^μ_λ Γ_μ^λ_σ) (C 3.67")

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 Chapter 4, "Generally Covariant Wave Equation for Grand Unified Field Theory" of Evans' "GENERALLY COVARIANT UNIFIED FIELD THEORY" [1] is completely questionable. However, that is a central point of M.W. Evans "Einstein Cartan Evans Theory".

Final Remark

In [1; Chap.9.2] Evans felt to have to attack the above problem again. The attack starts on p.178 - and collapses immediately due to the same reasons as in [1; Chap.4]. Namely the conclusion from (9.40) to (9.41) is wrong since the covariant derivative D_μ is to be applied to the non-tensor expression ∂^μq_ν^a where it is not well-defined. In addition o here has the meaning ∂_μ∂^μ which is not identical with o = ∂^μ∂_μ in [1; Chap.4.2]. Thus Evans' result differs a little bit due to the modified calculation. The question arises why the author didn't remark the difference of his results, which should have been the reason for searching for calculation errors and/or flaws of thinking.

References

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

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

[3] W. A. Rodrigues Jr. and Q.A.G. de Souza, An ambigous statement called
the "tetrad postulate", arXiv [math-ph/0411085]
Int. J. Mod. Phys. D 12, 2095-2150 (2005)

[4] S. M. Carroll, Lecture Notes in Relativity", arXiv [math-ph/0411085]

*) 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.