Last update: Febr 20, 2006

concerning the "Evans Lemma"

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

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

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

*D*^{ρ}*D*_{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*_{μ}*F*
=
∂_{μ}*F*

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

*D*_{μ} *F*^{ν}
=
∂_{μ}*F*^{ν}
+ Γ_{μ}^{ν}_{σ} *F*^{σ}

while a (0,1) tensor yields

*D*_{μ}*F*_{ν}
=
∂_{μ}*F*_{ν}
− Γ_{μ}^{σ}_{ν} *F*_{σ} .

Tetrad related quantities *F*_{a} and *F*^{a} give

*D*_{μ}*F*_{a}
=
∂_{μ}*F*_{a}
− ω_{μ}^{s}_{ν} *F*_{s} .

and

*D*_{μ}*F*^{a}
=
∂_{μ}*F*^{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*^{ρ}*D*_{ρ}
= o + *D*^{μ} Γ^{ρ}_{μρ}
+ 2 Γ^{λ}_{μρ} *D*_{λ},
(14)

In Equ.(16) applies the operator *D*^{μ}*D*_{μ} of Equ.(14) to
his object of interest, the tetrad coefficient *q*_{μ}^{a} :

*D*^{ρ}(*D*_{ρ}*q*_{μ}^{a})
:=
*D*^{ρ}*D*_{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*_{ρ}*q*_{ν}^{a})
:=
*D*^{ρ}*D*_{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*^{μ}Γ_{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")

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*^{ρ}*D*_{ρ} *q*_{ν}^{a}
independently.

The derivative *D*^{ρ} is
*g*^{ρλ}*D*_{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*_{ρ}*q*_{ν}^{a})
=
∂^{ρ}(*D*_{ρ}*q*_{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}
−
(∂^{ρ}Γ_{ρ}^{σ}_{ν}) *q*_{σ}^{a}
−
Γ_{ρ}^{σ}_{ν} ∂^{ρ}*q*_{σ}^{a}
+
(∂^{ρ}ω_{ρ}^{a}_{b}) *q*_{ν}^{b}
+
ω_{ρ}^{a}_{b} ∂^{ρ}*q*_{ν}^{b}
= 0 .

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

o *q*_{ν}^{a}
−
(∂^{ρ}Γ_{ρ}^{σ}_{ν}) *q*_{σ}^{a}
+
(∂^{ρ}ω_{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}
+
(∂^{ρ}ω_{ρ}^{a}_{b}
−
ω_{ρ}^{a}_{c} ω^{ρc}_{b})
*q*_{ν}^{b}

(2.7)

−
(∂^{ρ}Γ_{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.

Then all ωs vanish and we obtain

(2.8)
o *q*_{ν}^{a}
−
(∂^{ρ}Γ_{ρ}^{σ}_{ν}) *q*_{σ}^{a}
−
Γ_{ρ}^{σ}_{ν}
Γ^{ρτ}_{σ} *q*_{τ}^{a} = 0 .

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}
=
−
(∂^{ρ}Γ_{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}
=
∂_{μ}Γ_{ν}^{μ}_{ν}
−
∂_{ν}Γ_{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

[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://www.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}
(∂_{μ}*g*_{νρ}
+
∂_{ν}*g*_{μρ}
−
∂_{ρ}*g*_{μν}) ,

which yields
Γ_{μ}^{ν}_{ν}
=
½ *g*^{νρ}
∂_{μ}*g*_{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.