 Last update: 02.03.2005 16:00 CET
Remark to Equ. (2.42) added.

## Chapter 2: Duality and the Antisymmetric Metric (p.21 - 30)

#### Abstract

Chapter 2.1. contains two essential claims:
(i) The existence of an "antisymmetric metric", which MWE based on the claim that an antisymmetric matrix should be able to replace the usual symmetric matrix of differential geometry.
(ii) The well-known 3-D duality of polar and axial vectors could be generalized to 4-D.
However, it turns out here that
(i) the equations (2.8-9) in form of ω1 = ds2 = ω2 is only fulfilled for the (trivial) NULL-metric.
(ii) In 4-D there is no duality between 1-forms and 2-forms: The (Hodge-)dual of a 1-form in 4-D is a 3-form, e.g.
*dx0 = dx1Ůdx2Ůdx3 .
Thus, both claims of MWE are invalid.
Chapter 2.2. contains the generalization to the 4-D case: A completely wrong symmetric matrix (2.41) of the 4-D metric and, in addition the construction of the antisymmetric matrix (2.46) for Evans' "antisymmetric metric" for the 3-D case, which is invalid for non orthogonal coordinates. The "generalization" to the 4-D case refers to a duality in 4-D that does not exist.

We recommend that the author should try to correct his errors. Quite another theory will be the result.

MWE's original text  appears in black with intermediate comments in blue.

### 2.1. EUCLIDEAN SPACETIME. SPECIAL RELATIVITY

In this chapter it is proven that if there exists a symmetric metric in general relativity, then there must exist an antisymmetric metric: The fundamental reason is that one metric is implied by the other through duality in differential geometry . This result of differential geometry leads to the important new result in physics that if the gravitation be described through Riemannian geometry with the well-known Einstein field equation, Eq. (1.29), then there exists an equation of the same type for electrodynamics in general relativity. This equation is derived and explained in this chapter using differential geometry with a antisymmetric metric. Gravitation therefore is a manifestation of curved spacetime with a symmetric metric, and electromagnetism is a manifestation of spinning spacetime with an antisymmetric metric. >From differential geometry we can infer that electromagnetism is implied by gravitation in non-Minkowski, four-dimensional spacetime.

This result can be constructed from a consideration of Eq. (1.67), a duality in Euclidean space:

εij = − εijk εk,                                                                 (2.1)

ε12 = − ε123 ε3 = − ε123 = −1 due to (1.118-119), and ε12 = − ε3 = −1 due to (1.67). OK.

εij = εijk εk.                                                                 (2.2)

Consider the metric 3-vector defined in Euclidean space (Chap. 1) as

(qk) = (εk) = (1, 1, 1).                                                                 (2.3)

The symmetric metric tensor in Euclidean space can be defined by

1 0 0              −1 0 0
qk = qkr(S)qr,         (qkl(S)) = − 0 1 0         =   0 −1 0 .                                                             (2.4)
0 0 1              0 0 −1

(2.4) Þ (qk) = (−1, −1, −1) Þ qk qk = −3 Þ (2.5) , and εk = qk from (1.73): (qμ) = (0, q1, q2, q3) :
Tacitly: Latin indices run over 1,2,3 ; Greek indices run over 0,1,2,3.

Define the scalar metric quantity q by

q = qkqk = qkεk = −3.                                                                 (2.5)

The metric 3-vector qk is dual to the antisymmetric metric tensor in three space dimensions,

0 −1 1
(qij(A)) = (−εijkqk) = 1 0 −1 ;                                                                 (2.6)
−1 1   0

εijkqk = − εijk if k ≠ i,j = − εij Þ (2.6) OK.

and, from Eqs. (2.5) and (2.6) the following duality relation between the symmetric and antisymmetric metric tensor is obtained:

q = −˝ qij(A)εij = qkr(S)εrεk.                                                                 (2.7)

In three Euclidean-space dimensions it therefore has been shown that the existence of one metric implies that of the other through the fundamental duality relations (1.67). Because εk is dual to εij by geometry, then qkr(S)εr must be dual to qij(A). So the existence of the well-known symmetric metric (2.4) implies the existence of the antisymmetric metric (2.6) by three-dimensional Euclidean geometry.

This simple result is generalized in the chapter using the language of forms and differential geometry, where-upon it becomes clear that the homogeneous field equation of electromagnetism is a Bianchi identity on a closed two-form involving the antisymmetric metric in a non-Minkowski spacetime. The inhomogeneous field equation of electrodynamics can be derived by taking covariant derivatives of both sides of a novel field equation of electrodynamics whose structure is the same as that of the well-known Einstein field equation (1.29) of gravitation. The duality Eq. (2.7) implies that, if there exists a line element

ω1 = ds2 = qkr(S)dxrdxk,                                                                 (2.8)

a zero-form (scalar) of differential geometry, then there exists the zeroform

ω2 = ds2 = − ˝ qij(A) dxi Ů dxj ,                                                                 (2.9)

Therefore, the line element in three-dimensional differential geometry can be expressed as the zero-form ω1 or the zero-form ω2.

MWE claims here that ω1 = ds2 = ω2 . There is a simple possibility to check that:
We have the following tensor representations of the (antisymmetric) wedge product
dxiŮdxk = ˝ (dxiÄdxk − dxkÄdxi)
and of the (symmetric) algebraic product
dxi dxk = ˝ (dxiÄdxk + dxkÄdxi)
Using these representations one can readily show that ω1 and ω2 have the tensor representations
ω1 = qik(S) dxiÄdxk
and
ω2 = qik(A) dxiÄdxk .
However, since the products dxiÄdxk constitute a basis of the corresponding (0,2) tensors, their representation is unique. MWE's claim ω1 = ds2 = ω2 implies
qik(S) dxiÄdxk = ω1 = ω2 = qik(A) dxiÄdxk ,
and due to the uniqueness of basis representations we may conclude
qik(S) = qik(A)
Considering the symmetric part of that equation we obtain
qik(S) = sym.part (qik(A)) = 0,
and by considering the antisymmetric part likewise
qik(A) = anti-sym.part (qik(S)) = 0.
Thus, MWE's claim ω1 = ds2 = ω2 is fulfilled if and only if the metric is zero, which, of course, is of no interest.
Thus,

### MWE's claim of a metric given by the antisymmetric matrix (qik(A)) is wrong.

The two-form qij(A) is a closed two-form whose exterior derivative vanishes identically:

d Ů qij(A) = 0 = (d Ů q(A))ij .                                                                 (2.10)

The d operator is only defined for differential forms, e.g. for the 2-form
q(A) = qij(A) dxi Ů dxj = − (dx1 Ů dx2 + dx2 Ů dx3 + dx3 Ů dx1)
and due to the constant coefficients we have d Ů q(A) = O. It is easy to find a 1-form q with d Ů q = q(A), e.g.
q = − (x1 dx2 + x2 dx3 + x3 dx1), hence (q)k+1 = − xk , if we assume (q)4 = (q)1.

In Euclidean spacetime, the converse of the Poincaré lemma of differential geometry implies that:

q(A)ij = d Ů qj                                                                 (2.11)

That is evidently wrong, since the coefficients qj are constant due to (2.4), hence d Ů (qjdxj) = O,
while qij(A) dxi Ů dxj due to (2.6) is not O. However, we have found a correct solution of that simple problem above. Maybe that MWE's error here has no further consequences.

i.e., that the two-form q(A)ij is the exterior derivative of a one-form qj . Equation (2.10) is the Bianchi identity applied in flat or Euclidean space. These results of three-dimensional Euclidean space may be generalized to four-dimensional Minkowski spacetime, in which the symmetric metric tensor is defined by

qρ = qρσ(S)qσ.                                                                 (2.12)

qρσ(S) see (2.14)

The antisymmetric metric tensor then follows from the spacetime generalization of Eq. (2.7):

q = −1/6 qμν(A)εμν = qρσ(S) εσ ερ,                                                                 (2.13)

an equation that shows that if qρσ(S) is defined from Eq. (2.12) as

1   0   0   0
0 −1   0   0
qρσ(S) =                                                                               (2.14)
0   0 −1   0
0   0   0 −1

then qμν(A) is defined from Eq. (2.13) as

0 −1 −1 −1
1   0 −1   1
qμν(A) =                                 .                                               (2.15)
1   1   0 −1
1 −1   1   0

In the language of differential geometry, if there exists the spacetime Minkowski line element, the zero-form

ω1 = ds2 = qρσ(S)dxρdxσ,                                                                 (2.16)

then the line element ds2 exists as the dual zero-form

ω2 = ds2 = − 1/6 qμν(A) dxμŮdxν,                                                                 (2.17)

The equation ω1 = ds2 = ω2 is wrong as was shown above (see discussion of (2.8-9)) by a proof independent of the dimension of the space.

We'll discuss MWE's duality argument here: Why should the 1-form dxρ be dual to the 2-form dxμŮdxμ? That is an inadmissible "generalization" from the 3-D case, where we have dx1 dual to dx2Ůdx3, dx2 dual to dx3Ůdx1, dx3 dual to dx1Ůdx2, due to the definition of the Hodge duality operator *. That is known as the fact that in 3-D a polar vector can be converted into a axial vector and conversely. But in 4-D case the dual of a 1-form is a 3-form, e.g. *dx0 = dx1Ůdx2Ůdx3 (and so on cyclically). And no 2-form does determine a 1-form by duality or conversely. A 1-form has 4 coefficients while a 2-form has 6 coefficients due to its antisymmetry. Thus,

### 1-forms and 2-forms of 4-D cannot mutually correspond to each other by duality.

The Bianchi identity (2.10) is also a Bianchi identity of four-dimensional Minkowski spacetime. The two-form (2.17) is a closed two-form, and so is the exterior derivative of a one-form by the converse of the Poincaré lemma.

Due to the constant coefficients of ω2 a 1-form with exterior derivative ω2 can be easily determined, see the remark to (2.10).

The symmetric and antisymmetric metrics qρσ(S) and qμν(A) are covariants of special relativity, i.e., retain their form under a Lorentz transform. This means that the symmetric metric is diagonal symmetric before and after a Lorentz transform, and the antisymmetric metric is antisymmetric off-diagonal before and after a Lorentz transform. The symmetric and antisymmetric metrics in Euclidean spacetime are constants, so the Christoffel symbols, Riemann tensors, Ricci tensors and scalar curvatures derived from these metrics all disappear. It is well known that in Einstein’s general relativistic theory of gravitation, flat spacetime means that gravitation is absent, a result that is implied by the Einstein field equation (1.29). In the received view of electrodynamics, however, the electromagnetic field is an entity superimposed on flat spacetime. In the new view of electrodynamics forged in this book and elsewhere [1-5], the electromagnetic field is a manifestation of spinning spacetime. If the gravitational field is described through the symmetric metric in Eq. (1.29), the Einstein field equation, then the electromagnetic field is described through the antisymmetric metric dual to the symmetric metric using a field equation with the same structure as Eq. (1.29).

Summary of Chapter 2.1 MWE has constructed an antisymmetric matrix (qik(A)) by using the 3-D duality. However this cannot be generalized to 4-D spacetime. Additionally the interpretation of the matrix (qik(A)) as "antisymmetric metric" has failed.

### 2.2. NON-MINKOWSKI SPACETIME. GENERAL RELATIVITY

By developing the results in Sec. (2.1) in non-Minkowski spacetime, it is possible to develop a theory of general relativity in which a symmetric metric is dual to an antisymmetric metric through Eq. (2.17).

The gravitational field is defined by the Einstein field equation (1.29) through the symmetric metric, and the electromagnetic field is dual to this gravitational field through Eq. (2.17) developed for non-Minkowski spacetime. The two fields are therefore related to each other through a duality transformation of differential geometry and are two parts of the same thing. This means that one field can influence another, i.e., gravitation can influence electromagnetism and vice-versa. In order to extend Eq. (2.9) to a non-Euclidean threedimensional space, we consider the unit vectors and metric vectors in general curvilinear coordinates and extend the analysis to non-Minkowski spacetime. The metric 4-vector in this spacetime is written as an antisymmetric tensor which is used to define a two-form of differential geometry. Gravitation is then defined by an Einstein equation for the symmetric metric and electromagnetism by an Einstein equation for the antisymmetric metric. The homogeneous field equation of electromagnetism is the Bianchi identity analogous to Eq. (2.10) for Euclidean spacetime, and the inhomogeneous field equation of electromagnetism is obtained by taking covariant derivatives on both sides of the Einstein equation for electromagnetism. When the non-Minkowski metric is defined by Eq. (1.61), we recover the homogeneous and inhomogeneous field equations of O(3) electrodynamics. In general curvilinear coordinates both the symmetric and antisymmetric metrics are defined in terms of the same set of scale factors, and this result can be used in principle to measure the effect of one field on the other. Electromagnetism in this view is a theory of non-Minkowski spacetime, and one must use covariant derivatives in the field equations of electromagnetism as well as the field equations of gravitation. The same conclusion has been reached independently by Sachs  using Clifford algebra in general relativity. O(3) electrodynamics is an example where the covariant derivatives are defined in the complex circular basis [1-5] denoted ((1),(2),(3)).

Consider a region of non-Euclidean, three-dimensional space  such that each point is specified by three numbers (u1, u2, u3), the curvilinear coordinates. The transformation equations between the Cartesian and curvilinear coordinates are:

x = x(u1, u2, u3), u1 = u1(x, y, z),
y = y(u1, u2, u3), u2 = u2(x, y, z),                                                       (2.18)
z = z(u1, u2, u3), u3 = u3(x, y, z),

Curvilinear coordinates should be displayed as contravariant as is use in received differential geometry, i.e. with upper indices. (see e.g. Carroll [3; p.12 f.])

where the functions are single valued and continuously differentiable. There is therefore a one-to-one correspondence between the point (x, y, z) and (u1, u2, u3). The position vector in general curvilinear coordinates is r(u1, u2, u3), and the arc length is the modulus of the infinitesimal displacement vector:

ds = |dr| = |r/∂u1du1 + r/∂u2du2 + r/∂u3du3|                                                                 (2.19)

Here r/∂ui is the metric coefficient, whose modulus is the scale factor:

hi = |r/∂ui|.                                                                 (2.20)

The unit vectors of the curvilinear coordinate system are:

ei = 1/hi r/∂ui,                                                                 (2.21)

(ei)= (e1, e2, e3),                                                                 (2.22)

and so we can write

dr = h1du1e1 + h2du2e2 + h3du3e3.                                                                 (2.23)

If we write the unit vector in three dimensions as

ei = r/∂ui, / |r/∂ui|,     hi = |r/∂ui|,                                                                 (2.24)

then each component is unity, as in the Cartesian coordinates (Eq. (1.2)):

i = r/∂x / |r/∂x|, etc.,     qx = |r/∂x|, etc.                                                                 (2.25)

The unit vectors ei are unit tangent vectors to the curve ui at point P, i.e., the three unit vectors e1, e2, e3 of the curvilinear coordinate system are unit tangent vectors to the coordinate curves, are mutually orthogonal, and form the O(3) symmetry cyclic relations:

e1 × e2 = e3,     e2 × e3 = e1,     e3 × e1 = e2.                                 (2.26)

This means that each of the unit vectors e1, e2, e3 is dual to an antisymmetric rank-two tensor in three-dimensional non-Euclidean space:

εij = −εijkεk,                                                                 (2.27)
εij = εijkεk.                                                                 (2.28)

The rank-two tensors εij also form an O(3) symmetry cyclic relation. The curvilinear coordinate system is orthogonal if

e1 · e2 = 0,     e1 · e3 = 0,     e2 · e3 = 0.                                 (2.29)

CAUTION! Generally, coordinates are not orthogonal. Therefore the relations (2.26-29) are only available, if orthogonal coordinates are assumed, which is a important restriction.

The symmetric metric tensor qij(S) is then defined by the square of the arc length:

ds2 = dr · dr = qij(S) duiduj .                                                                 (2.30)

This is a product of one-forms of differential geometry . The one form duk is dual to the wedge product duiŮduj , a two-form, and qij(S) implies qij(A), where qij(A) is the antisymmetric metric in curvilinear coordinates. Therefore, the square of the arc length implies the following area zero-form in differential geometry:

dA = − ˝ gij(A) dui Ů duj .                                                                 (2.31)

This result means that the existence of a symmetric metric in curvilinear coordinates in three dimensions implies the existence of an antisymmetric metric.

We have already seen by discussing the implications of eqns. (2.8-9) that no (nontrivial) antisymmetric metric can exist, neither in 3-D nor in 4-D spacetime.

Generalization of this result to four dimensions in any non-Minkowski spacetime gives

ω1 = ds2 = qμν(S) duμduν,                                                                 (2.32)
ω2 = dA = − ˝ qμν(A) duμŮ duν.                                                                 (2.33)

Coordinate differentials are contravariant in literature. For sake of compatibility it would be better to follow that use.

A graphical summary of the concepts used to derive the important result (2.33) is given below in three space dimensions:

The diagram was skipped here .

The general vector field in curvilinear coordinates in three space dimensions  is

F = F1e1 + F2e2 + F3e3,                                                                 (2.34)

and so we may define the metric coefficients in terms of the infinitesimal of the displacement vector in curvilinear coordinates:

dr = r/∂u1du1 + r/∂u2du2 + r/∂u3du3                                                                 (2.35)

The scale factors are the moduli of the metric coefficients and the unit vectors are the metric coefficients divided by the scale factors:

hi = |qi|,     ei = qi/hi.                                                                 (2.36)

Therefore the elements of the symmetric metric tensor are

qij(S) = qi · qj = hi hj ei · ej ,                                                                 (2.37)

and the elements of the antisymmetric tensor are

qij(A) = qi × qj = hi hj ei × ej.                                                                 (2.38)

These results have been obtained independently by Sachs , using Clifford algebra. The square of the arc length in curvilinear coordinates in three dimensions is therefore

ds2 = qi · qj duuduj                                                                 (2.39)

and can be expressed through the 3 × 3 symmetric metric tensor:

h12   h1h2   h1h3
(qij(S)) = (qi qj) =   h2h1   h22   h2h3 .                                                     (2.40)
h3h1   h3h2   h32

Here the off-diagonal elements are wrong: The scalar products ei · ej (see (2.39) are missing.

The off-diagonal elements of this tensor vanish if and only if the curvilinear coordinates system is orthogonal. In four spacetime dimensions, the metric tensor is the tensor product of two four-dimensional metric vectors whose components are the four scale factors:

(qμ) = (h0, h1, h2, h3),     (qν) = (h0, h1, h2, h3)                                                                 (2.41)

giving the following 4 × 4 symmetric metric tensor in spacetime in curvilinear coordinates:

h02   h0h1   h0h2   h0h3
h1h0   h12   h1h2   h1h3
(qμν(S)) =                                                                 .                                 (2.42)
h2h0   h2h1   h22   h2h3
h3h0   h3h1   h3h2   h32

This equation is completely wrong like (2.40) and therefore marked red: The scalar products of the corresponding basis vectors of the components (2.41) are missing. Besides, the matrix (qμν(S)) is positive semidefinite:
qμν(S) duμduν = hμhν duμduν = (hμduμ)2 > 0 .
However, the metric of spacetime is indefinit.

In addition the matrix (2.42) has vanishing determinant, because all of its lines are parallel to the vector [h0,h1,h2,h3]. That means that a metric matrix given by (2.42) would not be invertible as is necessary for a metric.

#### Thus, the matrix (2.42) cannot be the metric tensor of spacetime.

The gravitational field in general relativity is identified through the Einstein equation derived from this symmetric metric tensor in non-Minkowski spacetime. There is a one-to-one relation between any set of curvilinear coordinates (u1, u2, u3) and any other set (u'1, u'2, u'3) so the metric qμν(S) is generally covariant by definition, i.e., does not change its form under transformation from one set of curvilinear coordinates to another. In general relativity, this means that the equations of gravitation are generally covariant, the same in form under any transformation of coordinates. The principle of general relativity asserts that all equations of natural philosophy must be generally covariant, including the field equations of electromagnetism. We now follow this well known principle rigorously and derive the generally covariant field equations of electromagnetism from the antisymmetric metric in four-dimensional non-Minkowski spacetime. These equations of electromagnetism derive essentially from the fact that we can form the cross products (2.38) as well as the dot products (2.37), i.e., we can always construct an antisymmetric metric from the same metric vectors are those used to construct the symmetric metric. This result is true for any non-Minkowski spacetime. Restrict attention to three space dimensions, and note that the antisymmetric metric is built up from the following area elements of curvilinear coordinate analysis :

dA1 = h2h3du2du3 |e2 × e3|,
dA2 = h3h1du3du1 |e3 × e1|,                                                       (2.43)
dA3 = h1h2du1du2 |e1 × e2|.

The axial surface vector in three dimensions is therefore

dA = h2h3du2du3 e1 + h3h1du3du1 e2 + h1h2du1du2 e3,                                 (2.44)

That is wrong for non-orthogonal coordinates that are supposed here. The author erroneously assumes the validity of the eqns. (2.26).

and is dual to the antisymmetric surface tensor in three dimensions in curvilinear coordinates:

0   0   0                            0   0   1                          0  −1  0
dAij =   h2h3du2du3   0   0  −1   + h3h1du3du1   0   0   0   + h1h2du1du2   1   0   0 .                 (2.45)
0   1   0                          −1   0   0                          0   0   0

I. The upper indices ij are wrong since having no pendants on the right-hand side.
II. The reshaping on the right hand side from (2.44) to (2.45) requires orthogonal coordinates.

The antisymmetric metric tensor in three space dimensions is

0   −h1h2   h1h3
qij(A) =   h2h1    0    −h2h3 .                                                                 (2.46)
−h3h1   h3h2    0

That holds only in case of orthogonal coordinates.

In order to generalize this result to four-dimensional spacetime in curvilinear coordinates, note that the existence of the zero-form

ω1 = ds2 = qμν(S) duμduν                                                                 (2.47)

implies the existence of the area zero-form

ω2 = dA = − ˝ qμν(A) duμŮduν ,                                                                 (2.48)

That "area zero-form" is a 2-form in received mathematics. But we have seen already (see (2.8-9)) that the (antisymmetric) 2-form ω2 = qμν(S) duμduν cannot define a (symmetric) metric ds2.

where qμν(A) is the antisymmetric metric in four non-Minkowski spacetime dimensions. The structure of the antisymmetric metric is therefore the same as that of the wedge product duμŮ duν in four dimensions, i.e., the antisymmetric metric is an antisymmetric 4×4 matrix formed from the antisymmetric product of two metric 4-vectors

0       −h0h1   −h0h2   −h0h3
h1h0       0       −h1h2     h1h3
qμν(A) =                                                                         .                                 (2.49)
h2h0      h2h1       0       −h2h3
h3h0    −h3h1     h3h2       0

The basis vectors ek must appear here. We have curvilinear coordinates.

Summary of Chapter 2.2 Serious errors: Completely wrong matrix (2.42) of the symmetric metric, even in case of orthogonal coordinates. The construction of the antisymmetric matrix (2.46) is wrong for non orthogonal coordinates. The "generalization" to the 4-D case refers to a duality in 4-D that does not exist.
Due to the seriousness of the detected errors the continuation of this examination makes no sense. The author should correct his errors and try it again. I'm sure:

### References

        M.W. Evans: Duality and the Antisymmetric Metric, in
THE GEOMETRIZATION OF PHYSICS Appendix D, November 30, 2004
http://www.aias.us/book01/chap2-cr3.pdf

        M.W. Evans: GENERALLY COVARIANT UNIFIED FIELD THEORY in
http://www.aias.us/book01/GCUFT-book-2.pdf

        S. M. Carroll: Lecture Notes on General Relativity,
http://arxiv.org/pdf/gr-qc/9712019