Last update: March 02, 2006

Summary

As we shall see in Sect.1 Evans introduces the tetrad coefficient
*q*_{a}^{μ}, which he believes to be
an essential tool of argumentation leading far beyond the limitations of General Relativity
because of giving the opportunity of modelling several other force fields of
modern physics in addition to gravitation.
In Sect.2 we'll see that Evans due to calculation errors has derived a wrong
"Evans Wave Equation" for the tetrad coefficients.
However, the most important point is, that Evans uses the tetrad coefficients for
"developing"
a wrong "Evans Field Theory" (Sect.3).
The main errors of that "theory"
are *invalid* field definitions:
They are simply invalid and therefore useless due to *type mismatch*.
That type mismatch is caused by the **bad habit** of suppressing
seemingly unimportant indices.
**There is no possibility of removing that tetrad indices a,b from Evans'
field theory, i.e. Evans' ECE Theory cannot be repaired.**

Preface

The following review of M.W. Evans'
"**E**instein **C**artan **E**vans Theory"
refers to Evans' book [1]. Quotes from Evans text will be displayed in
**black**
while the comments appear in **blue**.
Equation labels at the right hand side are referring to [1].

1. What Evans should have given first:

A clear description of his basic assumptions

Evans considers the spacetime as 4-dimensional manifold ** M**.
The tangent spaces

There is a pseudo-metric defined at the points P of ** M**
as a bilinear function
g :

(1.1)
*g*_{μν}
:= g(**e**_{μ},**e**_{ν}) ,

which is assumed to be nonsingular and of Lorentzian signature, i.e.
there exist vectors
**e _{a}** (

A linear transform L: *T*_{P} → *T*_{P}
that fulfils
g(L**e _{a}**,L

Each set of orthonormalized vectors
**e _{a}** (

(1.2)
**e**_{μ}
= *q*_{μ}** ^{a} e_{a}** .

From (1.1) and (1.2) we obtain due to the bilinearity of g

(1.3)
*g*_{μν}
= g(**e**_{μ},**e**_{ν}) ,
= *q*_{μ}^{a}*q*_{ν}** ^{b}**
g(

Therefore the matrix (*g*_{μν}) is symmetric.
And more general also g(**V**,**W**) = g(**W**,**V**)
for arbitrary vectors **V**, **W** of *T*_{P}.

A (non-Riemannian) linear connection is supposed, i.e. we have covariant derivatives *D*_{μ} of the
in direction of **e**_{μ} given by

(1.4)
*D*_{μ}*F*
:= ∂_{μ}*F*

for functions *F* (=(0,0)-tensors), while a (1,0)-tensor
*F*^{ν}
has the derivative

(1.5)
*D*_{μ}*F*^{ν}
:= ∂_{μ}*F*^{ν}
+
Γ_{μ}^{ν}_{ρ} *F*^{ρ}

and for a (0,1)-tensor *F*_{ν} we have

For coordinate dependent quantities the connection causes the additional terms
in Eqns.(1.5-6) with the coefficients
Γ_{μ}^{ρ}_{ν}

By the analogue way the connection gives rise to additional terms
with coefficients
ω_{μ}^{a}_{b}
for the covariant derivatives of tetrad dependent quantities, namely

(1.7)
*D*_{μ}*F*^{a}
:= ∂_{μ}*F*^{a}
+
ω_{μ}^{a}_{b} *F*^{b}

and

(1.8)
*D*_{μ}*F*_{a}
:= ∂_{μ}*F*_{a}
−
ω_{μ}^{b}_{a}
*F*_{b} .

2. Differential Equations for the tetrad coefficients

A well-known identity between coordinate frames and tetrads [5; (3.5)] says

(2.1)
∂_{μ}*q*_{ν}^{a}
+
ω_{μ}^{a}_{b}
*q*_{ν}^{b}
−
Γ_{μ}^{σ}_{ν}
*q*_{σ}^{a} = 0 ,

which is a first order linear system of pde for the tetrad coefficients.

We apply the operator
∂^{μ} = *g*^{μτ}∂_{τ}
to Equ.(2.1) to obtain with
o
:=
∂^{μ}∂_{μ}

(2.2)
o*q _{ν}*

or

(2.3)
o*q _{ν}*

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

(2.4)
∂^{μ}*q*_{ν}^{b}
=
Γ^{μρ}_{ν}*q*_{ρ}^{b}
−
*ω*^{μb}_{c}*
q*_{λ}^{c}
and
∂^{μ}*q*_{σ}^{a}
=
Γ^{μρ}_{σ}*q*_{ρ}^{a}
−
*ω*^{μa}_{c}*
q*_{σ}^{c} ,

to obtain

(2.5)
o*q _{ν}*

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 Wave Equation" (4.16) together with (4.8) in [1; Chap.4.2].

(o
+ ∂^{μ} Γ^{ρ}_{μρ}
+ Γ^{μρ}_{λ} Γ^{λ}_{μρ})
*q*_{ν}^{a}
= 0 .
(4.16")

Evans "multiplies" Einstein's Field equation

(3.1)
*R*^{ μν} − ½ *R* *g*^{μν}
= *T*^{ μν}

by *q*_{ν}^{b} η** _{ab}** to obtain

(3.2)
*R*_{a}^{μ} − ½ *R* *q*_{a}^{μ}
= *T*_{a}^{μ}

and suppresses the tetrad index **a**:

*R*^{μ} − ½ *R* *q*^{μ}
= *T*^{μ}
(3.18)

That leads to Evans' "2nd Newton Law":

*f*^{ μ}
= ^{∂Tμ}/_{∂τ}
(3.22)

where τ is the proper time.

However, (neglecting other counterarguments e.g. concerning the
"proper time") written *with the suppressed index* **a** we have

(3.3)
*f*^{ μ}
= ^{∂Taμ}/_{∂τ}
(**a** = **0**, **1**, **2**, **3**),

that are *four* force-vectors instead of one in GR.

Evans' Equ.(3.22) contains a

There is a further application of Equ.(3.2) or Evans' Equ.(3.18): We "wedge"
it by *q*_{b}^{ν} to obtain

(3.4)
*R*_{a}^{μ}
Ù
*q*_{b}^{ν}
−
½ *R* *q*_{a}^{μ}
Ù
*q*_{b}^{ν}
= *T*_{a}^{μ}
Ù
*q*_{b}^{ν}

and suppress the tetrad indices **a**,**b**

*R*^{μ}
Ù *q*^{ν}
−
½ *R* *q*^{μ}
Ù *q*^{ν}
= *T*^{μ}
Ù *q*^{ν} .
(3.25)

**Remark**

The wedge product used by Evans here is the wedge product of vectors
**A** = *A*^{μ}**e**_{μ}:

**A**Ù**B**
= ½
(*A*^{μ}*B*^{ν}
−
*A*^{ν}*B*^{μ})
**e**_{μ}Ù**e**_{ν}

written in short hand as

*A*^{μ}Ù*B*^{ν}
:=
*A*^{μ}*B*^{ν}
−
*A*^{ν}*B*^{μ} .

Evans remarks the term *R*^{μ}
Ù *q*^{ν}
being antisymmetric like the electromagnetic stress tensor
*G*^{μν} . Hence he feels encouraged
to try the following ansatz

*G*^{μν}
=
*G*^{(0)} (*R*^{μν(A)} −
½ *R* *q*^{μν(A)})
(3.29)

where

*R*^{μν(A)}
=
*R*^{μ}
Ù *q*^{ν} ,
*q*^{μν(A)}
=
*q*^{μ}
Ù *q*^{ν} .
(3.26-27)

Thus, Evans' ansatz (3.29) *with written tetrad indices* is

(3.5)
*G*^{μν}
=
*G*^{(0)} (*R*_{a}^{μ}Ù *q*_{b}^{ν}
−
½ *R* *q*_{a}^{μ}
Ù *q*_{b}^{ν}) .

However, by comparing the left hand side and the right hand side it is evident
that the ansatz cannot be correct due to *type mismatch*:
The tetrad indices are not available at the left hand side.

The tetrad indices **a**,**b** must be removed *legally* .
The only way to do so is to sum over **a**,**b** with some weight
factors χ^{ab}, i.e. to insert a factor
χ^{ab} on the right hand side of (3.29),
i.e. (3.5) in our detailed representation.
Our first choice for χ^{ab}
is the Minkowskian η^{ab}.
However, then the right hand side of (3.29) vanishes since we have

(3.6)
*q*_{a}^{μ}
Ù *q*_{b}^{ν}
η^{ab}
=
*q*_{a}^{μ}
*q*_{b}^{ν}
η^{ab}
−
*q*_{a}^{ν}
*q*_{b}^{μ}
η^{ab}
=
*g*^{μν}
−
*g*^{νμ}
= 0

and

(3.7)
*R*_{a}^{μ}
Ù *q*_{b}^{ν}
η^{ab}
=
*R*_{a}^{μ}
*q*_{b}^{ν}
η^{ab}
−
*R*_{a}^{ν}
*q*_{b}^{μ}
η^{ab}
=
*R*^{μν} − *R*^{νμ} = 0

due to the symmetry of the metric tensor *g*^{μν}
and of the Ricci tensor *R*^{μν} [4; (3.91)].

One could try to find a matrix (χ^{ab}) different from
the Minkowskian to remove the indices **a**,**b** from Evans equations
(3.25-29). That matrix should not depend on the special tetrad under
consideration i.e. be invariant under arbitrary
Lorentz transforms L:

(3.8)
*L*_{c}** ^{a}**
χ

However, due to the definition of the Lorentz transforms the matrices λ (η

Therefore we may conclude
that only a trivial **zero** em-field
*G*^{μν}
can fulfil the *corrected* Evans field ansatz.

the trivial zero case merely and is irreparably therefore.

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

[1a]
M.W. Evans, *A Generally Covariant Field Equation for Gravitation
and Electromagnetism*,

Foundations of Physics Letters Vol.16 No.4, 369-377

[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]

[5]
G. W. Bruhn and W. A. Rodrigues Jr., *Covariant Derivatives of Tensor Components*,

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