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:

DρDρ = o + Dμ Γμρρ ,                                                                 (4.7)

Dμ Γμρρ = ∂μ Γμρρ + Γμρλ Γμλρ                                                                 (4.8)

is the covariant derivative of the index contracted Christoffel symbol Γμρρ.

Here o is ∂μμ as can be seen from the context.

The operators Dμ and Dμ of Equ.(4.7) operate context-sensitive, e.g. the result of an application of the covariant derivative Dμ (or Dμ) depends on the index type of the quantity where it is applied to. We have

(1.1)                                                 DμF = ∂μF

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

(1.2)                                                 Dμ Fν = ∂μFν + Γμνσ Fσ

while a (0,1) tensor yields

(1.3)                                                 DμFν = ∂μFν − Γμσν Fσ .

Tetrad related quantities Fa and Fa give

(1.4)                                                 DμFa = ∂μFa − ωμsν Fs .

and

(1.5)                                                 DμFa = ∂μFa + ωμas Fs .

Remark
One should notice that the covariant derivatives Dμ and Dμ cannot be applied to quantities that are no tensor components in some sense. E.g. quantities like Dμ Γρνσ or Dμ ωρsν are not defined since the coefficients Γρνσ and ωρsν are no tensor components. Therefore greater parts of Evans calculations are senseless. Especially the Eqns. (4.7-8) contain the not defined quantity Dμ Γμρρ. Here we can eliminate the undefined quantity to obtain

DρDρ = o + ∂μ Γμρρ + Γμρλ Γμλρ .                                                                 (4.7')

However, even that equation must be handled with caution since the operator DρDρ 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

DρDρ = o + Dμ Γρμρ + 2 Γλμρ Dλ,                                                 (4.14)

In Equ.(4.16) applies the operator DμDμ of Equ.(4.14) to his object of interest, the tetrad coefficient qμa :

Dρ(Dρqμa) := DρDρ eμa = (o + Dμ Γρμρ + 2 Γλμρ Dλ) eμa                                
                                                                                                                                                (4.16)
= (o + Dμ Γρμρ) eμa = 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

Dρ(Dρqνa) := DρDρ qνa = (o + Dμ Γρμρ + 2 Γλμρ Dλ) qνa                                
                                                                                                                                                (4.16')
= (o + Dμ Γρμρ) qνa = 0 .                                                                

Finally we can eliminate here the undefined expression Dμ Γρμρ by means of Evans' Equ.(4.8) to obtain

(o + ∂μ Γρμρ + Γμρλ Γλμρ) qνa = 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 DρDρ qνa independently.

The derivative Dρ is gρλDλ. In addition we have in accordance with the above rules for the covariant derivative

(2.1)                                                 Dλ qνa = ∂λ qνa − Γλσν qσa + ωλas qνs .

A well-known frame relation gives

(2.2)                                                 ∂λ qνa − Γλσν qσa + ωλas qνs = 0 .

Therefore we have

(2.3)                                                 Dλ qνa = 0

By virtue of the last equations we obtain

(2.4)                 0 = Dρ (Dρqνa) = ∂ρ(Dρqνa) = o qνa − ∂ρρσν qσa) + ∂ρρab qνb)

where o is the d'Alembertian ∂ρρ. We apply the Leibniz rule to obtain

(2.5)                 o qνa − (∂ρΓρσν) qσa − Γρσνρqσa + (∂ρωρab) qνb + ωρabρqνb = 0 .

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

                o qνa − (∂ρΓρσν) qσa + (∂ρωρab) qνb                                                                
(2.6)
                                − Γρτνρστ qσa − ωρab qτb) + ωρacρτν qτc − ωρcb qνb) = 0 .

or, after rearranging

                o qνa + (∂ρωρab − ωρac ωρcb) qνb
(2.7)
                                − (∂ρΓρσν + Γρτν Γρστ) qσa + 2 ωρab Γρτν qτb = 0 .

This is a system of linear second order partial differential equations all of them having the same principal part o qνa 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 qνa = ∂μμ qνa avoiding double covariant differentiation.                 ¨

The spin-free case

Then all ωs vanish and we obtain

(2.8)                 o qνa − (∂ρΓρσν) qσa − Γρσν Γρτσ qτa = 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 Γνρρ of the Christoffels Γνρλ as if Γνρρ were a (1,0) tensor. However, it is well-known that the Christoffels Γνρλ are no (1,2) tensor, so why should the contraction Γνρρ 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 Γρμρ + Γμρλ Γλμρ) qνa = − (∂ρΓρσν) qσa − Γρσν Γρτσ qτa

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 := Rμνμν = ∂μΓνμν − ∂νΓμμν + Γμμν Γννν − Γνμν Γμνν ,                                 (4.22)

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

Remark

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

Rρσμν = ∂μ Γνρσ − ∂ν Γμρσ + Γμρλ Γνλσ − Γνρλ Γμλσ                                 (C 3.67)

hence

Rμσμν = ∂μ Γνμσ − ∂ν Γμμσ + Γμμλ Γνλσ − Γνμλ Γμλσ                                 (C 3.67')

and thus

R = gσν Rμσμν = gσν (∂μ Γνμσ + Γμμλ Γνλσ) − gσν (∂ν Γμμσ + Γνμλ Γμλσ)                 (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 Dμ is to be applied to the non-tensor expression ∂μqνa where it is not well-defined. In addition o here has the meaning ∂μμ which is not identical with o = ∂μμ 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://www.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

Γρσν = ½ gσρ (∂μgνρ + ∂νgμρ − ∂ρgμν) ,

which yields Γμνν = ½ gνρμgνρ . 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 Γμνν cannot be deduced.




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