PROOF OF THE FIRST CARTAN EQUATION

Last update: 18.03.2005, 13:00 CET

PROOF OF THE FIRST CARTAN EQUATION
USING THE TETRAD POSTULATE

by Myron W. Evans

with a Commentary by G.W. Bruhn and W.A. Rodrigues Jr.

The first Cartan equation is

T^a = d Ù q^a + ω^a_b Ù q^b (1)

i.e.

T^a_μν = ∂_μq_ν^a − ∂_νq_μ^a + ω_μ^a_b q_ν^b − ω_ν^a_b q_μ^b (2)

Here T^a is the torsion two-form; q_μ^b is the tetrad one-form; ω_μ^a_b is the spin connection.

The torsion tensor is:

T^λ_μν = q_a^λ T^a_μν . (3)

Use the tetrad postulate in the form

Γ^λ_μν = q_a^λ (∂_μq_ν^a + ω_μ^a_b q_ν^b) (4)

>From eqns (2) - (4):

T^λ_μν = q_a^λ (∂_μq_ν^a + ω_μ^a_b q_ν^b) − q_a^λ (∂_νq_μ^a + ω_ν^a_b q_μ^b)

T^λ_μν = Γ^λ_μν − Γ^λ_νμ (5)

Equ.(5) is the torsion tensor of Riemann geometry. Given the tetrad postulate (4) it has been proven to be the Cartan equation (1). QED

Commentary

The proof should start with Equ.(5), with the definition of the torsion tensor

T^λ_μν = Γ^λ_μν − Γ^λ_νμ . (5)

Using the compatibility relation of frames cf. [2; (3.9-10)] (not the tetrad postulate as is remarked by Evans) in the form

q_λ^a Γ^λ_μν = ∂_μq_ν^a + q_ν^b ω_μ^a_b (4')

and the relation

T^a_μν = q_λ^a T^λ_μν . (3')

we obtain from (5)

T^a_μν = (∂_μq_ν^a + q_ν^b ω_μ^a_b) − (∂_μq_ν^a + q_ν^b ω_μ^a_b) . (2')

Remark 1
The expressions in brackets are the type (II) covariant derivatives of q_ν^a (cf. [1; (3.2)])

D^(II)_μq_ν^a = ∂_μq_ν^a + q_ν^b ω_μ^a_b

hence

T^a_μν = D^(II)_μ q_ν^a − D^(II)_ν q_μ^a . (2'')

This can be rewritten with some easy algebra in form of an exterior derivative:

T^a = T^a_μνdx^μÙdx^ν = D Ù q^a := dÙq^a + ω^a_b Ù q^b . (1')

Remark 2
In contrast to Evans' denotation T^a is not the torsion two-form, but represents for a = 0, 1, 2, 3 four two-forms T⁰, T¹, T² and T³ .

Remark 3
We impose that the covariant derivative of the tetrad 1-form appearing in Equ. (1') is of type (II). Hence it is shown that different covariant derivative types of the tetrad tensor q_ν^a can appear depending on the context.

Summary

Cartan's first structure equation, which yields the torsion two-form fields T^a (a = 0, 1, 2, 3) are such that in the calculation of their components in a coordinate basis, the type (II) covariant derivative D^(II) of the components of the tetrad 1-form fields q^a (a = 0, 1, 2, 3) must be used.

References

[1] G.W. Bruhn and W.A. Rodrigues Jr.: Covariant Derivatives of Tensor Components,
http://www2.mathematik.tu-darmstadt.de/~bruhn/deblocking_dot.htm

[2] G.W. Bruhn: Covariant Derivatives and the Tetrad Postulate,
http://www2.mathematik.tu-darmstadt.de/~bruhn/covar_deriv.htm