Extended on Jan 29 and on Feb 22, 2007
Let's start with some well-known basics of Cartan geometry:
The moving frame compatibility relation
(C) ∂_{μ} q_{ν}^{a} − Γ_{μ}^{λ}_{ν} q_{λ}^{a} + ω_{μ}^{a}_{b} q_{ν}^{b} = 0 | dx^{μ} Ù dx^{ν}
gives rise to introduce the 1-forms
q^{a} := q_{ν}^{a} dx^{ν} and ω^{a}_{b} := ω_{μ}^{a}_{b} dx^{μ}
[2, (J.36-37)], [4, p.239 (5.14)]. The forms q^{a} are vector-valued. However, note that the forms ω^{a}_{b} are not tensorial since under LLT's they transform inhomogeneously [1, (3.134)], [2, (J.15)], [4, p.230 (1.11)].
We have
d Ù q^{a} +
ω^{a}_{b} Ù q^{b}
=
q_{λ}^{a}
Γ_{μ}^{λ}_{ν}
dx^{μ} Ù dx^{ν}
=
2 q_{λ}^{a}
Γ_{[μ}^{λ}_{ν]}
dx^{μ} Ù dx^{ν}
=
q_{λ}^{a}
T_{μ}^{λ}_{ν}
dx^{μ} Ù dx^{ν}
=
T_{μ}^{a}_{ν}
dx^{μ} Ù dx^{ν}
=: T^{a} ,
thus, as a direct consequence of (C) we obtain [1, (3.137)], [2, (J.28)]
Application of dÙ to (C1) yields
d Ù T^{a}
= 0 +
d Ù
(ω^{a}_{c} Ù q^{c})
=
(d Ù
ω^{a}_{c}) Ù q^{c}
−
ω^{a}_{b} Ù (d Ù
q^{b})
=
(d Ù
ω^{a}_{c}) Ù q^{c}
−
ω^{a}_{b} Ù (T^{b}
− ω^{b}_{c} Ù q^{c})
=
(d Ù
ω^{a}_{c}
+ ω^{a}_{b} Ù
ω^{b}_{c}) Ù q^{c}
−
ω^{a}_{b} Ù T^{b}
or
d Ù T^{a} + ω^{a}_{b} Ù T^{b} = (d Ù ω^{a}_{c} + ω^{a}_{b} Ù ω^{b}_{c}) Ù q^{c} .
Introducing here [1, (3.138)], [2, (J.29)], [4, p.240 (6.10)]
we obtain [1, (3.140)], [2, (J.31)]
Applying dÙ to (C2) yields
d Ù R^{a}_{b} = 0 +
d Ù (ω^{a}_{c} Ù
ω^{c}_{b})
=
(d Ù ω^{a}_{c}) Ù
ω^{c}_{b}
−
ω^{a}_{c} Ù
(d Ù ω^{c}_{b})
= (R^{a}_{c}
−
ω^{a}_{d} Ù ω^{d}_{c})
Ù ω^{c}_{b}
−
ω^{a}_{c} Ù
(R^{c}_{b} −
ω^{c}_{d} Ù ω^{d}_{b}) ,
thus
We introduce the exterior covariant derivative D Ù of p-forms X^{a}, X_{b} and X^{a}_{b} respectively ([1, (3.136)], [2, (J.31)]) by
to obtain for the 2-form R^{a}_{b} [3, (19.24a)]
(C1) can be rewritten as
R^{a}_{b} := d Ù ω^{a}_{b} + ω^{a}_{c} Ù ω^{c}_{b} = d Ù ω^{a}_{b} + (ω^{a}_{c} Ù ω^{c}_{b} − ω^{c}_{b} Ù ω^{a}_{c}) + ω^{c}_{b} Ù ω^{a}_{c}
or
(D) applied to (C3) yields [3, (19.24a)]
Remark. Evans writes the 2nd Bianchi identity in the shorthand form (C4') which means that he used Eq.(D) in that case.
However, he also writes the 2nd Maurer-Cartan equation (C2) in shorthand notation [5, (7.2) + (7.4) + (9.60) + entries #28,#31 on p.187 + (16.4) + (17.16)* + (17.26) . . . ] e.g.
evidently now assuming another meaning of the "exterior covariant derivative".
The question whether D'Ù can be considered as covariant is anwered by Arkadiusz Jadczyk in [6].
The usual Poincaré Lemma
(PL) d Ù d = 0
is valid due to the commutativity of partial differential operators ∂_{μ}, ∂_{ν} . In [5, (9.68)] Evans extends this Lemma to exterior covariant derivatives as given by (D):
So we'll check this Evans assertion by calculating (D Ù D) Ù X^{a} := D Ù (D Ù X^{a}) using the definition (D) of exterior covariant derivatives:
D Ù (D Ù X^{a})
=
d Ù (D Ù X^{a})
+
ω^{a}_{c} Ù
(D Ù X^{c})
=
d Ù (d Ù X^{a}
+
ω^{a}_{b} Ù X^{b})
+
ω^{a}_{c} Ù
(d Ù X^{c}
+
ω^{c}_{b} Ù X^{b})
=
0 +
d Ù
(ω^{a}_{b} Ù X^{b})
+
ω^{a}_{c} Ù
(d Ù X^{c})
+
ω^{a}_{c} Ù
ω^{c}_{b} Ù X^{b}
=
(d Ù
ω^{a}_{b}
+
ω^{a}_{c} Ù ω^{c}_{b})
Ù X^{b}
hence, using (C2)
Evans idea is to interprete geometric properties of spacetime as the reasons of electrodynamics in a certain analogy to Einstein who interpreted curvature of spacetime as the source of gravitation. So what remains from geometry is the torsion tensor T of spacetime to be brought in connection with the electromagnetic field tensor G.
On first view there are striking similarities: Both tensors are antisymmetric, and both can be derived from potentials.
(3.1) Geometry: T = D Ù q and Electrodynamics: G = D Ù A
Following Evans' (bad) habit of suppressing indices it is suggesting to assume proportionality between the fields G and T and between the potentials A and q.
(3.2) G = A^{(o)} T and A = A^{(o)} q
where A^{(o)} is some constant. And this is just that what Evans does.
Now we shall restore Evans' suppressed indices (cf Sec 1) to obtain
(3.1')
Geometry: T^{a} = D Ù q^{a}
and
Electrodynamics: G = D Ù A
(1st Maurer-Cartan structure relation)
(3.2') G = A^{(o)} T^{a} and A = A^{(o)} q^{a} .
where a = (0),(1),(2),(3), i.e. a can attain four possible values due to geometric reasons, while due to physical reasons for the electrodynamic quantities (as every experimentalist will assure at least for the vacuum case) there exists only one field tensor G and one electromagnetic potential A. Since the transformation behavior of tensors is indicated by their indices it is very dangerous and misleading to suppress indices. So by restoring the hidden indices we have
Possibly, Evans knows about this problem too. Therefore he assumes that the electromagnetic potentials and the corresponding fields consist of three orthogonal components
(3.3) A^{a} and G^{a} , (a = (1),(2),(3))
declaring that the actual fields (in case of free spacetime at least) are given by the sum of the components
(3.4) A = A^{(1)} + A^{(2)} + A^{(3)} , G = G^{(1)} + G^{(2)} + G^{(3)} ,
with missing zeroth components.
Hence Evans assumes the existence of a four-covector C_{a} such that
(3.5) A = C_{a} A^{a} , G = C_{a} G^{a}
is the actual potential and the actual field respectively. Since all tetrads can be transformed mutually into each other by local Lorentz transforms (LLTs) they are all equivalent: Hence the "composition" covector C = (C_{a}) must be the same for all these tetrads. The Eqs.(3.4) yield
(3.6) (C_{a}) = (0,1,1,1) .
On the other hand, the covector C = (C_{a}) must transform contravariantly, which is obviously violated by Equ.(3.6). Thus, we have:
[1] S.M. Carroll, Lecture Notes on General Relativity, arXiv 1997
[2] S.M. Carroll, Spacetime and Geometry, Addison Wesley 2004
[3] E. Zeidler in Teubner Taschenbuch der Mathematik Teil II, 8.Aufl. 2003
[4] Y. Choquet-Bruhat, Géométrie différentielle et systèmes extérieurs, Dunod Paris 1968
[5] M.W. Evans, GENERALLY COVARIANT UNIFIED FIELD THEORY, Arima 2006
[6] A. Jadczyk, Remarks on Evans' "Covariant" Derivatives,
http://www.mathematik.tu-darmstadt.de/~bruhn/byArk260107.html