*7.2. Torsion in a Local Basis*

We would like to express the torsion form in terms of tetrads and the gauge field.

In Reference[9], a formula is given and it is obtained by applying the previous definition of the torsion form under the canonical isomorphism Ω*<sup>k</sup> G* (*P*, *V*) ∼= Ω*<sup>k</sup>* (*M*, V); therefore yielding

$$
\tilde{\Theta}^a = (d\_A e)^a = de^a + A^{ab} \eta\_{bc} \wedge e^c. \tag{55}
$$

*7.3. 1st Bianchi Identity*

**Proposition 4.** *Following our previous definitions, we have*

$$d\_{\omega}\Theta = \Omega \wedge\_f \theta \, \tag{56}$$

*which is called the first Bianchi identity.*

**Proof.** For this proof, we prefer using Equation (26).

We consider three vector fields *<sup>u</sup>*, *<sup>v</sup>*, *<sup>w</sup>* ∈ <sup>Γ</sup>(*TP*). By definition, it follows

<sup>20</sup> Torsion can be defined for every principal bundle, but physics arises when considering the frame bundle.

$$\begin{aligned} d^{\mathbb{d}} \Theta(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) &= d \Theta(\boldsymbol{h} \boldsymbol{u}, \boldsymbol{h} \boldsymbol{v}, \boldsymbol{h} \boldsymbol{w}) \\ &= (d \boldsymbol{\omega} \wedge\_f \theta - \boldsymbol{\omega} \wedge\_f d\theta)(\boldsymbol{h} \boldsymbol{u}, \boldsymbol{h} \boldsymbol{v}, \boldsymbol{h} \boldsymbol{w}) \quad \text{(because of Equation (54))} \\ &= d \boldsymbol{\omega} \wedge\_f \theta(\boldsymbol{h} \boldsymbol{u}, \boldsymbol{h} \boldsymbol{v}, \boldsymbol{h} \boldsymbol{w}) \quad \text{(because of Equation (19))} \\ &= \Omega \wedge\_f \theta(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}). \end{aligned} \tag{57}$$

The last equality holds because of tensoriality of *θ* and the second remark in Remark 2.

This proposition is a natural consequence of the property of the covariant differential expressed in Equation (28).

*7.4. Torsion Tensor*

**Definition 21** (Torsion tensor)**.** *Given two vector fields <sup>X</sup>*,*<sup>Y</sup>* <sup>∈</sup> <sup>Γ</sup>(*TM*) *and a* <sup>1</sup>*-form <sup>τ</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> (*M*)*, we define the torsion tensor field Q as the tensor field of type-*( 1 2 ) *such that*

$$Q(X,Y;\tau) := \tau(Q(X,Y)) = \tau\left(\bar{e}(d\_A e(X,Y))\right). \tag{58}$$

*It is evidently antisymmetric in X*,*Y, by definition.*

**Proposition 5.** *We have the following formula:*

$$Q(X,Y) = \nabla\_X Y - \nabla\_Y X - [X,Y] \tag{59}$$

*and, in components, it reads*

$$Q\_{\mu\nu}^{\quad\sigma} = \Gamma^{\sigma}\_{\mu\nu} - \Gamma^{\sigma}\_{\nu\mu} - \mathbb{C}^{\sigma}\_{\mu\nu\prime} \tag{60}$$

*where C<sup>σ</sup> µν* <sup>=</sup> <sup>0</sup> *in a holonomic basis for X and Y and* <sup>∇</sup> *is the covariant derivative*<sup>21</sup> *.*

**Proof.** Recalling the definition of torsion

$$Q = \overline{e} \cdot (d\_{\omega} e) = \overline{e}\_{a} (d\_{\omega} e)^{a},\tag{61}$$

it follows

$$\begin{split} \bar{e}\_{a}(d\_{\omega}e)^{a} &= \bar{e}\_{a}^{\sigma} (\partial\_{[\mu}e^{a}{}\_{\nu]} + \omega\_{[\mu b}^{a}e^{b}\_{\nu]}) d\mathbf{x}^{\mu} \wedge d\mathbf{x}^{\nu} \otimes \partial\_{\sigma} \\ &= (\Gamma\_{\mu \nu}^{\sigma} - \Gamma\_{\nu \mu}^{\sigma}) d\mathbf{x}^{\mu} \wedge d\mathbf{x}^{\nu} \otimes \partial\_{\sigma} \quad (\Gamma\_{\mu \nu}^{\sigma} = \bar{e}\_{a}^{\sigma} (D\_{\mu}e\_{\nu}^{a}).) \\ &= Q\_{\mu \nu}{}^{\sigma} d\mathbf{x}^{\mu} \wedge d\mathbf{x}^{\nu} \otimes \partial\_{\sigma}. \end{split} \tag{62}$$

then *Qµν <sup>σ</sup>* = Γ *σ µν* − <sup>Γ</sup> *σ νµ*.

We have now set up all the background for building our theory and for discussing field equations of ECSK theory.

<sup>21</sup> See Reference [10] for references about this.
