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


Insertion on April 8, 2006 after a short email discussion where Myron wrote to me:

"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.


Further Errors in Evans' GCUFT Manuscript

by Gerhard W. Bruhn, Darmstadt University of Technology



1. Introduction

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 qμ or qa instead of qaμ, 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:

qμν (S) = qμqν     (= qμa qνb ηab).                                                                 (2.73)

(ditto (2.172)) If one would try to calculate the derivative Dρ qμν (S) = Dρ qμqν from Equ.(2.73) by applying the Leibniz rule (without the explanation in brackets) then the suppression of ηab would immediately lead to the erroneous suppression of the derivative Dρηab 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 qaμ be the tetrad coefficients, i.e. qaμ ea = ∂μ, ea (a=0,1,2,3) being the tetrad basis vectors. Evans believes that the condition

Dρ qaμ = 0     (or equivalently Dρ qμa = 0)

would imply the metric compatibility condition for the metric gμν (= qμν(s) in Evans notation), see [5]:

Dρqμν(s) = Dρqμν(S) = 0.                                                                 (2.170)

However, Evans' reference [1] is ambiguous: The covariant derivation operator Dρ 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'))

Dρqμa = ∂ρqμa + Γμρσqσa                                                                 (1.59')

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

(I)                                                 ∂νqμa − ωbνa qμb + Γμνλqλa = 0

we obtain from

Dνqμ = ∂νqμ + Γμνλ qλ = 0                                                         (2.182)

the result

ωaμb = 0     for all index combinations.

Thus, Dρqμ = 0 is valid, if and only if the manifold M possesses a tetrad field such that ωaμb = 0 everywhere on M. Such special manifolds are called teleparallel.

The conditions Dρqμa = 0 and Dρqaμ = 0 where Dρqaμ is defined by (2.182)/(1.59') are not satisfied in general.


The other version of Dμ appears first at p.149:

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

Dμ qaν = ∂μqaν + ωaμb qbν − Γλμνqaλ = 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]:

Dρqμ = 0.                                                                 (2.179)

Dρqμ = 0.                                                                 (2.180)

where Dρ is given by

Dρqμa = ∂ρqμa + Γμρσ qσa                                                 (1.59')

and its counterpart

Dρqaμ = ∂ρqaμ − Γσρμ qaσ                                                 (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 Dμ ηab does not vanish, [5; (7)] is wrong, see [4]. So nothing has been proved:

The metric compatibility (2.170) is invalid in general.



2. Review of some pages


Page 46:

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

Dρqμν(s) = Dρqμν(S) = 0.                                                                 (2.170)

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

Dρqμa = ∂ρqμa + Γμρσqσa                                                                 (1.59')

Evans' aim is to derive Dρqμ = 0, i.e. Dρqaμ = 0 .

However we know from the Introduction, that the condition Dρqμ = Dρqaμ = 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

qμν(S) = qμν,                                                                 (2.171)

qμν(S) = qμqν,                                                                 (2.172)

the equation of metric compatibility becomes

Dρ(qμqν) = (Dρqμ)qν + qμ(Dρqν) = 0,                                                 (2.173)

where the Leibnitz rule . . .

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

− 2 Dρqμ + qμqνDρqν = 0.                                                                 (2.175)

Now multiply both sides of this equation by qμqν:

Error! The multiplication with qν is unadmissible since ν is a summation dummy index.

qμqνDρqμ = 2 Dρqν.                                                                 (2.176)

Similarly, it can be shown that

qνqμDρqν = 2 Dρqμ.                                                                 (2.177)

When μ = ν, these equations become . . .

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

Hence the following result is invalid.

Dρqμ = − Dρqμ,                                                                 (2.178)

whose only solution is

Dρqμ = 0.                                                                 (2.179)

Recall that qμ is the tetrad qaμ with unwritten index a, and qμ is the inverse tetrad qμa Equations (2.179) and (2.180) are the tetrad postulate of differential geometry [11]. . . .

Dρqμ = 0.                                                                 (2.180)

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


Page 46 below:

The equation of metric compatibility for qμ, written out in terms of the Christoffel symbol, is

Dνqμ = ∂νqμ + Γμνλ qλ = 0.                                                         (2.182)

Multiplication of both sides of this equation by qλ gives

qλ Dνqμ = qλνqμ + qλqλ Γμνλ = 0.                                                 (2.183)

Not admissibly: λ is summation index!


Page 65: (with eaμ := qaμ)

(4.1) The tetrad postulate: Dνeaμ = 0,                                                                 (4.6)

where Dν 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]?

The claim is not true for the covariant derivatives defined by (1.59') and (2.182)].


page 68:

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

Dρ eaμ = 0,                                                                 (4.15)

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

Dρ(Dρeaμ) := DρDρ eaμ = ( + Dμ Γρμρ + 2 Γλμμ Dλ) eaμ = ( + Dμ Γρμρ) eaμ = 0.                 (4.16)

where  := ∂ρρ = gρσσρ .

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

Ñρ eaμ := ∂ρ eaμ + ωaμb ebρ − Γσμb eaρ

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

Dρ eaμ := ∂ρ eaμ − Γσρμ eaσ

where the "tetrad postulate" Dρeaμ = 0 is invalid in general.


Page 70:

Dρqμν(S) = 0                                                                                 (4.27)

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

q(S)μν = qμqν = ηabqaμqbν 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

Dρ(qμqν) = qμ(Dρqν) + (Dρqμ)qν = 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.

Thus, the "wave equation" (4.30) is not true in general.


Page 70 once more:

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 qμ:

Dqμ/ds := dqμ/ds + Γμνλ dxν/ds qλ = 0,                                                 (4.35)

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


Page 71:

(ds)² = qμqνdxμdxν                                                                 (4.36)

Wrong index positions. η missing.


Page 72:

∂qμ/ ∂xν = − Γμνλ qλ.                                                                 (4.56)

(1.59) used again.


Page 79:

In (4.115): (1.59') assumed again:


Page 109:

. . . and the equation of metric compatibility for qμ is

Dνqμ = ∂νqμ + Γμνλqλ = 0.                                                                 (5.85)

again (1.59').


Page 119:

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

Dμqν = ∂μqν + Γνμλ qλ = 0                                                                 (6.1)

That's the special case (1.59').

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

qμν = qμqν = qaμ qbν 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 gμν.

                                                qa0 qb0   qa0 qb1   qa0 qb2   qa0 qb3
                                                qa1 qb0   qa1 qb1   qa1 qb2   qa1 qb3
                                qabμν =
                                                qa2 qb0   qa2 qb1   qa2 qb2   qa2 qb3
                                                qa3 qb0   qa3 qb1   qa3 qb2   qa3 qb3

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 ea, i.e. to consider ea eb qabμν. There is only one possibility to associate a geometrical meaning to the products ea eb, namely to consider their scalar products ea · eb = ηab. Thus, we are lead to the only tetrad independent geometrical object ηab qaμ qbν. Its symmetric part yields the metric gμν = ηab qaμqbν while its antisymmetric part vanishes. Hence we have the result that another geometrical object, say

Evans' (nontrivial) antisymmetric metric does not exist.

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

Dνqμν = 0                                                                 (6.3)

used in Riemann geometry [3] to relate the Christoffel symbol Γν μλ and metric tensor qμν.

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


Page 173:

The tetrad can be defined by the equation

Va = qaμ Vμ,                                                                 (9.5)

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

xa = qaμ xμ,                                                                 (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

qa = qaμ qμ.                                                                 (9.7)

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

qμν(S) = qaμ qbν ηab,                                                                 (9.8)

where ηab 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:

qcμν(A) = qaμ qbν,                                                                 (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 R³ to 4-dimensional manifolds: In R³ equipped with an orthonormal "triad" {ea | a=1,2,3} one can define the cross product of the vectors vμ = qaμea and vν = qaνea by

vμ × vν := qaμea × qbνeb = ½ (qaμqbν − qbμqaν) ea×eb

where ea×eb =: ec 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 R³. However, Evans' definition (9.9) is not the generalization of that cross product to the Minkowski R1+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

qμν(A) := ηab (qaμqbν − qaνqbμ) ,

however, this definition vanishes trivially due to the symmetry of ηab: qμν(A) = 0.

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

qabμν = qaμqbν = qaμ qbν .                                                                 (9.10)

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

In the Minkowski space M4 there exists no "cross product" × : M4 × M4M4 .

. . . 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,

(μ) = qaμ (a),                                                                 (9.11)

for covariant basis vectors, or

θ(a) = qaμ θ(μ)                                                                 (9.12)

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


Page 174:

The tetrad is also the generalization of the Lorentz transformation to general relativity:

xa'μ = qa'μ xμ.                                                                 (9.17)

Nonsense! A tetrad is no transform.

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


Page 175:

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

Dμqaν = ∂μqaν + ωaμb qbν − Γλμνqaλ = 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)

μqaμ + ωaμb qbν − Γλμνqaλ = 0

is a well-known tetrad identity. Accepting this would make a great part of the preceeding chapters irrelevantly.


3. Remarks on the Wedge Product


Page 53:

Therefore gravitation is described by the symmetric metric qμqν and electromagnetism by the anti-symmetric metric defined by the wedge product qμqν.


Page 54:

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

qμν (S) = qμqν (= qμa qνb ηab = gμν)                                                                 (3.4)

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

qμν (A) = qμ qν (= qμa qνb ηab = O)                                                                 (3.5)

where the metric four-vector is:

qν = (h0, h1, h2, h3)                                                                 (3.6)

If the term ηab 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.


Page 55:

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

ω1 = ds² = qij (S)dui duj ,                                                                 (3.12)

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

ω1 = ds² = qij (S)dui duj ,                                                                 (3.12')

In the following we correct that permanent error tacitly in Evans' text.

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

ω2 = dA = − ½ qij (A)dui duj .                                                                 (3.13)

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

ω1 = ds² = qμν (S) duμ duν,                                                                 (3.14)

ω2 = *ω1 = − ½ qμν (A)duμ duν .                                                                 (3.15)

In differential geometry the element duσ is dual to the wedge product duμ duν.

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 ω2 = *ω1 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 ω1. The equation ω2 = *ω1 is therewith senseless.

The symmetric metric tensor is:

                                                h0²   h0h1   h0h2   h0h3
                                                h1h0   h1²   h1h2   h1h3
                qμν (S) =                                                                                                                 (3.16)
                                                h2h0   h2h1   h2²   h2h3
                                                h3h0   h3h1   h3h2   h3²

It must be remarked here that (3.16) yields det(qμν (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

ω2 = − ½ qabμν (A)duμ duν ,

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

[4]    G.W. Bruhn, Evans Proving Metric Compatibility?,
                http://www.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