Last Update: Sept 22, 2005, 8:00 pm

*"Well Gerhard you always say that any calculation is wrong.
So if you say
Schneeberger is wrong you are simply behaving in the same old way.
This is very boring. You have no credibility and so I request you not to send me any further absurd e mail, or any e mail. ...
If you go on like this you will certainly get a haircut in the Tower. ...
British Civil List scientist."*

Myron, excuse me, you are always continuing with producing a series of further errors and inconsistencies,
and so your critics will continue their job risking "a haircut in the Tower" −
if you *had* the power, if ...

Let's go.

First of all we have to point out a misleading habit of M.W. Evans that makes his
texts ambiguous and difficult for reading: Evans suppresses indices,
e.g. he writes q_{μ} or q^{a} instead of q^{a}_{μ}, when
the suppressed index is not used at the moment. More, he suppresses coefficients that are not
needed presently, a very dangerous habit.

Example from page 34:

q^{μν (S)} = q^{μ}q^{ν}
(= q^{μ}_{a} q^{ν}_{b} η^{ab}).
(2.73)

(ditto (2.172)) If one would try to calculate the derivative
D_{ρ} q^{μν (S)} = D_{ρ} q^{μ}q^{ν}
from Equ.(2.73) by applying the Leibniz rule (without the explanation in brackets)
then the suppression of η^{ab} would immediately lead
to the erroneous suppression of
the derivative D_{ρ}η^{ab} in cases where that is not allowed [5; (7)].
Another "nice" error directly caused by that bad habit is reported in [4; Sec.6].

Let q^{a}_{μ} be the tetrad coefficients, i.e.
q^{a}_{μ} e_{a} = ∂_{μ}, e_{a} (a=0,1,2,3)
being the tetrad basis vectors.
Evans believes that the condition

D_{ρ}
q^{a}_{μ}
= 0 (or equivalently
D_{ρ}
q^{μ}_{a}
= 0)

D_{ρ}q_{μν}^{(s)}
= D_{ρ}q^{μν}^{(S)} = 0.
(2.170)

However, Evans' reference [1] is ambiguous:
The covariant derivation operator D_{ρ} is available in Evans' text
in *two* versions [1; p.56-58] and [1; p.91]:

In the first part of his book manuscript [1] (Chap.2 - 7) he uses the version (1.59) (after tacit correction of an index exchange error at Γ here named (1.59'))

D_{ρ}q^{μ}_{a}
=
∂_{ρ}q^{μ}_{a}
+
Γ^{μ}_{ρσ}q^{σ}_{a}
(1.59')

which, as we shall see, restricts the spacetime manifold to an uninteresting special case: Using the well-known tetrad identity

(I)
∂_{ν}q^{μ}_{a}
− ω^{b}_{νa}
q^{μ}_{b}
+ Γ^{μ}_{νλ}q^{λ}_{a}
= 0

we obtain from

D_{ν}q^{μ}
= ∂_{ν}q^{μ}
+ Γ^{μ}_{νλ} q^{λ}
= 0
(2.182)

the result

ω^{a}_{μb}
= 0 for all index combinations.

Thus, D_{ρ}q^{μ} = 0 is valid, if and only if
the manifold ** M** possesses a tetrad field such that
ω

**The other version** of D_{μ} appears first at p.149:

The Evans wave equation and lemma are derived [1-3] from the tetrad postulate of differential geometry [9,10]:

D_{μ} q^{a}_{ν}
= ∂_{μ}q^{a}_{ν}
+ ω^{a}_{μb}
q^{b}_{ν}
− Γ^{λ}_{μν}q^{a}_{λ}
= 0
(8.13)

without any further word of introduction or explanation of the contrast of (8.13) to (1.59'), where we had learned on p.46:

Equations (2.179) and (2.180) are the *tetrad postulate* of differential
geometry [11]:

D_{ρ}q_{μ} = 0.
(2.179)

D_{ρ}q^{μ} = 0.
(2.180)

where D_{ρ} is given by

D_{ρ}q^{μ}_{a}
=
∂_{ρ}q^{μ}_{a}
+
Γ^{μ}_{ρσ} q^{σ}_{a}
(1.59')

and its counterpart

D_{ρ}q^{a}_{μ}
=
∂_{ρ}q^{a}_{μ}
−
Γ^{σ}_{ρμ} q^{a}_{σ}
(1.59")

respectively.

An **important disadvantage of that new covariant derivative** is that
now Evans' proof of the metric compatibility (2.170)
becomes erroneous
since in that case D_{μ} η_{ab} does not vanish,
[5; (7)] is wrong, see [4]. So nothing has been proved:

... consider the equation of metric compatibility in geometry [11]:

D_{ρ}q_{μν}^{(s)}
= D_{ρ}q^{μν}^{(S)} = 0.
(2.170)

Note that Evans according to [5] assumes the metric compatibility
D_{ρ}g_{μν}=0 here,
where D_{ρ} is given by his Equ.[1; (1.59) on p.11] in corrected version:

D_{ρ}q^{μ}_{a}
=
∂_{ρ}q^{μ}_{a}
+
Γ^{μ}_{ρσ}q^{σ}_{a}
(1.59')

Evans' aim is to derive D_{ρ}q_{μ} = 0, i.e.
D_{ρ}q^{a}_{μ} = 0 .

However we know from the Introduction, that the condition
D_{ρ}q_{μ}
=
D_{ρ}q^{a}_{μ}
= 0
in the sense of (1.59') does not hold in general. Hence Evans' following proof must be erroneous:

Using the definition of the symmetric metric

q_{μν}^{(S)}
= q_{μν},
(2.171)

q_{μν}^{(S)} = q_{μ}q_{ν},
(2.172)

the equation of metric compatibility becomes

D_{ρ}(q_{μ}q_{ν})
= (D_{ρ}q_{μ})q_{ν}
+ q_{μ}(D_{ρ}q_{ν}) = 0,
(2.173)

where the Leibnitz rule . . .

Multiply both sides of Eq. (2.173) by q^{ν}, to obtain
from the wrong Equ.(2.149) (see [3; Sec.6]

− 2 D_{ρ}q_{μ}
+ q_{μ}q^{ν}D_{ρ}q_{ν} = 0.
(2.175)

Now multiply both sides of this equation by q^{μ}q_{ν}:

**Error!** The multiplication with q_{ν} is *unadmissible* since
ν is a summation dummy index.

q_{μ}q_{ν}D_{ρ}q_{μ}
= 2 D_{ρ}q_{ν}.
(2.176)

Similarly, it can be shown that

q^{ν}q_{μ}D_{ρ}q_{ν}
= 2 D_{ρ}q_{μ}.
(2.177)

When μ = ν, these equations become . . .

**Wrong!** The *fixed* index μ cannot be equated with the *summation dummy*
index ν.

D_{ρ}q_{μ} = − D_{ρ}q_{μ},
(2.178)

whose only solution is

D_{ρ}q_{μ} = 0.
(2.179)

Recall that q_{μ} is the tetrad q^{a}_{μ}
with unwritten index a, and q^{μ} is the inverse
tetrad q^{μ}_{a}
Equations (2.179) and (2.180) are the *tetrad postulate* of differential
geometry [11]. . . .

D_{ρ}q^{μ} = 0.
(2.180)

Note that *tetrad postulate* here means the Equations (2.179) and (2.180),
which are **not fulfilled in general.**

The equation of metric compatibility for q^{μ}, written out in terms of the
Christoffel symbol, is

D_{ν}q^{μ}
= ∂_{ν}q^{μ}
+ Γ^{μ}_{νλ} q^{λ}
= 0.
(2.182)

Multiplication of both sides of this equation by q_{λ} gives

q_{λ} D_{ν}q^{μ}
= q_{λ} ∂_{ν}q^{μ}
+ q_{λ}q^{λ} Γ^{μ}_{νλ}
= 0.
(2.183)

**Not admissibly**: λ is summation index!

**Page 65:** (with e^{a}_{μ} := q^{a}_{μ})

(4.1) The tetrad postulate:
D_{ν}e^{a}_{μ}
= 0,
(4.6)

where D_{ν} denotes the covariant derivative [2], is true for any connection,
whether or not it is metric compatible or torsion free.

The reference [1] is ambiguous: [1; p.56-58] or [1; p.91]?

Equation (4.5), the wave equation for the vielbein as eigenfunction, follows from the tetrad postulate [2]:

D_{ρ} e^{a}_{μ}
= 0,
(4.15)

which holds whether or not the connection is metric-compatible or torsionfree. Differentiating Eq. (4.15) covariantly gives Eq. (4.5):

D^{ρ}(D_{ρ}e^{a}_{μ})
:=
D^{ρ}D_{ρ} e^{a}_{μ}
=
( + D^{μ} Γ^{ρ}_{μρ}
+ 2 Γ^{λ}_{μμ}
D_{λ})
e^{a}_{μ}
= ( + D^{μ} Γ^{ρ}_{μρ})
e^{a}_{μ}
= 0.
(4.16)

where := ∂^{ρ}∂_{ρ}
= g^{ρσ}∂_{σ}∂_{ρ} .

The reference "tetrad postulate [1]" points to [1; p.91 (3.128)].
However, then Equ.(4.16) would be **wrong** since the Carroll derivative

Ñ_{ρ} e^{a}_{μ}
:=
∂_{ρ} e^{a}_{μ}
+
ω^{a}_{μb} e^{b}_{ρ}
−
Γ^{σ}_{μb} e^{a}_{ρ}

would cause ω-terms. Hence we conclude that Evans here meant "tetrad postulate" in the sense of Equ.(1.59")

D_{ρ} e^{a}_{μ}
:=
∂_{ρ} e^{a}_{μ}
−
Γ^{σ}_{ρμ} e^{a}_{σ}

where the "tetrad postulate" D_{ρ}e^{a}_{μ}
= 0 is **invalid in general**.

D_{ρ}q_{μν}^{(S)} = 0
(4.27)

A third type of wave equation can be obtained using the definition [1]

q^{(S)}_{μν}
= q_{μ}q_{ν}
= η_{ab}q^{a}_{μ}q^{b}_{ν}
in extended notation.
(4.29)

Covariant differentiation of products is defined by the Leibniz theorem [2]; therefore the metric compatibility of the symmetric metric tensor, Eq. (4.27), implies that

D_{ρ}(q_{μ}q_{ν})
= q_{μ}(D_{ρ}q_{ν})
+ (D_{ρ}q_{μ})q_{ν} = 0 .
(4.30)

**From where do we have metric compatibility?**
Metric compatibility does *not hold generally*
while the case of "tetrad postulate" in the sense of (1.59') where metric compatibility
would hold is only the case of a
very special kind of manifolds.

The equation of metric compatibility (4.32) can be derived independently as a
solution of the equation of parallel transport [2] written for the inverse metric
four-vector q^{μ}:

^{Dqμ}/_{ds}
:= ^{dqμ}/_{ds}
+ Γ^{μ}_{νλ}
^{dxν}/_{ds}
q^{λ} = 0,
(4.35)

**Invalid** in general since the invalid equation (1.59) is underlayed.

(ds)² = q^{μ}q^{ν}dx_{μ}dx_{ν}
(4.36)

Wrong index positions. η missing.

∂q^{μ}/
∂x^{ν} = − Γ^{μ}_{νλ} q^{λ}.
(4.56)

(1.59) used again.

. . . and the equation of metric compatibility for q^{μ} is

D_{ν}q^{μ}
= ∂_{ν}q^{μ} +
Γ^{μ}_{νλ}q^{λ}
= 0.
(5.85)

again (1.59').

A novel and fundamental wave equation of general relativity can be deduced [1] from the metric compatibility condition

D_{μ}q^{ν} = ∂_{μ}q^{ν}
+ Γ^{ν}_{μλ} q^{λ} = 0
(6.1)

That's the special case (1.59').

on the metric four-vector q_{μ} used in the standard definition [2,3] of the metric
tensor of the Einstein field equation

q_{μν} = q_{μ}q_{ν}
= q^{a}_{μ}Ä q^{b}_{ν}
in extended notation
(6.2)

as the outer product of two metric vectors.

The metric is not at all the "outer product of two metric vectors":
The extended vision of the tensor of the outer product of two metric vectors
shows its *dependence on the choice of the tetrad* by the upper indices a,b that does not
appear in the metric g_{μν}.

q^{a}_{0}Ä q^{b}_{0}
q^{a}_{0}Ä q^{b}_{1}
q^{a}_{0}Ä q^{b}_{2}
q^{a}_{0}Ä q^{b}_{3}

q^{a}_{1}Ä q^{b}_{0}
q^{a}_{1}Ä q^{b}_{1}
q^{a}_{1}Ä q^{b}_{2}
q^{a}_{1}Ä q^{b}_{3}

q^{ab}_{μν} =

q^{a}_{2}Ä q^{b}_{0}
q^{a}_{2}Ä q^{b}_{1}
q^{a}_{2}Ä q^{b}_{2}
q^{a}_{2}Ä q^{b}_{3}

q^{a}_{3}Ä q^{b}_{0}
q^{a}_{3}Ä q^{b}_{1}
q^{a}_{3}Ä q^{b}_{2}
q^{a}_{3}Ä q^{b}_{3}

The question is whether a tetrad independent geometrical object could be constructed
from this coefficient scheme.
The dependence on the tetrad indices a,b can be removed by multiplication with
the tetrad basis vectors e_{a}, i.e. to consider
e_{a} e_{b} q^{ab}_{μν}.
There is only one possibility to associate a geometrical meaning to the products
e_{a} e_{b}, namely to consider their scalar products
e_{a} · e_{b} = η_{ab}. Thus, we are lead to the only
tetrad independent geometrical object
η_{ab} q^{a}_{μ}Ä q^{b}_{ν}.
Its symmetric part yields the metric
g_{μν}
= η_{ab} q^{a}_{μ}q^{b}_{ν}
while its antisymmetric part vanishes. Hence we have the result that another geometrical
object, say

Equation (6.1) states that the
covariant derivative of the metric vector q^{ν} vanishes. It has been shown that
the metric compatibility condition (6.1) leads to the usual metric compatibility
condition of the metric tensor

D_{ν}q_{μν} = 0
(6.3)

used in Riemann geometry [3] to relate the Christoffel symbol Γ^{ν}
_{μλ} and metric
tensor q_{μ}^{ν}.

However, Equ.(6.1) covers the uninteresting special case (1.59') of a teleparallel manifold.

The tetrad can be defined by the equation

V^{a} = q^{a}_{μ} V^{μ},
(9.5)

where V^{a} is any contravariant vector in the orthonormal basis, indexed a, and
V^{μ} is any contravariant vector in the base manifold indexed µ. For example,
V can represent position vectors x, so the tetrad becomes

x^{a} = q^{a}_{μ} x^{μ},
(9.6)

and has sixteen independent components, the sixteen irreducible representations of the Einstein group [8-12]. The vector V can also represent metric vectors q [3-7], so the tetrad can be defined by

q^{a} = q^{a}_{μ} q^{μ}.
(9.7)

This means that the symmetric metric [2-7] can be defined in general by the dot product of tetrads:

q_{μν}^{(S)}
= q^{a}_{μ} q^{b}_{ν} η_{ab},
(9.8)

where η_{ab} is the metric diag(−1, 1, 1, 1) of the orthogonal space [2].
The antisymmetric metric is defined in general by the wedge product of tetrads:

q^{c}_{μν}^{(A)}
= q^{a}_{μ} Ù q^{b}_{ν},
(9.9)

This definition is *dubious*. The index c is *not defined*,
it should depend on the indices a,b. Evans tries here to generalize the cross-product of
**R**³ to 4-dimensional manifolds: In **R**³ equipped with an orthonormal
"triad" {**e**_{a} | a=1,2,3} one can define the cross product
of the vectors
**v**_{μ} =
q^{a}_{μ}**e**_{a}
and
**v**_{ν} =
q^{a}_{ν}**e**_{a}
by

**v**_{μ} × **v**_{ν}
:=
q^{a}_{μ}**e**_{a}
×
q^{b}_{ν}**e**_{b}
=
½
(q^{a}_{μ}q^{b}_{ν}
−
q^{b}_{μ}q^{a}_{ν})
**e**_{a}×**e**_{b}

where **e**_{a}×**e**_{b} =: **e**_{c}
if a,b,c are cyclically taken from (1,2,3).
It is well-known that this definition is invariant under orthogonal transforms of the **R**³.
However, Evans' definition (9.9) is not the generalization of that cross product to the
Minkowski **R**^{1+3} space. The admissible transforms should preserve the Minkowski
orthonormality of the tetrad basis vectors, i.e. the definition should be invariant under Lorentz
transforms, and such a definition does not exist.

The only meaningful counterpart of (9.8) would be the definition

q_{μν}^{(A)}
:=
η_{ab}
(q^{a}_{μ}q^{b}_{ν}
−
q^{a}_{ν}q^{b}_{μ}) ,

however, this definition vanishes trivially due to the symmetry of η_{ab}:
q_{μν}^{(A)} = 0.

and the general metric tensor with sixteen independent components is defined by the outer product of tetrads:

q^{ab}_{μν}
= q^{a}_{μ}q^{b}_{ν}
= q^{a}_{μ} Ä q^{b}_{ν}
.
(9.10)

The dot product is the gauge invariant gravitational field, the cross product the gauge invariant electromagnetic field, . . .

In the Minkowski space **M**^{4} there exists no
"cross product" × :
**M**^{4}
× **M**^{4}
→ **M**^{4} .

. . . and the outer product combines the two fields and defines the way in which one influences the other.

see the above remarks on the outer product of metric vectors

The tetrad can also be defined [2] by basis vectors; for example,

ê_{(μ)} = q^{a}_{μ} ê_{(a)},
(9.11)

for covariant basis vectors, or

θ^{(a)} = q^{a}_{μ} θ^{(μ)}
(9.12)

"Covariant basis vectors" are usually called *basis 1-forms* [2: p.89].

x^{a'}_{μ} = q^{a'}_{μ} x^{μ}.
(9.17)

S.M. Carroll writes in [2; p.88]: ". . . at each point of the manifold we
introduce a set of basis vectors ê_{a}. . . The set of vectors
comprising an orthonormal basis is sometimes known as a **tetrad** . . ."

The well-known tetrad postulate [2-7] is

D_{μ}q^{a}_{ν}
= ∂_{μ}q^{a}_{ν}
+ ω^{a}_{μb}
q^{b}_{ν}
− Γ^{λ}_{μν}q^{a}_{λ}
= 0
(9.24)

and is the basis of the Evans lemma and Evans wave equation [3-7] of differential geometry, the most powerful and general wave equation known in general relativity, and thus in physics.

The right part of Equ.(9.24)

∂_{μ}q^{a}_{μ}
+ ω^{a}_{μb}
q^{b}_{ν}
− Γ^{λ}_{μν}q^{a}_{λ}
= 0

Therefore gravitation is described by the
symmetric metric q^{μ}q^{ν} and electromagnetism by the anti-symmetric metric defined
by the wedge product q^{μ}Ùq^{ν}.

The symmetric metric tensor is defined by the symmetric tensor product of two metric four-vectors:

q^{μν} ^{(S)}
= q^{μ}q^{ν}
(= q^{μ}_{a} q^{ν}_{b} η^{ab}
= g^{μν})
(3.4)

and the anti-symmetric metric tensor by the wedge product:

q^{μν} ^{(A)}
= q^{μ} Ù q^{ν}
(= q^{μ}_{a} Ù
q^{ν}_{b} η^{ab}
= O)
(3.5)

where the metric four-vector is:

q^{ν} = (h^{0}, h^{1}, h^{2}, h^{3})
(3.6)

If the term η^{ab} would be omitted in the extended notation
of (3.5) then the "antisymmetric metric" would not be invariant under
changes of the tetrad by Lorentz transforms in the tangent space.

The symmetric metric tensor is then defined through the line element, a one form of differential geometry:

ω_{1} = ds² = q^{ij (S)}du_{i} du_{j} ,
(3.12)

Differentials are due to convention contravariant and to be indexed
at the *upper* position:

ω_{1} = ds² = q_{ij}^{ (S)}du^{i} du^{j} ,
(3.12')

and the anti-symmetric metric tensor through the area element, a two form of differential geometry:

ω_{2} = dA = − ½
q_{ij}^{ (A)}du^{i} Ù du^{j} .
(3.13)

These results generalize as follows to the four dimensions of any non-Euclidean space-time:

ω_{1} = ds² = q_{μν}^{ (S)} du^{μ} du^{ν},
(3.14)

ω_{2} = *ω_{1} = − ½
q_{μν}^{ (A)}du^{μ} Ù du^{ν} .
(3.15)

In differential geometry the element du^{σ} is dual to the wedge product du^{μ} Ù du^{ν}.

Two objections:

(1) The last statement is only true in *3-dimensional* differential geometry in the sense of Hodge's duality..

(2) The symbol * in ω_{2} = *ω_{1} is not defined here. On p.139
* denotes the Hodge dual operator. Therefore we remark here that the Hodge dual operator *
is not defined for the dot-2-form ω_{1}. The equation ω_{2} = *ω_{1}
is therewith **senseless**.

The symmetric metric tensor is:

h_{0}² h_{0}h_{1} h_{0}h_{2} h_{0}h_{3}

h_{1}h_{0} h_{1}² h_{1}h_{2} h_{1}h_{3}

q_{μν}^{ (S)} =
(3.16)

h_{2}h_{0} h_{2}h_{1} h_{2}² h_{2}h_{3}

h_{3}h_{0} h_{3}h_{1} h_{3}h_{2} h_{3}²

It must be remarked here that (3.16) yields det(q_{μν}^{ (S)}) = 0,
which makes (3.16) unsuitable for being a metric. And (3.15) defines an "antisymmetric metric"
that is *not invariant under Lorentz transforms* in the tangent space: Since in extended notation
we have

ω_{2} = − ½
q^{ab}_{μν}^{ (A)}du^{μ} Ù du^{ν} ,

which transforms *twofold contravariantly* under changes of the tetrad.
An antisymmetric metric, whatsoever it is, should be *invariant* under those transforms.

References

[1]
M.W. Evans, *Generally Covariant Unified Field Theory,
the geometrization of physics*,

http://www.aias.us/Comments/Evans-Book-Final.pdf

[2]
Sean M. Carroll, *Lecture Notes on General Relativity*,

http://arxiv.org/pdf/gr-qc/9712019

[3]
G.W. Bruhn, *Myron W. Evans' Most Spectacular Errors*,

http://www2.mathematik.tu-darmstadt.de/~bruhn/MWEsErrors.html

[4]
G.W. Bruhn, *Evans Proving Metric Compatibility?*,

http://www2.mathematik.tu-darmstadt.de/~bruhn/MetrCompProof.html

[5]
M.W. Evans, *Generally Covariant Unified Field Theory,
the geometrization of physics*,

http://www.aias.us/Comments/ametriccompatibilityfromtetradpostulate.pdf