Last Update: Jan 17 2007


Comments on

M.W. Evans: "The Spinning and Curving of Spacetime:
The Electromagnetic and Gravitational Fields In The Evans Unified Field Theory"

by Gerhard W. Bruhn, Darmstadt University of Technology


Abstract

Some serious objections to Evans' GCUFT are found in Chap.17 of the book [4] and presented here:
- The misunderstanding of the tetrad concept. (Sect.1)
- The failing of Evans' outer product of tetrads to construct an antisymmetric metric. (Sect.2)
- The severe inconsistencies of Evans' two "Tetrad Postulates" and
   the "Metric Compatibility Condition", which turns out to be invalid in general. (Sect.3)
- The non Lorentz invariance of the associated tetrad to a plane wave. (Sect.4)
- The non-reality of the metric associated to Evans' Complex Circular Base. ((Sect.5))


Introduction

The paper [1] that is to appear in Springer's FPL is copy of Chap.17 of just that book [4] that Springer's wise referees declined a few months ago. With exception of the equation numbering we could not find any differences between book Chap.17 [4] and paper [1]. Even some evident misprints survived the FPL refereeing, see Appendix 2. − Evans denotes that chapter as a "cornerstone" of his theory. Nevertheless we shall point out several serious flaws in the following sections with consequences for the whole GCUFT.

For a later discovered error in the Introduction of Evans' paper the reader is kindly requested to click at
                A Remark on Evans' 2nd Bianchi Identity

All equation numbers without reference [...] refer to Evans web-published book manuscript [4]. For convenience we have quoted those equations here. Sometimes slight modification of an Evans equation were necessary: In that case an apostrophe ' is added to the equation number: (xxx) is the original equation and (xxx') our modification.

Original quotations from Evans' book manuscript appear in black , our comments are colored.


1. Misunderstanding the Tetrad Concept


Quote from M.W. Evans [4; p.302f.]:

The type of Riemann geometry almost always used by Einstein and others [2] for generally covariant gravitational field theory is a special case of the more general Cartan differential geometry [2] in which the connection is no longer symmetric and which the metric is in general the outer or tensor product of two more fundamental tetrads qaμ.

gμν = qaμ qbν ηab ,                                                                 (17.10)

where η ab is the diagonal, constant metric in the orthonormal or flat spacetime of the tetrad, indexed a:

ηab = diag (−1, 1, 1, 1) .                                                                 (17.11)

The sentence "the metric is in general the outer or tensor product of two more fundamental tetrads qaμ" shows at first a strange understanding of the tetrad concept that was criticized already elsewhere [3]: A tetrad is a coordinate independent frame, i.e. set of four tangent vectors ea (a=0,1,2,3) of the spacetime manifold defined for the diverse points of spacetime. The relation to the coordinate dependent tangent vectors vectors eμ = ∂μ (μ=0,1,2,3)

eμ = qaμ ea

defines the tetrad coefficients qaμ which hence are different from the tetrad {ea | a=0,1,2,3}. Therefore it is nonsense to speak of "two" (different???) tetrads qaμ.


2. The Antisymmetric Metric

Quote from the break before equ.(9)/(17.10): . . . in which the connection is no longer symmetric (ok.) and which the metric is in general the outer or tensor product of two (???) more fundamental tetrads qaμ. Thus, in differential geometry the metric tensor is in general an asymmetric matrix. Any asymmetric matrix is always the sum of a symmetric matrix and an antisymmetric matrix [3], so it is possible to construct an antisymmetric metric tensor. The symmetric metric used by Einstein to describe gravitation is therefore the special case defined by the inner or dot product of two tetrads [2]:

The outer product qaμÄ qbν as proposed by M.W. Evans depends on the choice of the tetrad field attached to the manifold, while a metric should be independent of the choice of the tetrads. Therefore the tetrad indices a,b must be removed. The only possibility to do that is to multiply the outer product by the corresponding tetrad vectors, namely by the scalar products ea· eb = ηab. This way we obtain the tetrad independent tensor ηab qaμÄ qbν the symmetric part of which is the well-known symmetric metric tensor

gμν = ηab qaμqbν = ηab ½ (qaμÄ qbν + qaνÄ qbμ)

while its antisymmetric part

ηab qaμ qbν = ηab ½ (qaμÄ qbν − qaνÄ qbμ)

vanishes due to the symmetry of ηab. Hence the only "geometric" tensor that can be derived from the outer product qaμÄ qbν is the symmetric metric tensor gμν.

A (nontrivial) antisymmetric metric does not exist.

Or, correcting Evans' text from the quote above: In differential geometry the metric tensor is NOT asymmetric.

That is confirmed by the fact that Equ.(17.10) yields a symmetric metric tensor gμν = gνμ.


3. The Tetrad Postulate and the Metric Compatibility

We come to another quote from p.306:

In the Einstein theory the tetrad postulate [2] of differential geometry:

Dν qaμ = 0                                                                 (17.32)

is specialized to the metric compatibility condition [2] for a symmetric metric:

Dν gμρ = Dν gμρ = 0.                                                         (17.33)

That means that Evans believes in the unrestricted validity of the "tetrad postulate" (17.32). In [5] he writes:

The attached is a simple and powerful demonstration that the tetrad postulate is a fundamental property of Cartan differential geometry, a property that IMPLIES the less general metric compatibility condition of Riemann geometry used by Einstein and Hilbert in their famous theory of general relativity (1915 to 1916). The tetrad postulate leads directly to metric compatibility and the former is more fundamental, because it relies on no assumptions concerning the connection.

And in [4; p.65]:

The tetrad postulate: Dνeaμ = 0, ... , where Dν denotes the covariant derivative [2], is true for any connection, whether or not it is metric compatible or torsion free.

However, Evans' notation is not unique: The quantity eaμ has later in the book converted to qaμ and denotes tetrad coefficients. And more important: Also the derivative Dνqμa is not uniquely defined: In the first part of his book manuscript [4] (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 Evans' first "tetrad postulate"

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

the result

ωaμb = 0     for all index combinations.

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

The "tetrad postulate" [4; (2.182)] is not satisfied in general.


The second version of Dν appears first at [4; 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').

An important disadvantage of that second covariant derivative (8.13) is that now Evans' proof of the metric compatibility (2.170) (cf. [6])

Applying the Leibniz rule to gμν = ηab qaμ qbν yields

Dρ gμν = ηab (Dρqaμ) qbν + qaμ (Dρqbν)

According to S.M. Carroll [2; p.91] one has

Dρqaμ = ∂ρqaμ + ωaρb qbμ − Γσρμ qaσ = 0 ,

which due to the preceding equation yields the metric compatibility Dρ gμν = 0 , q.e.d..

becomes erroneous since in that case Dμηab does not vanish, [6; (7)] is wrong. So nothing has been proved:

The condition of metric compatibility Dρ gμν = 0 is invalid in general.

Conclusion

Wherever M.W. Evans refers to "metric compatibility" in his GCUFT (a search in the GCUFT book manuscript yields about 20 spots) it should be remembered that this is an important additional condition on the "γ" connection of the spacetime manifold that is not fulfilled in general.

The author M.W. Evans should feel challenged to check the consequences of that important restriction. Since Dρqaμ has an ambiguous meaning in his book manuscript it is necessary to reconsider the consequences for the diverse spots of GCUFT under concern. Clear, however, is that Evans' first "tetrad postulate" (2.182), Dν qaμ = 0, and the metric compatibility condition (17.33), Dρ gμν = 0, are not valid in general.

Therefore all nice conclusions from these equations in GCUFT (at about 20 spots) have lost their justification now and have become dubious.


4. Evans' Complex Circular Base

Another Quote from M.W. Evans [1; p.307]:

The tetrad is defined for the electromagnetic field by:

Va = qaμ Vμ                                                                 (17.35)

where Vμ is a vector in the base manifold, and where Va is a vector in the tangent bundle. The tetrad is therefore the four by four invertible transformation matrix [2] between base manifold and tangent bundle.

The wrong tetrad definition has already been criticized above and in [3]. In addition this means that Evans believes in having two different vectors under consideration. Thus, this quote shows Dr Evans' complete misunderstanding of the vector concept: Both (Vμ) and (Va) are descriptions of the same vector V in the tangent space at some point P of the manifold under consideration, related to two different frames in the tangent space at P: (Vμ) are the coefficients of some vector V relative to the coordinates, i.e. to the coordinate related tangent basis eμ = ∂μ := ∂/∂xμ (μ=0,1,2,3) while (Va) is the same relative to the coordinate independent (freely definable) basis of given tetrad vectors ea (a=0,1,2,3) (the Greek index μ indicating that the coordinate related basis eμ = ∂μ (μ=0,1,2,3) is referred to, and the Latin index a refers to the tetrad basis {ea | a=0,1,2,3}. Equ.(17.35) is nothing but the coordinate transform between the two different frames in the tangent space at some point P of the manifold.

In the sequel Evans considers the vector potential A of a circularly polarized plane wave related to his Complex Circular Basis using the "electromagnetic phase" Φ = ω(t − z/c) defined in Equ.(1.37):

Circular polarization, discovered experimentally by Arago in 1811, is described geometrically by elements of Aaμ from Eq. (17.12), i.e. by the following complex valued tetrad elements:

A(1)x = (A(0)/2½) eiΦ,                                                                 (17.36)
A(1)y = −i (A(0)/2½) eiΦ,                                                            (17.37)

where Φ is the electromagnetic phase. The complex conjugates of these elements are:

A(2)x = (A(0)/2½) eiΦ,                                                               (17.38)
A(2)y = i (A(0)/2½) eiΦ.                                                             (17.39)

Therefore these tetrad elements are individual components of the following vectors:

A(1) = (A(0)/2½) (iij) eiΦ,                                                                 (17.40)
A(2) = (A(0)/2½) (i + ij) eiΦ,                                                               (17.41)

representing a spinning and forward moving frame. This frame is multiplied by A(0) to give the generally covariant electromagnetic potential field. In 1992 it was inferred by Evans [22] that these vectors define the Evans spin field, B(3), of electromagnetism:

B(3) * = − ig A(1) × A(2),                                                                 (17.42)

Evans here refers to a "triad" e(1), e(2), e(3) of complex unit vectors attached to a plane wave [4; (1.43), (5.4)ff.]

whose unit vectors are related to the orthonormal Cartesian unit vectors i, j, k of the tangent space by

e(1) = 2−½ (iij),         e(2) = 2−½ (i + ij),         e(3) = k.                                                 (5.4)

. . .

Let the orthonormal tangent space rotate and translate with respect to the space i = 1, 2, 3 by introducing the phase Φ = ωt − κZ of the wave equation [5-10]. Here ω is an angular frequency at instant t and κ is a wave-vector at point Z. The orthonormal tangent space is thereby defined by the metric vectors

q(1) = e(1)e,         q(2) = e(2)e−iΦ,         q(3) = e(3)                                                 (5.7)

whose magnitudes are . . . follows an erroneous line of formulas (5.8).

We insert (5.4) into (5.7) to obtain

q(1) = 2−½ (iij)e,         q(2) = 2−½ (i + ij)e−iΦ,         q(3) = e(3) = k

Evans' expositions show his idea of the special tetrad frame to be used, a twisted Complex Circular Basis, now renamed with e instead of q:

e(1) := 2−½ (iij) eiΦ,         e(2) := 2−½ (i + ij) eiΦ,         e(3) := k                         (17.40'-41')

which is no tetrad but an orthonormal spatial triad, that must be completed to a tetrad by introducing a unit vector eo in time direction having vanishing scalar products with the other basis vectors e(a) (a=1,2,3).


5. The metric and Evans complex Circular Basis

A further objection:

According to Evans the Eqns.(17.36-39) define the "complex valued tetrad elements", i.e. the coefficients qaμ (cf. (17.35) and the Appendix 1). Therefore we may conclude from (17.36-39)

q11 = 2−½ eiΦ,                 q12 = −i 2−½ eiΦ,                                                            (17.36'-37')
q21 = 2−½ eiΦ,               q22 = i 2−½ eiΦ,                                                             (17.38'-39')

due to the invariance of e(0) and e(3) to be completed with

q0μ = δ0μ ,                 q3μ = δ3μ .

Then due to Evans' equation

gμν = qaμqbν ηab                                                                 (17.10)

we should obtain a real metric tensor (gμν), however, the complex Eqns. (17.36-39) yield a non-real result for (17.10):

Equ. (17.10) is not applicable for complex tetrad coefficients qaμ.

The reason of that error is that Evans violates the rules of real manifolds as spacetime is considered in General Relativity. The tetrad vectors ea (a=0,1,2,3) have to be defined as real linear combinations of the coordinate related tangent vectors ∂μ, i.e. the coefficients qaμ of the inverse representation ∂μ = qaμ ea must be real also [2; p.31 ff.]. Non-real linear combinations as the basis vectors of Evans' Complex Circular Basis are inadmissible.


Conclusion

That is a serious error, which should make the author reconsider the concept of his Complex Circular Basis which is of severe influence for the whole GCUFT book.


Remark 1 The first way out of the dilemma could appear to be replacing Equ.(17.10) with

gμν = qaμ qbν* ηab ,                                                                 (17.10*)

however, that would yield a Hermitean metric, which would in general be non-real likewise.

Remark 2 Let Q be the matrix (qaμ), QT its transposed and M = diag(1,−1,−1,−1) the Minkowski matrix (see Appendix 1). Then, as can easily be seen from Eqns. (17.36'-39') above, we have QT M Q* = M. That means that Evans' introduction of his twisted Complex Circular Basis by (17.40'-41'), (eo, i, j, k) → (e(0), e(1), e(2), e(3)), is nothing but a (complex) Lorentz transform and hence the spacetime under consideration is flat. Of course, the modified metric (17.10*) has to be used here. Instead of all that trouble with Evans' twisted Complex Circular Basis (17.36-41) a real twisted basis (eo, i cos Φ + j sin Φ, − i sin Φ + j cos Φ, k) could be introduced with the same (useless) result and using a real metric (17.10) only.

Remark 3 Another problem would be the definition of the well suited tetrad vectors e(a) (a=0,1,2,3) in more general cases than plane waves, say e.g. for the case of the superposition of two plane waves with different directions of propagation.


References

[1]     M.W. Evans, The Spinning and Curving of Spacetime:
        The Electromagnetic and Gravitational Fields In The Evans Unified Field Theory

        Foundations of Physics Letters Vol. 18, No. 5, p.431-454

[1a]   M.W. Evans, The Spinning and Curving of Spacetime:
        The Electromagnetic and Gravitational Fields In The Evans Unified Field Theory

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

[2]    Sean M. Carroll, Lecture Notes on General Relativity,
                http://arxiv.org/pdf/gr-qc/9712019

[3]    G.W. Bruhn and A. Lakhtakia, Commentary on Myron W. Evans' paper
                "The Electromagnetic Sector ..."
,
                http://www.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html

[4]    M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics,
                http://www.aias.us/Comments/Evans-Book-Final.pdf

[5]    M.W. Evans, METRIC COMPATIBILITY FROM TETRAD POSTULATE,
                http://www.aias.us/Comments/comments01172005b.html

[6]    M.W. Evans, METRIC COMPATIBILITY FROM TETRAD POSTULATE,
                http://www.aias.us/Comments/ametriccompatibilityfromtetradpostulate.pdf



Appendix 1

Equ. (17.40'-41') in extended matrix notation

                æ e(0) ö        æ   1             0             0            0    ö    æ eo ö
                |  e(1)  |        |    0     2−½ eiΦ    −i 2−½ eiΦ     0     |    |   i   |
                |        |   =   |                                                      |    |       |
                |  e(2)  |        |    0     2−½ eiΦ    i 2−½ eiΦ    0     |    |   j   |
                è e(3) ø        è   0             0             0            0    ø    è  k  ø

i.e.             (e(a))   =                             Q                                 (eμ)

defines the complex tetrad coefficient matrix Q = (qaμ). The reader is kindly asked to convince himself of the validity of the equation (17.10*),

(gμν) = (qaμ ηab qbν*) = QT M Q* = M = (ημν)

and that Evans' Equ.(17.10) gives a non-real result.

The analogous effect occurs for the real twisted tetrad system given by

                æ e(0) ö        æ   1             0             0            0    ö    æ eo ö
                |  e(1)  |        |    0         cos Φ       sin Φ        0     |    |   i   |
                |        |   =   |                                                      |    |       |
                |  e(2)  |        |    0       − sin Φ      cos Φ        0     |    |   j   |
                è e(3) ø        è   0             0             0            0    ø    è  k  ø

with the difference that we need not modify the real metric equation (17.10). However, in both cases it turns out that the underlying metric (gμν) is Minkowski, so for Evans' purpose of introducing a metric with torsion we can speak with Shakespeare:

Much Ado About Nothing!




Appendix 2


Identical Misprints in book [4] and article [1]

"Fa = ... ≠ 0" should be "Fa = ... = 0" in Equ.(17.65).

Wrong equation reference on p.311: (17.68) should be (17.69).

Equ.(17.91) should probably run as

R = κκ* = (T232)² + (T132)² .




Appendix 3

Lorentz Transform of a Triad Associated With a Plane Wave

We consider a plane linearly polarized em-wave with the phase function Φ = ω(t − x·n/c). Φ must be invariant under Lorentz transforms. That yields the transformation rules for two inertial systems S and S', S' travelling against S with velocity v in x direction:

ω = ω' γ (1 + β nx')
ω nx = ω' γ (β + nx')
ω ny = ω' ny'
ω nz = ω' nz'

where β = v/c, γ = 1/(1 − β²)½ .

For a linearly polarized plane em-wave travelling in z-direction of S, we obtain nx = ny = 0 , ny' = 0 and nx' = − β, hence ω = ω'/γ . The propagation direction n = k is transformed to n' = − β i' + 1/γ k' .

We assume the E field || i+j and the H field || ij, hence Ez = Hz = 0 while E := Ex = Ey and H := Hx = − Hy where due to Maxwell E = H.

The transformation of the fields E and H yields

Ex' = Ex = E,                 Hx' = Hx = H
Ey' = γ (Ey − βHz) = γ Ey = γ E ,                 Hy' = γ (Hy + βEz) = γ Hy = − γ H
Ez' = γ (Ez + βHy) = βγ Hy = − βγ H,                 Hy' = γ (Hz − βEy) = − βγ Ey = − βγ E

Therewith we obtain

E' · H' = (1 − γ² − β²γ²) EH = 0 EH = 0

i.e. the transformed field vectors E' and H' are orthogonal also. In addition we have n'· E' = β (E − H) = 0 and n'· H' = β (H − E) = 0 .

Thus, the S' vectors n', E', H' are pair wise orthogonal like the S vectors n, E, H.

Consider the orthonormal "triad" e(1), e(2), e(3) associated with the wave vectors: In S we have

                                                                E || e(1) := (i+j)/2½ ,
                                                                H || e(2) := (ij)/2½ ,
                                                                n || e(3) := k .

However the Lorentz transform L: S → S' does neither preserve the orthonormality of the triad nor its association to the plane wave: We obtain

                                                                L e(1) := 2−½−1i'+j')         not || E',
                                                                L e(2) := 2−½−1i'j')         not || H',
                                                                L e(3) = k'                           not || n'

where the orthonormality of the vector L e(1) and L e(2) and the parallelity to the wave vectors is evidently lost.

The association of the triad e(1), e(2), e(3) to the plane wave is no physical property.