Last update: 01.02.2005 16:00

## The AntiSymmetric Metric

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

#### 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 = qμa Vμ                                                                 (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 Vμ.

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

gμν(S) = qμa qνb ηab                                                      (D.2)

In eqn. (D.2) ηab is the metric of the tangent bundle. Eqn. (D.2) defines a symmetric metric gμν(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:

Va = q0a V0 + q1a V1 + q2a V2 + q3a V3                            (D.3)

Similarly, eqn. (D.2) is:

gμν(S) = qμ0 qν0 η00 + . . . + qμ3 qν3 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):

gμν(S) gμν(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:

qμνc = qμa Ù qνb                                                        (D.6)

The wedge product is a generalization to any dimension of the vector cross product in 3-D. In eqn. (D.6) qμνc 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:

gμνc(A) = qμνc                                                                (D.7)

a To begin with the term qμνc: 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:
« dxρ is dual to the wedge product dxμ Ù dxν »,
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 qμa Ù qνb?
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 qaμ be the representation coefficients of the tetrad vectors e0, e1, e2, e3 relative to the “natural” base vectors êμ := ∂μ := /uμ (w.r.t. coordinates uμ):
(1)                                                                   ea = qaμ êμ .

Then the matrix (qμa) is the inverse of the vielbein-coefficient matrix (qaμ). Therefore the notation “tetrad” for the numbers qμa is rather misleading; at most the (inverse) matrix (qaμ) could be called so (better not!).
So what is qμa Ù qνb ?
For each fixed value of the index a the numbers qμa are the coefficients of the 1-form (covector) qa := qμa duμ. For 1-forms B = bμduμ, C = cμ duμ (but not for vectors) the wedge product Ù is defined by the rules:

(i)        duμ Ù duν = − duνÙduμ                                                (anti-commutativity)

(ii)       BÙC = (bμduμ)Ù(cνduν ) := (bμcν) duμÙduν                      (linearity).

From these rules one obtains

(2)                               BÙC = (bμ duμ)Ù(cν duμ) = ½ (bμcνbνcμ) duμÙ duν,

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

(3)                                           (BÙC)μν = bμa cνb bμb cνa.

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

(4)                         qaÙqb = (qμa duμ)Ù(qνb duμ) = ½ (qμa qνb qμb qνa) duμÙ duν.

Not qμaÙqνb is defined, but the wedge product qaÙqb of the 1-forms qa := qμa duμ and qb := qνb duμ. And (qaÙqb)μν is given by its coefficients relative to the 2-forms basis {duμÙ duν | μ , ν = 0, . . . , 4}, which due to antisymmetry de facto contains 6 elements:

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

The coefficients of qaÙqb relative to the basis forms duμÙduν above are:

(5)                                           (qaÙqb)μν   = qνa qμb qμb qνa .

Consequently the 2-forms qaÙqb depend antisymmetrically on the indices a,b:
qaÙqb = − qbÙqa and especially qaÙqa = 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 (qaÙqb):

O             q0Ùq1               q0Ùq2               q0Ùq3

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

− q0Ùq3       − q1Ùq3             − q2Ùq3                  O

Each entry here is a 2-form qaÙqb = ½ (bμa cνb bμb cνa) duμÙduν. That antisymmetric matrix (6) is what Evans calls "antisymmetric metric".

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

gμνab = qμa qνb                                                             (D.8)

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

gμνab duμÄduν = (qμa duμ)Ä(qνb duν) = qaÄqb = qμa qνb  duμÄ duν

which for each fixed pair of indices a,b is a (0,2) tensor. These tensors can be represented by the matrix scheme (qaÄqb), 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)                                 duμÙduν = ½ (duμÄduν − duνÄduμ),

it turns out that the matrix (6), i.e. (qaÙqb ), is the antisymmetric part of the matrix (7), of (qaÄqb ).

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:

qab = qa qb                                                                  (D.9)

Correctly written:

qab = qaÄqb , where qa := qa du μ, qb := qνb duν,

i.e. the matrix (qab) is given by (7).

gc(A) = qa Ù qb                                                             (D.10)

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

g(S) = qaqbηab                                                             (D.11)

This notation shows clearly that gab is a tensor; gc(A) is a vector; g(S) is a scalar.

I agree with the first statement: For fixed a, b the term gab = qaÄqb = qμaduμÄqνbduν is a tensor.
However, as was shown above, gc(A) is not defined due to duality error.
To discuss the term given by (D.11) let Äs be defined by BÄsC := ½ (BÄC + CÄB) for arbitrary 1-forms B,C. For g(S) we have to consider

(9)       ηab qaÄsqb = ηab (qμa duμ)Äs(qνb duν) = (ηab qμa qνb) duμÄsduν = gμν duμÄs duν.

Due to the symmetry of (gμν) we have additionally gμν duμÄsduν = gμν duμÄduν , hence

(10)                                          ηab qaÄsqb = gμν duμÄduν .

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

qab = qab(S) + qab(A)                                                   (D.12)

Furthermore, qab(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, gc(A) is the antisymmetric part of qab:

gc(A) = ½ Îabc qab(A)                                                   (D.13)

The antisymmetric part of (qab ) 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)

### qab(A) cannot be given by (D.13).

But we know the decomposition (D.12) already from (6),(7):
(qaÄqb ) = (qaÄsqb ) + (qaÙqb )

In eqn. (D.11):

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

thus:

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

= q0q0q1q1q2q2q3q3                                                   (D.15)

and so:

g(S) = Trace qab                                                      (D.16)

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

g(S) = gμν duμÄduν.

For Trace qab we obtain the sum of the main diagonal elements in (7):

Trace qab = qμ0 qν0 duμÄ duν + qμ1 qν du μÄ duν + qμ2 qν2 duμÄduν + qμ3 qν3 duμÄ duν

= (qμ0 qν0 + qμ1 qν1 + qμ2 qν2 + qμ3 qν3 )   duμÄduν ,

therefore (D.16) implies

gμν   = qμ0 qν0 + qμ1 qν1 + qμ2 qν2 + qμ3 qν3 ,

while (D.2) yields

gμν = qμa qνb ηab = qμ0 qν0 qμ1 qν1 qμ2 qν2 qμ3 qν3 ,

### (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.

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.

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

ω2 = − qμν(A) duμ Ù duν                                           (D.17)

That should read: ω2 = − qμν(A) duμ Ù duν with upper indices at the differentials du.

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

ω2 = − ½ qμν(A) qμν(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:

### 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 (ηab) We try to overcome Evans' difficulties with the construction of an "antisymmetric metric" by writing gμν(S) duμÄduν as product of two 1-forms:
gμν(S) duμÄduν = qμa qνb ηab duμÄduν = (qμa duμ) Ä (ηab qνb duν)
and then replacing Ä with the wedge product Ù:
We obtain in our example:
(qμa duμ) Ù (ηab qνb duμ)
and this yields
(qμa duμ) Ù (ηab qνb duν) = (ηab qμb qνb) duμ Ù duν = gμν(S) duμ Ù duν = O
due to the symmetry of gμν(S) and the antisymmetry of duμ Ù duν. Thus, the above decomposition did not yield a non trivial antisymmetric metric. Due to arithmetical reasons all other decompositions will fail as well.