Gerhard W. Bruhn, Darmstadt University of Technology

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

ε_ij = ε^ijk ε_k.                                                                 (2.2)

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

(q^k) = (ε_k) = (1, 1, 1).                                                                 (2.3)

The symmetric metric tensor in Euclidean space can be defined by

1 0 0              −1 0 0                                                                      
q^k = q^kr(S)q_r,         (q_kl(S)) = − 0 1 0         =   0 −1 0 .                                                             (2.4)
0 0 1              0 0 −1                                                                      

Define the scalar metric quantity q by

q = q^kq_k = q^kε_k = −3.                                                                 (2.5)

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

0 −1 1                                                                          
(q^ij(A)) = (−ε^ijkq_k) = 1 0 −1 ;                                                                 (2.6)
−1 1   0                                                                          

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

q = −½ q^ij(A)ε^ij = q^kr(S)ε_rε_k.                                                                 (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 ε_k is dual to ε^ij by geometry, then q^kr(S)ε_r must be dual to q^ij(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 = q^kr(S)dx_rdx_k,                                                                 (2.8)

ω₂ = ds² = − ½ q^ij(A) dx_i Ù dx_j ,                                                                 (2.9)

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

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

d Ù q^ij(A) = 0 = (d Ù q(A))_ij .                                                                 (2.10)

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

q^(A)_ij = d Ù q_j                                                                 (2.11)

i.e., that the two-form q^(A)_ij is the exterior derivative of a one-form q_j . 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

q^ρ = q^ρσ(S)q_σ.                                                                 (2.12)

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

q = −1/6 q^μν(A)ε_μν = q^ρσ(S) ε_σ ε_ρ,                                                                 (2.13)

an equation that shows that if q^ρσ(S) is defined from Eq. (2.12) as

1   0   0   0                                                                
0 −1   0   0                                                                
q^ρσ(S) =                                                                               (2.14)
0   0 −1   0                                                                
0   0   0 −1                                                                

then q^μν(A) is defined from Eq. (2.13) as

0 −1 −1 −1                                                                
1   0 −1   1                                                                
q^μν(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 = q^ρσ(S)dx_ρdx_σ,                                                                 (2.16)

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

ω₂ = ds² = − 1/6 q^μν(A) dx_μÙdx_ν,                                                                 (2.17)

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.

The symmetric and antisymmetric metrics q^ρσ(S) and q^μν(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).

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 [8] 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 [6] 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),                                                                

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:

h_i = |^∂r/_∂ui|.                                                                 (2.20)

The unit vectors of the curvilinear coordinate system are:

e_i = 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

e_i = ^∂r/_∂ui, / |^∂r/_∂ui|,     h_i = |^∂r/_∂ui|,                                                                 (2.24)

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

i = ^∂r/_∂x / |^∂r/_∂x|, etc.,     q_x = |^∂r/_∂x|, etc.                                                                 (2.25)

The unit vectors e_i are unit tangent vectors to the curve u_i 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:

ε^ij = −ε^ijkε^k,                                                                 (2.27)
ε_ij = ε^ijkε_k.                                                                 (2.28)

The rank-two tensors ε^ij 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)

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

ds² = dr · dr = q^ij(S) du_idu_j .                                                                 (2.30)

This is a product of one-forms of differential geometry [7]. The one form du_k is dual to the wedge product du_iÙdu_j , a two-form, and q^ij(S) implies q^ij(A), where q^ij(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 = − ½ g^ij(A) du_i Ù du_j .                                                                 (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.

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

ω₁ = ds² = q^μν(S) du_μdu_ν,                                                                 (2.32)
ω₂ = dA = − ½ q^μν(A) du_μÙ du_ν.                                                                 (2.33)

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

The general vector field in curvilinear coordinates in three space dimensions [6] 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:

h_i = |q_i|,     e_i = q_i/h_i.                                                                 (2.36)

Therefore the elements of the symmetric metric tensor are

q_ij^(S) = q_i · q_j = h_i h_j e_i · e_j ,                                                                 (2.37)

and the elements of the antisymmetric tensor are

q_ij^(A) = q_i × q_j = h_i h_j e_i × e_j.                                                                 (2.38)

These results have been obtained independently by Sachs [8], using Clifford algebra. The square of the arc length in curvilinear coordinates in three dimensions is therefore

ds² = q_i · q_j du_udu_j                                                                 (2.39)

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

h₁²   h₁h₂   h₁h₃                                                                
(q_ij^(S)) = (q_i q_j) =   h₂h₁   h₂²   h₂h₃ .                                                     (2.40)
h₃h₁   h₃h₂   h₃²                                                                

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:

(q^μ) = (h⁰, h¹, h², h³),     (q_ν) = (h₀, h₁, h₂, h₃)                                                                 (2.41)

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

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

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.

That holds only in case of orthogonal coordinates.

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 = q_μν(S) du_μdu_ν cannot define a (symmetric) metric ds².

The basis vectors e_k 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:

Quite another theory will be the result.