Last update: 01.04.2005, 10:00 CEST

01.04.2004: Remark inserted in Sect.2

Evans starts the "proof" of his "Evans Lemma" by applying a "covariant differentiation"
operator *D*^{μ} to his so-called "tetrad postulate", the compatibility
relation of frames (see e.g. [2; Sect.4])

(1.1)
∂_{μ}*q _{λ}*

without telling the reader how the operator

*D*^{μ}
(∂_{μ}*q _{λ}*

Further more he assumes that the operator *D*^{μ} would
anihilate the tetrad coefficients *q*_{ν}^{a}.
Since the existence of such an operator is at least dubious, we shall give an independent
derivation below (see Sect. 2).

However, before let's have a look at the "Evans Lemma" itself:
After having introduced
the "scalar curvatures" *R*_{1} and *R*_{2} by

− *R*_{1}
*q*_{λ}^{a}
:=
(*D*^{μ}*ω*^{a}_{μb})
*q _{λ}*

and

− *R*_{2}
*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

o *q*_{λ}^{a}
= *R*
*q*_{λ}^{a}
,
(9.49)

where

*R* = *R*_{1} + *R*_{2}.
(9.50)

We remark that the "curvatures" *R*_{1} and
*R*_{2}depend on the tetrad coefficients, while
the main equation (9.49) pretends to be a single linear equation for
each single tetrad coefficients. Actually, however, equ. (9.50)
represents a *nonlinear system of partial differential equations* for
the 16 tetrad coefficients *q*_{λ}^{a}.
Especially there is no reason to consider equ. (9.49) as an eigenvalue problem
of the d’Alembertian operator o, since the "eigenvalues" *R*
are non constant and solution dependent. Therefore all conclusions contained
in the following quotation from Evans' book are senseless:

*
"Given the tetrad postulate, the lemma shows that scalar curvature R
is always an eigenvalue of the wave equation (9.49) for all spacetimes, in other
words, R is quantized. The eigenoperator is the d’Alembertian operator
*o

The compatibility relation of frames

(2.1)
∂_{μ}*q _{λ}*

yields

(2.2)
∂^{μ}
(∂_{μ}*q _{λ}*

**Remark**

M.W. Evans uses two *different* d'Alembertian operators *with the same notation*:
In [1; (2.188)] we read
o
:= ∂^{μ}∂_{μ},
while in [1; (9.42)]
o
:= ∂_{μ}∂^{μ}
is used. For matter of distinction we index both versions:

o_{1}
:= ∂^{μ}∂_{μ} and
o_{2}
:= ∂_{μ}∂^{μ}.

We'll use here o_{1}.
The use of o_{2} instead of
o_{1} causes additional terms;
see Sect.9 in

http://www.mathematik.tu-darmstadt.de/~bruhn/MWEsErrors.html .

With
o_{1}
:=
∂^{μ}∂_{μ}
we obtain

(2.3)
o_{1}*q _{λ}*

or

(2.4)
o_{1}*q _{λ}*

where the first derivatives of *q*_{λ}^{b}
and *q*_{ν}^{a}
can be removed by

(2.5)
∂^{μ}*q _{λ}*

to obtain

(2.6)
o_{1}*q _{λ}*

The result is a *system of linear partial differential equations* for the
tetrad coefficients, all of which having the principal part
of the wave equation. That is the substitute for the wrong
"Evans Lemma" (9.49-50) in Chap. 9.2 of Evans' book.

[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: Covariant Derivatives and the Tetrad Postulate:

http://www.mathematik.tu-darmstadt.de/~bruhn/covar_deriv.htm