Review of Lar Felker's book
"The Evans Equations of Unified Field Theory" Rev.3.2

by Gerhard W. Bruhn, Darmstadt University of Technology


Last update: February 21, 2006

Supplements:      
Febr 13: p.138/139    
Febr 15: pp.144-175  
Febr 16: p.156          
Febr 21: p.157          


To be supplemented occasionally


Quotes from Felker's book in black, comments in blue


Preface

The book is written for laymen's introduction to Myron W. Evans' CGUFT, nowadays also called ECE Theory (= EVANS' Cartan Einstein Theory). So certain simplifications and reductions are necessary, however, the text should be as correct as possible. Instead it contains a collection of erroneous and strange formulations up to severe irremovable errors.

Who is responsible for that disaster? Of course, first of all the author himself. A lot of akward and cryptic formulations could have been avoided, if the author would have been equal to his task. However, even more responsibility is to Myron W. Evans whose ideas are only reflected by Lar Felker. And Evans expressly recommends Felker's book on his website.

What I have listed below are only the tips of icebergs without any intention of completeness.

The most akward formulations could be removed without consequences, because the book is not of mathematical stringency. E.g., the remarks on p.26 about C = A × B are simply wrong and could be skipped. The bad formulations, however, give evidence of the author's mathematical ignorance and are embarrasing therefore.

When reviewing the book one should imagine that poor willing readers try to understand and interprete these enigmatic and misleading formulations. They waste their time and their confusion grows from page to page.

Download of L. Felker's original pdf-text from MW Evans' website (2.8 MByte)


Quotes and Comments


p.14

4 As this book was being written, it was hard to keep up with developments. For example, explanation of the Faraday homopolar generator which has been a mystery for 130 years has been published.

The author should have a look into the relevant physics text books. Then he will detect that the mysteries of the Faraday homopolar generator are his own mysteries only.


p.26

a × b is called the cross product. If a and b are 2-dimensional vectors, the cross product gives a resulting vector in a third dimension

"Perpendicular to both 3-D vectors a and b" would be correct, i.e. both a and b and the resulting vector a×b are 3-D vectors.

Generalizing to 4 dimensions, if A and B are vectors in the xyz-volume, then vector C = A × B will be perpendicular or “orthogonal” to the xyz-volume.

WRONG: A vector C = A × B is not defined in 4-D space. See p.341 Cross product

The wedge product is the four dimensional cross product.

The wedge product AÙB in 4-D space is no vector.


p.35

Imagine that we take a penny and a truck, we take them out into a region with very low gravity, and we attach a rocket to each with 1 kg of propellant. We light the propellant. Which – penny or truck – will be going faster when the rocket stops firing? We are applying the same force to each. The penny. At the end of say 1 minute when the rockets go out, the penny will have traveled farther and be going faster. It has less mass and could be accelerated more. If instead we drop each of them from a height of 100 meters, ignoring effects of air resistance, they will both hit the ground at the same time. Gravity applies the same force to each, as did the rocket. Thus, gravity seems like a force but is not a force like the rocket force. It is a false force. It is actually due to curved spacetime. Any object – penny or truck – will follow the curved spacetime at the same rate. Each will accelerate at the same rate. If gravity were a force, different masses would accelerate at different rates.

An erroneous example: The author should apply his "theory" to 1 penny compared with a bag of  1.000.000 pennies (instead of the truck).


p.47

The tetrad is introduced here and will be developed further in later chapters. The tetrad is a 4×4 matrix of 16 components that are built from vectors in the base manifold and the index as shown in Figure 2-9.

NO! A tetrad is a "4-bein" (tetra = 4), i.e. a set of 4 vectors e0,e1,e2,e3 in the tangent space of each point of the spacetime manifold with certain additional geometric properties. See the remarks to p.49.


p.49

It is built from four 4-vectors in the base manifold - our real universe - and four 4-vectors in the mathematical tangent space.

No, both sets of vectors belong to the tangent space. One set is given by the tangent vectors of the coordinate axes (usually denoted by 0,1,2,3, which have the disadvantage of not being orthonormal relative to the metric in general) the other set (the tetrad) consists of 4 vectors e0,e1,e2,e3, orthonormal relative to the metric. Since both sets belong to the same linear space there exist linear representations of one system by the other one.

ea = qaμ μ   and conversely   μ = qμa ea .

The coefficients qaμ and qμa of these representations can be called "tetrad coefficients", not being identical with the tetrad itself. Both systems of vectors can be used as a reference basis. Each vector V in the tangent space can be represented relative to both sets of reference.

V = Vμ μ = Va ea .

The coordinates of a vector V, Vμ and Va respectively, relative to the two reference systems are linearly interrelated with the tetrad coefficients as coefficients:

Va = Vμ qμa   and conversely   Vμ = Va qaμ .

NB. The tetrad-matrices (qμa) and (qaμ) are mutually inverses, hence both have non-vanishing determinants.


p.51

Four vectors make up what we call a "4-vector".

No! Each element of a 4-dimensional linear vector space is a "4-vector." That is a result of Felker's permanent misunderstanding of the vector concept. A vector

V = Vμ μ = Va ea .

has different four coordinates (not "vectors") Vμ (μ=0,1,2,3) and Va (a=0,1,2,3) relative to different frames. Felker erroneously considers these coordinates as "vectors" itself; hence he can state «Four vectors make up what we call a "4-vector».

Which is the meaning of the η's in (3)? On p.52 we learn:
A four-vector is multiplied by the factors –1, 1, 1, 1 to find the metric four-vector.

ea metric = ηa ea (without summation over the tetrad index a)

where η0=−1 and ηa=+1 for a=1,2,3. Completely misleading.

What Felker really wants to introduce here is a third kind of vector description: Let the matrix (ηab) be the Minkowski matrix diag(−1, 1, 1, 1). Then for arbitrary vectors V = Va ea Felker's "metric" four-vector is given by Va= ηabVb. In other words: Felker's "metric" four-vector is given by the covariant representation of V.

Due to his erroneous misunderstanding of the vector concept Felker does not remark that he is always involved with the same vector.


p.52

Completely awful!


p.61

ψ(r,t) is the wave function of a particle defined over r space and t time. If squared it becomes a probability. In quantum theory ψ² is used to find the probability of an energy, a position, a time, an angular momentum.

ψ(r,t) as solution of the Schroedinger equation is complex-valued in general. How can it be that ψ² is a probability, i.e. real-valued?


p.83

Figure 4-4
The dot or scalar product is a scalar, just a distance without direction. It can be used to get work or other values also.

The scalar product yields no distance in general. The Figure 4-4 is misleading nonsense.


p.84

Two four-vectors define a four dimensional space and the product of two is in the space.

How that? They (A and B) define a two-dimensional space, the space of all of their linear combinations aA + bB, with the coefficients a,b.
Which product? The scalar or dot product would yield scalars (i.e.no 4-D vectors), the cross product is not defined and the wedge product yields no vectors, and also the outer product (see p. 89-90) doesn't fit. So what?

If an inner product is defined for every point of a space on a tangent space, all the inner products are called the Riemann metric.

OOOOOH!!! How awful! Do whatever you like, however, don't write a book on a topic you have not understood.


p.85

The inner product of two tetrads is: qμν = qaμ qbν ηab. This is a tensor, the symmetric metric, gμν, which is the distance between two points or events in 4-dimensional Riemann space – the space of our universe.

What is the inner product of TWO tetrads. From where do we obtain two (different?) tetrads? And the symmetric metric should be the distance between two points or events. What a nonsense!

If three vectors are multiplied, the result is a volume in three dimensions. See Figure 4-5.

WRONG! Not for the double cross-product a×(b×c) as displayed on p.86 Figure 4-6.


p.86

Figure 4-6
The scalar or mixed product is a volume.

OK, but that volume is not given by the double cross-product a × (b × c) displayed in that figure.


p.87

Divergence
The divergence of a vector tells the amount of flow through a point.

So calculate div (a×r) where a is constant and r denotes the radius vector.
Is there any flow of  F=a×r "through a point" or not?


p.95

Evans uses this to decompose the tetrad into parts. Those parts define
gravitation and electromagnetism. He uses a purely mathematical function that
is well known and generalizes to the physical process.
     qμa =     qμa (S)     +      qμa (A)
           = (symmetric) + (antisymmetric)
           = gravitation   +  electromagnetism
           = curvature    +      torsion
           = distance      +       turning

WRONG!!! The matrix of the tetrad coefficient doesn't transform symmetrically relative to the two indices μ and a. Thus the well known splitting of a matrix Q (cf. p.295: Antisymmetric tensor)
into its symmetric part           Q(S) = ½ [Q + QT]
and its antisymmetric part      Q(A) = ½ [Q − QT]
where QT denotes the transposed matrix of Q, does not apply here. The reason is that both matrices Q and QT have different transformation behavior (co- and contravariantly with respect to the index μ), and hence the addition of both matrices doesn't yield a quantity with physical meaning (apples and pears cannot be added), while the lines

           = gravitation   +  electromagnetism
           = curvature    +      torsion
           = distance      +      turning
contain physical/geometrical quantities. Cf. the chart on p.175.


p.96

The Tetrad qaμ
. . . . . . . .
In the tetrad matrix each element is the product of two vectors. One vector is in the base manifold and one in the index. q00 indicates q0 times q0.

Which "vectors" could be meant by q0 and q0 ? Felker cannot distinguish between vector and vector coordinate (see the remarks to p.51). Here "vector" means "vector coordinate" q0 or q0 respectively. So we have an outer product of two vectors in the sense of p.89/90 under consideration yielding the tetrad matrix (qaμ). However, such a matrix cannot be invertible since having parallel line vectors, in contradiction to the necessary invertibility of the tetrad matrices (see the remarks to p.49).

Felker's claim of an outer product decomposition of the tetrad matrix cannot be true.


p.97

Let Va be the four-vector in the tangent orthonormal flat space and Vμ be the corresponding base four-vector, then:

Va = qaμ Vμ                                                                 (7)

qaμ is then the tetrad matrix. Here a and μ stand for 16 vectors as shown in the matrix above. Each element of qaμ, the tetrad, is Va Vμ.

The sentence "Here a and μ stand for 16 vectors as shown in the matrix above" means that the tetrad matrices are 4×4 and therefore have 16 numbers (not vectors) as entries.

May we write the last sentence as the equation qaμ = Va Vμ or what? Then the author should explain how that could be deduced from Equ.(7); i.e. from its written out version

Va = qa0 V0 + qa1 V1 + qa2 V2 + qa3 V3

where the indices written in italics refer to the coordinate basis of rhe tangent space. Until we get the author's reply we declare the sentence "Each element of qaμ, the tetrad, is Va Vμ" to be nonsense. And he should explain what is meant by "a and μ stand for 16 vectors".

Here is only one vector, namely V, which has 4 coordinates V0,V1,V2,V3 relative to the coordinate basis 0,1,2,3 and other 4 coordinates V0,V1,V2,V3 relative to the tetrad basis e0,e1,e2,e3:

V = Vμμ = Vaea .

The two different vector coordinate sets are interrelated by the equations (7).

Most of the trouble above is caused by the fact that the author confounds a vector with its coordinates relative to different vector bases.


p.109

Figure 4-18 Vector Multiplication

contains several enigmatic and misleading formulations: Which is the meaning of the geometrical symbols?
E.g.: What is meant by

↑ · ↑ · η = distance ?

Perhaps VaWbηab for two vectors ↑ = V = Vaea, ↑ = W = Waea and the Minkowski matrix η = diag(−1, 1, 1, 1) ?
However, that is no distance at all; even in the case V = W we would obtain the square of the Minkowskian length of V, namely the value

− V0 2; + V1 2; + V2 2; + V3 2;

And what shall the reader learn from

Hodge Dual *A = dual of A
Dual of a 3-vector is a plane. ?


p.137/138

The most general asymmetric metric tensor is defined by the outer or tensor product of two tetrads: Here the tetrad is equivalent to metric four-vectors

qabμν = qaμ qbν ,     qabμν = qabμν(S) + qabμν(A)                                 (2)

An asymmetric form has both symmetry and antisymmetry within it.

None of the above quantities can be identical with the metric tensor gμν of spacetime as they have two additional upper tetrad indices a,b, which are missing at gμν. The correct relation is

gμν = ηab qaμ qbν.

Here, due to the symmetry of (ηab) = diag(−1 , 1, 1, 1) the matrix gμν turns out to be symmetric.

Thus, a (nontrivial) antisymmetric part of the metric gμν does not exist.

The anti-symmetric metric tensor is qabμν(A). This defines an area, dA, with rotational potential. The antisymmetric metric tensor is defined by the wedge product of two tetrads:

qabμν(A) = qaμÙ qbν                                                 (3)

Instead of (3) we have to consider

(3')                                                 ηab qabμν(A) = ηab qaμÙ qbν

which vanishes due to the symmetry of ηab.

Thus, the antisymmetric part of the metric gμν is ZERO.

Symmetry indicates centralized potentials – spherical shapes.
Anti-symmetry always involves rotational potentials – the helix.
Asymmetry indicates both are contained in the same shape.

NO COMMENT!


p.144

The Evans solution, Raμ – ½ Rqaμ = kTaμ, is more fundamental than Einstein’s equation, Gμν = Rμν − ½ gμν R = − kTμν.

That is not true. As will readily be shown here both equations are equivalent.

Proof We "multiply" the Einstein equation by qbν ηab to obtain Evans' equation. Conversely "multiplying" Evans' equation by ηab qbν yields the Einstein equation, i.e. both equations are equivalent. That means:

Evans' equation does not deserve an own name.
It is identical with the Einstein equation except a trivial calculation.


p.146

The electromagnetic and weak fields are described by a postulate similar to Einstein’s R = kT:

Aaμ = A(0) qaμ                                                 (12)

A(0) is a fundamental electromagnetic potential in volts-seconds per meter.

cf. Remarks to p.175


p.151/152

Field Descriptions

The Evans Lemma gives quantum field/matter theory from general relativity. Gravitation is described by the Lemma when the field is the tetrad, qaμ.

The other three fields – electromagnetism, weak, and strong - are described when the field is the tetrad multiplied by an appropriate scaling factor and is in the appropriate representation space. For example, the fundamental electromagnetic field has the antisymmetry of equation (6) and is described by equation (12).

Equation (6) on p.140 is

gabμν (A) = qaμ ^ qbν                                                 (6)

Evans felt remembered of the antisymmetry of the electromagnetic fundamental tensor Fμν = − Fνμ

The strong and weak fields are described respectively as:

Saμ = S(0) qaμ                                                 (13)

Waμ = W(0) qaμ                                                 (14)

Electromagnetism is defined by the torsion form. It is spinning of spacetime itself, not an object imposed upon the spacetime. The weak field is also a torsion form. It is related to electromagnetism. Thus in the neutron's conversion to a proton the torsion form will be involved. We see an electron leave the neutron and a proton remains. Now it is more clear that there was an electrical interaction that has an explanation. See Chapter 12 on the electroweak theory.

That all may be or not: Pure speculations. see p.175


p.156

"Multiplication" of Einstein's field equation

Gμν = Rμν − ½ R gμν = kTμν .                         (EFE)

by qbν ηab yields the eqivalent equation

Gaμ = Raμ – ½ Rqaμ = kTaμ .                         (9,16)

Multiplication by qbν leads to Evans' "General Field Equation"

Gaμ qbν = kTaμ qbν ,                        

which due to the invertibility of (qbν) is eqivalent to the former versions of Einstein's field equation. The "multiplication" by ηab leads back to the Einstein equation (EFE).

In addition Evans considers the antisymmetric part of (4.3) to be obtained by wedging (9,16) by qbν:

GaμÙqbν = kTaμÙqbν .                         (8)

Now Evans has a problem: There are two upper indices a and b (=0,1,2,3) where not any should be. His way out is contracting GaμÙqbν and TaμÙqbν to Gcμν and Tcμν respectively.

Gcμν = k Tcμν .                         (p.156)

And specifying nothing about c.

Thus, Equ.(p.156) has no meaning at all.

Remark The only correct way of removing the indices a,b is to "multiply" by ηab. Due to ηab qaμÙqbν = 0 that yields the very convincing equation 0 = 0.


p.157

Evans tried at least twice to derive his "Evans wave equation", in the Chapters 4.2 (p.67 f.) and 9.2 (p.178 f.) of his GCUFT book I. However, both attempts are wrong due to elementary calculation errors. Even an unlearned reader can recognize that the equations (4.21-23) on p.69/70 of [2] cannot be true due to wrong occurrence of the indices λ, μ and ν in these equation: A formal (and very helpful) condition of tensor calculus is, that at each summand an index can at least appear twice, one in the lower and the other one in the upper index position. So check the equations (4.21-23) on p.69/70 of [2]! And a correct calculation yields a result that does not agree with Evans' ideas.

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

The result:

The Evans Wave Equation is invalid.


p.174

Of great interest is that the symmetries of the matrix in mathematics are applied to physical quantities:

qaμ = qaμ(S) + qaμ(A)

The splitting is coordinate dependent, hence "physically not meaningful".

Gravitation can be symmetric or antisymmetric, however it is always curving.

Electromagnetism can be symmetric or antisymmetric, however it is always turning.

Summary

The wave equation of general relativity and unified field theory is:

(o + kT) qaμ = 0

By substituting appropriate representations for the tetrad, the various equations of physics can be derived. Gravitation, electromagnetism, the weak force, and the strong force can all be represented. This simple looking equation can be expanded as given in the chart at the end of the chapter. The wave equation was derived from general relativity using differential geometry.

The factorization of the symmetric and antisymmetric metrics from the asymmetric tetrad is basic differential geometry. It gives four forces – a discovery opening new insights into physics. This is qaμ = qaμ(S) + qaμ(A).

The splitting is coordinate dependent, hence "physically not meaningful".

Applied to physics, four potential fields are represented as shown in the chart.

The standard model is not generally covariant – it does not allow calculations of interactions among particles in different gravitational fields, say near a black hole. The standard model does not allow for electromagnetism's and gravitation's mutual effects to be defined.


p.175

The Evans unified equations allow both these processes to be accomplished.

The use of the mathematical representation space for the tangent spacetime “a” is of the essence. This is the Palatini variation of gravitational theory. The tetrad can now be expressed as any of the four fields of physics – G, A, W, and S. Thus four forms of energy can be described within the unified field qaμ :

While the structure is not fully developed yet, the logic is clear. Gravitation is centralized and symmetric. Mass curves space in a spherical shell around it. The electromagnetic wave, the photon, is antisymmetric spinning spacetime. The electron (charge) is centralized spin. Antisymmetric curvature is a matter for study.

TYPE                                 POTENTIAL FIELD

Gravitational curvature
Symmetric = Centralized                qaμ(S)
Einstein gravitation

Gravitational curvature
Antisymmetric = turning
Unexplored. The strong                 qaμ(A)
field, dark matter?

Electromagnetic Field
Antisymmetric EM                 Aaμ(A) = A(0) qaμ(A)
Photon, EM waves

Electrodynamics
Symmetric EM                       Aaμ(S) = A(0) qaμ(S)
Charge, the electron

MW Evans felt motivated to these four Ansatzes by the shining example of the Einstein equation Gμν = − k Tμν (cf. p.146). However, the ansatzes have a shortcoming that is not present at Einstein's formula: Each potential should generate a single physical field only. But there is the tetrad dependent index a (= 0, 1, 2, 3), which gives rise to four fields at once. More, the generated fields depend on the choice of the tetrad, i.e. they are not well determined. Last not least the physical motivation of the potentials is questionable.


p.184

We have seen that G qaμ = kT qaμ. This says that curvature, G, and stress energy, T, are related through the tetrad matrix. The unified field is the tetrad qaμ.

That's intellectual fraud: Due to EINSTEIN we know the equation G = kT. To multiply that by the factor qaμ is trivial. And saying that both G and T are related by qaμ is rubbish! G and T were related by Einstein without Evans' factor qaμ.

The tetrad is an asymmetric square matrix. This can be broken into its symmetric and antisymmetric parts:

qaμ = qaμ(S) + qaμ(A)                                                 (3)

The splitting is coordinate dependent, hence "physically not meaningful".

See "Antisymmetric tensor" in Glossary: There a double contravariant tensor is split into its symmetric and its antisymmetric part. Both parts are double contravariant again. The same procedure applied to the tetrad matrix (qaμ) would destroy the type of the matrix, i.e. its transformation behavior, completely, because of the different characters of upper and lower indices.


p.296

Basis vectors

Unit vectors e(1), e(2), e(3), e(4) which define a mathematical tangent space.

Read "Unit vectors e0, e1, e2, e3 ...". The brackets at the indices on p.296-297 should be removed for sake of consistency.

The basis vectors of a reference frame are a group of four mutually orthogonal vectors. Basis vectors establish a unit vector length that can be used to determine the lengths of other vectors. They obey:

e0² + e1² + e2² + e3² = +1

Due to the following equations we have e0² + e1² + e2² + e3² = −1 + 1 + 1 + 1 = 2 .

That is, they form a four dimensional sphere. In addition:

e0·e0 = −1 ,   e1·e1 = e2·e2 = e3·e0 = +1

ea·eb = 0 if a ≠ b; a, b are 0, 1, 2, or 3.

OK. That could be taken as the definition of the dot product for the linear space spanned by the vectors e0, e1, e2, e3, i.e. in the tangential space of the spacetime-manifold at some of its points. The vectors e1, e2, e3 are defined on p.297 by means of the parameter representation R = R(u1,u2,u3) (of the spacetime manifold???). By that way we see that the orthogonality relations

ea·eb = 0 if a ≠ b; a, b are 0, 1, 2, or 3.

suppose the mutual orthogonality of the parameter lines. However, there is no place for e0:

Where and which is the definition of e0?

Following that question we shall recognize that the parameter representation R = R(u1,u2,u3) is not sufficient to represent the spacetime manifold under consideration.


p.297

Orthonormal means they are both orthogonal and normalized. If one finds the orthonormal vectors at an event point, then one can define the vectors at any other event point in the spacetime. The spacetime is curved and some method of calculating and visualizing the spacetime is needed. Unit vectors are tangent to a curve at a point. For example the three unit vectors of a curved coordinate system are mutually orthogonal ("perpendicular") and cyclically symmetric with O(3) symmetry. With e(1,2,3) the unit basis vectors and u(1,2,3) the coordinates at any point.

The three coordinates u1, u2, u3 can parametrize a three-dimensional manifold only by R = R(u1,u2,u3). However, spacetime has four dimensions. Below only three tangential unit vectors e1, e2, e3 appear.

Where is the fourth tangent vector e0, and how is it defined?

e(1)·e(2) = 0, e(1)·e(3) = 0 and e(2)·e(3) = 0

and

e(1)×e(2) = e(3), e(1)×e(3) = e(2) and e(2)×e(3) = e(1)

The vectors e(k) are 4-vectors. And the operation × is not defined in 4-D. cf. p.341 Cross product.

e(n) = 1/hi ×R/∂ui     read ei = 1/hi ×R/∂ui

and the arc length is:

ds = |dR| = |∂R/∂u1×du1 + ∂R/∂u2×du2 + ∂R/∂u3×du3|

Here the operators × should be removed as being no cross product.

By squaring the last equation one would obtain

ds² = |dR|² = ∂R/∂uj duj ·R/∂uk duk = gjkdujduk

to be summed w.r.to j,k over 1,2,3. That is merely the metric of a three-dimensional manifold while a metric of the four-dimensional spacetime has to be defined. Not even the case of the (flat) Minkowskian spacetime can be treated by that line element ds.

A definition of the metric of the 4-dimensional spacetime manifold is missing.

Questions:
To which space X does R(u1,u2,u3) belong? A space wherein derivatives ∂R/∂uk are defined and belong to the space. How many dimensions has that space?

By the representation R(u1,u2,u3) MWE tries to embed the spacetime manifold in a greater space X, so as a sphere surface is embedded in the Euclidean X = R³. By doing so, in case of the sphere surface the metric can be inherited from the surrounding X = R³ space. However, in MWE's case there is no metric that could be inherited, and that's MWE's problem. The metric that should be induced on spacetime from X should be locally Minkowskian, however, I cannot see any way to do so. (In the theory of manifolds one doesn't need an embedding of the manifold and defines the metric independently.)

Evans error: Instead he tries to define a dubious pseudo-Euclidean metric; see MWE's difficulties concerning his metric below.


p.329

M(A) = M(S) + M(A-S)

Asymmetric equals Symmetric plus Anti-Symmetric

The splitting above from M(A) = M(S) + M(A-S) is no splitting into symmetric and antisymmetric parts.


p.331

Metric tensor

ds² = gμν xμ xν = qμqν

read ds² = gμν dxμdxν (= qμqν ?).

The latter part in (...) is evidently wrong as it depends on the summation indices μ and ν. Probably the author has meant the equation

ds² = gμν dxμdxν = qμqν dxμdxν .

However, the last equation would imply that the matrix must have a vanishing determinant, because all line vectors are parallel to the line [q0 q1 q2 q3].

The metric tensor is defined as the outer product of two metric vectors. Form the outer product of two four vectors, i.e. multiply a column four vector by a row four vector, and you have a 4×4 matrix, with sixteen components.

However, in that case we obtain det(gμν) = 0 due to the above argument.

. . . The symmetric metric is always defined as the tensor or outer product of two vectors.

We have quite generally det (ATÄB) = 0 for two arbitrary n-vectors A, B and their outer product ATÄB. However, the matrix g = det(gμν) is not allowed to vanish since the metric matrix must be invertible. Thus:

The symmetric metric (gμν) cannot be defined as outer product of two 4-vectors.


p.332

We want a real number to define the distance. The metric tensor takes two vectors which define the curvature that occurs and calculates the real distance between them. We cannot have a negative distance so the metric must be "positive definite" – it must be real.

WRONG! Even the special metric of Minkowkian spacetime (cf. p.333), the matrix (ημν), is indefinite, i.e. ds² = − c²dt² + dx² + dy² + dz² attains negative values inside the light cone and positive values outside while it vanishes at the cone itself. And the determinant of the Minkowskian metric, det(ημν) has the value −1. Thus:

The Minkowskian metric (ημν) is a counter example of a metric matrix that cannot be written as an outer product of two 4-vectors.


References

[1]     Laurence G. Felker, The Evans Equations of Unified Field Theory; Rev. 3.2 Oct.2005          http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[2]     Myron W. Evans, GENERALLY COVARIANT UNIFIED FIELD THEORY:
         THE GEOMETRIZATION OF PHYSICS
; Web-Preprint,
         http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[3]     G.W. Bruhn, Remarks on the "Evans Wave Equation" ;
         http://www.mathematik.tu-darmstadt.de/~bruhn/EvansWaveEqu.html


To be supplemented occasionally.



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