Last update: 01.02.2005 16:00

See the Appendix attached on 01.02 2005.

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Comments on M.W. Evans’ preprint

The AntiSymmetric Metric

(Appendix D in http://www.aias.us/book01/GCUFT-book-2.pdf )

Gerhard W. Bruhn, Darmstadt University of Technology

Abstract

Evans has some difficulties when trying to apply (Hodge-)duality for his definition of his so-called "antisymmetric metric", because in 4-D the dual of a 2-form is no longer a 1-form as in the 3-D case. But this difficulties could be eliminated. What remains is a fatal error: It turns out that his definition (D.6) of the "antisymmetric metric" contradicts the basic usual definition (D.2) of the symmetric metric.

MWE's original text [1] appears in black with intermediate comments in blue.

It is well known in differential geometry that the tetrad is defined by:

V ^a = q_μ^a V^μ (D.1)

Here V ^a is a four-vector defined in the Minkowski spacetime of the tangent bundle at point P to the base manifold. The latter is the general 4-D spacetime in which the vector is defined by V^μ.

The metric tensor used by Einstein in his field theory of gravitation (1915) is (Carroll):

g_μν^(S) = q_μ^a q_ν^b η_ab (D.2)

In eqn. (D.2) η_ab is the metric of the tangent bundle. Eqn. (D.2) defines a symmetric metric g_μν^(S), through an inner or dot product of two tetrads.

It is seen in eqn. (D.1) that there is summation over repeated indices. This is the Einstein convention. One index μ is a subscript (covariant) on the right hand side of eqn. (D.1).

Thus, written out in full eqn. (D.1) is:

V^a = q₀^a V⁰ + q₁^a V¹ + q₂^a V² + q₃^a V³ (D.3)

Similarly, eqn. (D.2) is:

g_μν^(S) = q_μ⁰ q_ν⁰ η₀₀ + . . . + q_μ³ q_ν³ η₃₃ (D.4)

In eqn. (D.4) it is seen that all possible combinations of a,b are summed.

Another example is given by Einstein in his famous book “The Meaning of Relativity” (Princeton, 1921-1954):

g_μν^(S) g^μν(S) = 4 (D.5)

It is seen that the double summation over μ and ν in eqn. (D.5) produces a scalar (the number 4). In differential geometry a scalar is a zero-form.

It is seen from the basic and well known definition (D.2) that is possible to define the wedge product of two tetrads:

q_μν^c = q_μ^a Ù q_ν^b (D.6)

The wedge product is a generalization to any dimension of the vector cross product in 3-D. In eqn. (D.6) q_μν^c is a two-form of differential geometry, i.e. a tensor antisymmetric in μ and ν. It is a vector-valued two-form due to the presence of the index c. This is the antisymmetric metric:

g_μν^c(A) = q_μν^c (D.7)

a To begin with the term q_μν^c: From (D.2) we see that all indices run over 0, 1, 2, 3, therefore we have a,b = 0,...,3. Which value has c? c = c(a,b)???
That is the duality error, an error, which appears also in [2], where MWE “generalizes” the 3-D case to the 4-D case: On p.23 he writes:
« dx_ρ is dual to the wedge product dx_μ Ù dx_ν »,
which means (tacitly) that the index ρ is different from μ and ν (≠μ). In the 3-D case ρ is uniquely determined by μ and ν (≠μ). However, this cannot be generalized to the 4-D case, since then for given μ and ν (≠μ) two possible indices are left, and the index ρ is no longer uniquely determined.

The same duality error appears in [2; p.27 and p.30], where again the 3-D case shall be generalized.
From all we must conclude:

The value of c is not defined.

b What is q_μ^a Ù q_ν^b?
It is not a wedge product of two (different?) tetrads:
Let’s recall: A tetrad is a quadruple of four (tetra) vector fields e₀, e₁, e₂, e₃ on the manifold, which may be subjected to certain additional conditions, e.g. the mutually orthonormality. It was introduced into Differential Geometry as “repère mobile” (moving n-hedral of a curve, vielbein) by Darboux and generalized by Elie Cartan (~1920). Let q_a^μ be the representation coefficients of the tetrad vectors e₀, e₁, e₂, e₃ relative to the “natural” base vectors ê_μ := ∂_μ := ^∂/_∂u^μ (w.r.t. coordinates u^μ):
(1) e_a = q_a^μ ê_μ .

Then the matrix (q_μ^a) is the inverse of the vielbein-coefficient matrix (q_a^μ). Therefore the notation “tetrad” for the numbers q_μ^a is rather misleading; at most the (inverse) matrix (q_a^μ) could be called so (better not!).
So what is q_μ^a Ù q_ν^b ?
For each fixed value of the index a the numbers q_μ^a are the coefficients of the 1-form (covector) q^a := q_μ^a du^μ. For 1-forms B = b_μdu^μ, C = c_μ du^μ (but not for vectors) the wedge product Ù is defined by the rules:

(i) du^μ Ù du^ν = − du^νÙdu^μ (anti-commutativity)

(ii) BÙC = (b_μdu^μ)Ù(c_νdu^ν ) := (b_μc_ν) du^μÙdu^ν (linearity).

From these rules one obtains

(2) BÙC = (b_μ du^μ)Ù(c_ν du^μ) = ½ (b_μc_ν − b_νc_μ) du^μÙ du^ν,

which can be written as (cf. [1; p.22 (1.80)])

(3) (BÙC)_μν = b_μ^a c_ν^b − b_μ^b c_ν^a.

Carroll’s introduction of the wedge product in [1; p.21-23] is somewhat misleading since not referring to the basis {du^μ | μ = 0, . . . , 4} of the 1-forms.

The answer: We can write
(4) q^aÙq^b = (q_μ^a du^μ)Ù(q_ν^b du^μ) = ½ (q_μ^a q_ν^b − q_μ^b q_ν^a) du^μÙ du^ν.

Not q_μ^aÙq_ν^b is defined, but the wedge product q^aÙq^b of the 1-forms q^a := q_μ^a du^μ and q^b := q_ν^b du^μ. And (q^aÙq^b)_μν is given by its coefficients relative to the 2-forms basis {du^μÙ du^ν | μ , ν = 0, . . . , 4}, which due to antisymmetry de facto contains 6 elements:

du⁰Ùdu¹, du⁰Ùdu² du⁰Ùdu³, du¹Ùdu ², du¹Ùdu ³, du²Ùdu³.

The coefficients of q^aÙq^b relative to the basis forms du^μÙdu^ν above are:

(5) (q^aÙq^b)_μν = q_ν^a q_μ^b − q_μ^b q_ν^a .

Consequently the 2-forms q^aÙq^b depend antisymmetrically on the indices a,b:
                q^aÙq^b = − q^bÙq^a and especially q^aÙq^a = O, where O is the 2-Null-form.

By that way we obtain six 2-forms q⁰Ùq¹, q¹Ùq², q²Ùq³, q³Ùq⁰, q⁰Ùq², q¹Ùq³. These six 2-forms can be arranged in an antisymmetric matrix scheme (q^aÙq^b):

O q⁰Ùq¹ q⁰Ùq² q⁰Ùq³

− q⁰Ùq¹ O q¹Ùq² q¹Ùq³
(6)
− q⁰Ùq² − q¹Ùq² O q²Ùq³

− q⁰Ùq³ − q¹Ùq³ − q²Ùq³ O

Each entry here is a 2-form q^aÙq^b = ½ (b_μ^a c_ν^b − b_μ^b c_ν^a) du^μÙdu^ν. That antisymmetric matrix (6) is what Evans calls "antisymmetric metric".

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The antisymmetric metric is part of the more general tensor metric formed by the outer product of two tetrads:

g_μν^ab = q_μ^a q_ν^b (D.8)

Again the same shortcoming: To be correct we have to supplement the basis forms:

g_μν^ab du^μÄdu^ν = (q_μ^a du^μ)Ä(q_ν^b du^ν) = q^aÄq^b = q_μ^a q_ν^b du^μÄ du^ν

which for each fixed pair of indices a,b is a (0,2) tensor. These tensors can be represented by the matrix scheme (q^aÄq^b), which due to the non-commutativity of the tensor product is not symmetric.

q⁰Äq⁰ q⁰Äq¹ q⁰Äq² q⁰Äq³

q¹Äq⁰ q¹Äq¹ q¹Äq² q¹Äq³
(7) .
q²Äq⁰ q²Äq¹ q²Äq² q²Äq³

q³Äq⁰ q³Äq¹ q³Äq² q³Äq³

Indeed, if we express the wedge product by the tensor product by the rule

(8)                                 du^μÙdu^ν = ½ (du^μÄdu^ν − du^νÄdu^μ),

it turns out that the matrix (6), i.e. (q^aÙq^b ), is the antisymmetric part of the matrix (7), of (q^aÄq^b ).

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It is seen that the indices μ and ν are always the same on both sides, so can be left out for clarity of presentation (see Carroll). Thus we obtain:

q^ab = q^a q^b (D.9)

Correctly written:

q^ab = q^aÄq^b , where q^a := q^a du ^μ, q^b := q_ν^b du^ν,

i.e. the matrix (q^ab) is given by (7).

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g^c(A) = q^a Ù q^b (D.10)

Correctly: The right-hand side is q^aÙq^b, while the left-hand side is not defined as we have seen above. But we recall here that the matrix (q^aÙq^b) is the antisymmetric part of the matrix (q^aÄq^b). So we can use (q^aÙq^b) instead of the undefined side of (D.10).

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g^(S) = q^aq^bη_ab (D.11)

This notation shows clearly that g^ab is a tensor; g^c(A) is a vector; g^(S) is a scalar.

I agree with the first statement: For fixed a, b the term g^ab = q^aÄq^b = q_μ^adu^μÄq_ν^bdu^ν is a tensor.
However, as was shown above, g^c(A) is not defined due to duality error.
To discuss the term given by (D.11) let Ä_s be defined by BÄ_sC := ½ (BÄC + CÄB) for arbitrary 1-forms B,C. For g^(S) we have to consider

(9) η_ab q^aÄ_sq^b = η_ab (q_μ^a du^μ)Ä_s(q_ν^b du^ν) = (η_ab q_μ^a q_ν^b) du^μÄ_sdu^ν = g_μν du^μÄ_s du^ν.

Due to the symmetry of (g_μν) we have additionally g_μν du^μÄ_sdu^ν = g_μν du^μÄdu^ν , hence

(10) η_ab q^aÄ_sq^b = g_μν du^μÄdu^ν .

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It is well known that any tensor is the sum of a symmetric and antisymmetric component:

q^ab = q^ab(S) + q^ab(A) (D.12)

Furthermore, q^ab(S) is the sum of an off-diagonal symmetric tensor and a diagonal tensor. The sum of the elements of the diagonal tensor is known as the trace.

Thus, g^c(A) is the antisymmetric part of q^ab:

g^c(A) = ½ Î^abc q^ab(A) (D.13)

The antisymmetric part of (q^ab ) contains 6 essential entries:

                                                                q⁰¹, q⁰², q⁰³, q¹², q¹³, q²³.

However, (D.13) defines a quantity that has only 4 entries (c = 0, 1, 2, 3). Since 6≠4 (at least in received mathematics) we see that (whatever c(a,b) may be)

q^ab(A) cannot be given by (D.13).

But we know the decomposition (D.12) already from (6),(7):
                                                                (q^aÄq^b ) = (q^aÄ_sq^b ) + (q^aÙq^b )

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In eqn. (D.11):

1 0 0 0                                                                
0 −1 0 0                                                                
η_ab = (D.14)
0 0 −1 0                                                                
0 0 0 −1                                                                

thus:

g^(S) = q⁰q⁰η₀₀ + q¹q¹η₁₁ + q²q²η₂₂ + q³q³η₃₃

= q⁰q⁰ − q¹q¹ − q²q² − q³q³ (D.15)

and so:

g^(S) = Trace q^ab (D.16)

As we have seen above in (9),(10) we have

g^(S) = g_μν du^μÄdu^ν.

For Trace q^ab we obtain the sum of the main diagonal elements in (7):

Trace q^ab = q_μ⁰ q_ν⁰ du^μÄ du^ν + q_μ¹ q_ν¹  du ^μÄ du^ν + q_μ² q_ν² du^μÄdu^ν + q_μ³ q_ν³ du^μÄ du^ν

= (q_μ⁰ q_ν⁰ + q_μ¹ q_ν¹ + q_μ² q_ν² + q_μ³ q_ν³ ) du^μÄdu^ν ,

therefore (D.16) implies

g_μν = q_μ⁰ q_ν⁰ + q_μ¹ q_ν¹ + q_μ² q_ν² + q_μ³ q_ν³ ,

while (D.2) yields

g_μν = q_μ^a q_ν^b η_ab = q_μ⁰ q_ν⁰ − q_μ¹ q_ν¹ − q_μ² q_ν² − q_μ³ q_ν³ ,

which is a contradiction. So

(D.16) must be wrong.

The deeper reason for that contradiction is that (D.6) and what Evans derives from it fails to be invariant under local Lorentz transforms while (D.2) has this property [3; p.90].

We can trace that error back to (D.8) where the "antisymmetric metric" is introduced as a claim:

The introduction of the "antisymmetric metric" by (D.8) yields a contradiction.

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From eqn. (D.9), (D.13) and (D.16) it is seen that the existence of the antisymmetric metric is implied by the existence of the symmetric metric.

As we have seen the error was introduced with (D.8) which turned out to be incompatible with the basic metric equation (D.2).

So nothing is implied by the other equations.

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In the notation of eqn. (2.33) of Evans, Chapter 2:

ω₂ = − q_μν^(A) du_μ Ù du_ν (D.17)

That should read: ω₂ = − q_μν^(A) du^μ Ù du^ν with upper indices at the differentials du.

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From the definition of the wedge product, eqn. (D.6), eqn. (D.17) is:

ω₂ = − ½ q^μν(A) q_μν^(A) (D.18)

and by comparison with Einstein’s eqn. (D.5), it is seen that ω₂ is a scalar.

(D.18) yields a scalar, a number, that’s true. But (D.17) is a 2-form. And a 2-form in 4-D has 3+2+1 = 6 independent coefficients corresponding to the six basis 2-forms. Therefore it cannot agree with one number; it’s always the same:

Apples are no pears!

References

[1] 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

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

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

Appendix

Evans' problem with the contradiction (D.2)./.(D.16) is caused by the fact that the definition (D.8) doesn't contain the Lorentz metric (η_ab) We try to overcome Evans' difficulties with the construction of an "antisymmetric metric" by writing g_μν^(S) du^μÄdu^ν as product of two 1-forms:
g_μν^(S) du^μÄdu^ν = q_μa q_νb η_ab du^μÄdu^ν = (q_μa du^μ) Ä (η_ab q_νb du^ν)
and then replacing Ä with the wedge product Ù:
We obtain in our example:
(q_μa du^μ) Ù (η_ab q_νb du^μ)
and this yields
(q_μa du^μ) Ù (η_ab q_νb du^ν) = (η_ab q_μb q_νb) du^μ Ù du^ν = g_μν^(S) du^μ Ù du^ν = O
due to the symmetry of g_μν^(S) and the antisymmetry of du^μ Ù du^ν. Thus, the above decomposition did not yield a non trivial antisymmetric metric. Due to arithmetical reasons all other decompositions will fail as well.