**8. Field Equations and Conservation Laws**

We present here field equations for ECSK theory22. Thus, we will neither assume the possibility of a propagating torsion (and we will always keep non-identically vanishing Riemann curvature [14]) nor display a lagrangian for a totally independent torsion field; rather, we will only set the Palatini–Cartan lagrangian for gravity, as done in Reference [15], and a matter lagrangian as the source. This theory is known as Einstein–Cartan–Sciama–Kibble gravity (ECSK).

In the present case, torsion reduces to an algebraic constraint. This is a consequence of making torsion join the action of the theory as only contained in the Ricci scalar because of a non-torsion-free connection and not with an independent coupling coefficient. In works like References [16–19], torsion is present as an independent part (independent coupling coefficient) of the action and it does propagate.

This is why the ECSK is considered as the most immediate generalization of General Relativity with the presence of torsion.

Therefore, we wish to eventually obtain an action of two independent objects, tetrads and connection, where this latter action should give rise to equations for curvature when varying tetrads and for torsion when varying the connection.

We will focus more on the geometrical side of these equations and we will not dwell on deepening matter interaction (couplings, symmetry breaking, etc.), as done for instance in References [20–24].

### *8.1. ECSK Equations*

ECSK theory with cosmological constant belongs to the Lovelock–Cartan family, which describes the most general action in four dimensions such that this action is a polynomial on the tetrads and the spin connection (including derivatives), is invariant under diffeomorphisms and local Lorentz transformations, and is constructed without the Hodge dual<sup>23</sup> .

Recalling that we will refer to *A* as *ω* and stressing that it must be only the antisymmetric part, the notation for *d<sup>A</sup>* becomes *dω*.

We will be dealing with a variational problem given by an action of the kind

$$\mathcal{S} = \mathcal{S}\_{P\mathcal{C}} + \mathcal{S}\_{matter} \tag{63}$$

where the Palatini–Cartan action is

$$S\_{\rm PC}[e,\omega] = \int\_M \text{Tr}\left[\frac{1}{2}e \wedge e \wedge F\_{\omega} + \frac{\Lambda}{4!}e^4\right].\tag{64}$$

The wedge product is defined over both space–time and internal indices as a map24<sup>∧</sup> : <sup>Ω</sup>*<sup>k</sup>* (*M*, <sup>Λ</sup>*p*V) <sup>×</sup> Ω*l* (*M*, <sup>Λ</sup>*q*V) <sup>→</sup> <sup>Ω</sup>*k*+*<sup>l</sup>* (*M*, <sup>Λ</sup>*p*+*q*V) and the trace is a map Tr : <sup>Λ</sup>4V → <sup>R</sup>, normalized such that (for *<sup>v</sup><sup>i</sup>* elements of a basis in V) Tr[*v<sup>i</sup>* ∧ *v<sup>j</sup>* ∧ *v<sup>k</sup>* ∧ *v<sup>l</sup>* ] = *εijkl*. The choice of the normalization of the trace works as a choice of orientation for *M* (since the determinant of a matrix in *O*(3, 1) may be ±1). Therefore, we reduce the total improper Lorentz group O(3, 1) to the only orientation preserving part, which is still not connected, SO(3, 1). This gives an invariant volume form on *M*. In this way, we consider sections of Λ*kT* <sup>∗</sup>*<sup>M</sup>* <sup>⊗</sup> <sup>Λ</sup>*p*V.

<sup>22</sup> Some classical works about ECSK theory and General Relativity with torsion, like References [11–13].

<sup>23</sup> See Reference [25] for details.

<sup>24</sup> Such that, for *<sup>α</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>k</sup>* (*M*, <sup>Λ</sup>*p*V) and *<sup>β</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>l</sup>* (*M*, <sup>Λ</sup>*q*V), we have *<sup>α</sup>* <sup>∧</sup> *<sup>β</sup>* = (−1) (*k*+*p*)(*l*+*q*)*<sup>β</sup>* <sup>∧</sup> *<sup>α</sup>*.

**Be aware:** Later on, we will make explicit some indices and keep implicit some others; for this purpose, we will specify what kind of wedge product we are dealing with, even though it will be evident because it will be among the implicit indices.

We recall the definition of *F<sup>ω</sup>* and deduce the identity for its variation

$$
\delta\_{\omega} F\_{\omega} = d\_{\omega} \delta \omega\_{\prime} \tag{65}
$$

where we stress that, despite *ω* being non-tensorial, *δω* is instead, and therefore, it transforms under the adjoint action (as *F<sup>ω</sup>* does) like in Equation (33). That holds because *δω* may be regarded as a difference of two spin connections.

The action for the matter is of the kind

$$S\_{matter}[e,\omega,\varphi] = \kappa \int\_{M} \text{Tr}[L(e,\omega,\varphi)],\tag{66}$$

where *L* is an invariant lagrangian density form with the proper derivative order in our variables, *ϕ* is a matter field, and *κ* is a constant.

Such matter lagrangian is supposed to be source for both curvature and torsion equations, namely it will be set for fulfilling some conditions fitting the theory.

Therefore, varying the actions in Equations (64) and (66) and considering Equation (65), we have<sup>25</sup>

$$\begin{aligned} \int\_M \text{Tr}[\delta e \wedge (e \wedge F\_{\omega} + \frac{\Lambda}{\mathfrak{J}!} e^3)] &= \int\_M \text{Tr}[\mathbf{x} \frac{\delta L}{\delta e} \wedge \delta e] \\ \int\_M \text{Tr}[\frac{1}{2} d\_{\omega} (e \wedge e) \wedge \delta \omega] &= \int\_M \text{Tr}[\mathbf{x} \frac{\delta L}{\delta \omega} \wedge \delta \omega]\_{\text{\textquotedblleft}} \end{aligned} \tag{67}$$

which is equivalent to

$$\begin{aligned} \varepsilon\_{abcd} e^b \wedge F\_{\omega}^{cd} + \frac{\Lambda}{\mathfrak{J}!} \varepsilon\_{abcd} e^b \wedge e^c \wedge e^d &= \mathfrak{k} \frac{\delta \text{Tr}[L]}{\delta e^b} := \mathfrak{k} T\_d\\ \frac{1}{2} \varepsilon\_{abcd} d\_{\omega} (e^c \wedge e^d) &= \mathfrak{k} \frac{\delta \text{Tr}[L]}{\delta \omega^{ab}} := \mathfrak{k} \Sigma\_{ab} \end{aligned} \tag{68}$$

where the wedge product here is only between differential forms.

Setting Λ = 0 and in performing the derivative, Equation (68) can be rewritten as

$$\begin{aligned} \varepsilon\_{abcd} \mathbf{e}^b \wedge F^{cd}\_{\omega} &= \kappa T\_a \\ \varepsilon\_{abcd} \tilde{\mathbf{Q}}^c \wedge \mathbf{e}^d &= \kappa \Sigma\_{ab\prime} \end{aligned} \tag{69}$$

where we have set *Q*˜ = *dωe*.

These are equations for the ECSK theory in their implicit form26, where *T* and Σ are related to, respectively, the *energy momentum* and the *spin* tensor, once pulled back.

By making all the indices explicit, as given in Reference [20], and properly setting *κ* according to natural units27, Equation (69) takes the following form

$$\begin{aligned} \mathcal{G}\_{\mu\nu} &= 8\pi T\_{\mu\nu} \\ \mathcal{Q}\_{\mu\nu}^{\ \sigma} &= -16\pi \Sigma\_{\mu\nu} \mathcal{I} \end{aligned} \tag{70}$$

<sup>25</sup> Omitting equations of motion *<sup>δ</sup><sup>L</sup> δϕ* = 0 for the matter field, which have to be satisfied for conservation laws anyway.

<sup>26</sup> Without making space–time indices explicit.

<sup>27</sup> All fundamental constants = 1.
