 Last update: 18.03.2005, 13:00 CET

## PROOF OF THE FIRST CARTAN EQUATION USING THE TETRAD POSTULATE

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

The first Cartan equation is

Ta = d Ů qa + ωab Ů qb                                                             (1)

i.e.

Taμν = ∂μqνa − ∂νqμa + ωμab qνbωνab qμb                                         (2)

Here Ta is the torsion two-form; qμb is the tetrad one-form; ωμab is the spin connection.

The torsion tensor is:

Tλμν = qaλ Taμν .                                                             (3)

Use the tetrad postulate in the form

Γλμν = qaλ (∂μqνa + ωμab qνb)                                                             (4)

>From eqns (2) - (4):

Tλμν = qaλ (∂μqνa + ωμab qνb) − qaλ (∂νqμa + ωνab qμb)

Tλμν = Γλμν − Γλνμ                                                             (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

Tλμν = Γλμν − Γλνμ .                                                             (5)

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

qλa Γλμν = ∂μqνa + qνb ωμab                                                   (4')

and the relation

Taμν = qλa Tλμν .                                                             (3')

we obtain from (5)

Taμν = (∂μqνa + qνb ωμab) − (∂μqνa + qνb ωμab) .                                         (2')

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

D(II)μqνa = ∂μqνa + qνb ωμab

hence

Taμν = D(II)μ qνaD(II)ν qμa .                                                   (2'')

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

Ta = TaμνdxμŮdxν = D Ů qa := dŮqa + ωab Ů qb .                                         (1')

Remark 2
In contrast to Evans' denotation Ta 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 qνa can appear depending on the context.

### References

    G.W. Bruhn and W.A. Rodrigues Jr.: Covariant Derivatives of Tensor Components,