Last update: 01.02.2005 16:00

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

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}*
=

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

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

*g _{μν}*

In eqn. (D.2) *η*_{ab}
is the metric of the tangent bundle. Eqn. (D.2) defines a symmetric
metric *g _{μν}*

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}
=
q_{0}^{a}
V^{0 }
+
q_{1}^{a}
V^{1 }
+
q_{2}^{a}
V^{2 }
+
q_{3}^{a}
V^{3}*

Similarly, eqn. (D.2) is:

*g _{μν}*

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}*

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}*

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

*g _{μν}^{c}*

** a**
To begin with the term

That is the

«

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)

The same

From all we must conclude:

** b**
What is

It is

Let’s recall: A

(1)

Then the matrix
(*q _{μ}*

So what is

For each

(i)
d*u ^{μ}*
Ù
d

(ii)
**B**Ù**C**
=
(*b _{μ}*d

From these rules one obtains

(2)
**B**Ù**C**
=
(*b _{μ}*

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

(3)
(**B**Ù**C**)* _{μν}*
=

Carroll’s introduction of the wedge product in [1; p.21-23] is somewhat
** misleading** since

**The answer**:
We can write

(4)
q^{a}Ùq^{b}
=
(*q _{μ}*

**Not** *q _{μ}*

d*u ^{0}*Ùd

The coefficients of q** ^{a}**Ùq

(5)
(q** ^{a}**Ùq

Consequently the 2-forms
q** ^{a}**Ùq

q

By that way we obtain six 2-forms
q** ^{0}**Ùq

**O**
q** ^{0}**Ùq

− q** ^{0}**Ùq

(6)

− q

− q** ^{0}**Ùq

Each entry here is a 2-form
q** ^{a}**Ùq

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

*g _{μν}^{ab}*
=

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

*g _{μν}*

which for each fixed pair of indices **a,b** is a (0,2) tensor.
These tensors can be represented by the matrix scheme
(q** ^{a}**Äq

q** ^{0}**Äq

q** ^{1}**Äq

(7) .

q

q** ^{3}**Äq

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

(8)
d*u ^{μ}*Ùd

it turns out that the matrix (6), i.e. (q

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}*
=

Correctly written:

q^{ab}
=
q** ^{a}**Äq

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

*g*^{c(A)}
=
*q ^{a}*
Ù

Correctly: The right-hand side is
q** ^{a}**Ùq

*g*^{(S)}
=
*q ^{a}q^{b}η_{ab}*
(D.11)

This notation shows clearly that
*g ^{ab}*
is a tensor;

I agree with the first statement: For fixed **a, b** the term
g^{ab}
=
q** ^{a}**Äq

However, as was shown above,

To discuss the term given by (D.11) let Ä

(9)
*η*** _{ab}**
q

Due to the symmetry of (*g _{μν}*) we have additionally

(10)
*η*_{ab}*
*q** ^{a}**Ä

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

*q ^{ab}*
=

Furthermore, *q ^{ab}*

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

*g*^{c(A)}
= ½ *Î ^{abc}
q^{ab}*

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

q^{01}, q^{02}, q^{03},
q^{12}, q^{13}, q^{23}.

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)

But we know the decomposition (D.12) already from (6),(7):

(q** ^{a}**Äq

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)}
=
*q*^{0}*q*^{0}*η*_{00}
+
*q*^{1}*q*^{1}*η*_{11}
+
*q*^{2}*q*^{2}*η*_{22}
+
*q*^{3}*q*^{3}*η*_{33}

=
*q*^{0}*q*^{0}
−
*q*^{1}*q*^{1}
−
*q*^{2}*q*^{2}
−
*q*^{3}*q*^{3}
(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}*
d

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

Trace q^{ab}*
*
=
*q _{μ}*

* ^{
}*
=
(

therefore (D.16) implies

*g _{μν}^{
}*
=

while (D.2) yields

*g _{μν}*
=

which is a *contradiction*. So

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:

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).

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

*ω*_{2}
=
− *q _{μν}*

That should read:
*ω*_{2}
=
− *q _{μν}*

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

*ω*_{2}
=
− ½ *q ^{μν}*

and by comparison with Einstein’s
eqn. (D.5), it is seen that *ω*_{2}
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

[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

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}*

and then replacing Ä with the wedge product Ù:

We obtain in our example:

(

and this yields

(

due to the symmetry of g