The Einstein-Cartan theory of gravitation

In the present Chapter we treat some classical field theories in the space $\mathcal{S}$, which are symmetric with respect to the Lorentz group, do not satisfy the equity principle and do not contain a fundamental length $\ell$ (see Section 2.4). They allow an alternative formulation based on the spacetime $\mathcal{M}$, but some fields may acquire a geometric meaning that is not present in the spacetime formulation. A careful study of these theories is necessary for the search of new theories with a higher symmetry group, that is the object of Chapter 7.

We start with the Einstein-Cartan theory of gravitation [37,38], that generalizes Einstein's theory by allowing nonvanishing values of the torsion and an influence of the spin density on the geometry. It may be considered as a step towards a theory in which all the structure coefficients have a physical relevance. However, in this theory the torsion is not a really new dynamical variable, since it is given as an algebraic function of the spin density. The physical consequences of the Einstein-Cartan theory and of General Relativity cannot be distinguished by means of the presently available experimental techniques. In a first approach, we disregard the presence of electromagnetic fields and of other internal gauge fields, namely we consider a 10-dimensional space $\mathcal{S}$.

A Lagrangian form that describes the Einstein-Cartan theory in the space $\mathcal{S}$ has been proposed in ref. [3]. A simpler and more elegant Lagrangian has been proposed independently by Ne'eman and Regge in refs. [51,52]. In the following we consider a slight generalization of the Ne'eman-Regge Lagrangian form and we decompose it into a hard part $\lambda^H$, that depends linearly on the structure coefficients through the exterior derivatives $d \omega^{\alpha}$, and an additional part $\lambda^A$, which does not contain them and could equally well be considered as a part of the matter Lagrangian $\lambda^M$. For instance, one can include a cosmological constant $k_4$ in the potential (5.61) of the Higgs scalar field.

We put $\lambda^G = \lambda^H + \lambda^A$ with

$$\lambda^H = (k - k_1) \epsilon_{ikjl} \, d \omega^{[ik]} \wedge \omega^j \wedge \omega^l$$

$$+ 2 k_1 \epsilon_{ikjl} \, d \omega^{i} \wedge \omega^{[kj]} \wedge ... ...psilon_{ikjn} g_{lm} d \omega^{[ik]} \wedge \omega^{[jl]} \wedge \omega^{[mn]},$$ (5.1)

$$\lambda^A = k_3 \epsilon_{ikjl} \, g_{mn} \, \omega^{[im]} \wedge \omega^{[nk]} \wedge \omega^j \wedge \omega^l - k_4 \eta,$$ (5.2)

where $\eta$ is defined in Section 0.3, $k = k_3 = (32 \pi G)^{-1}$ and $G$ is Newton's gravitational constant. In Section 6.1 we need this formula with $k_3 \neq k$.

One can show that these equations give the most general Lorentz invariant 4-form depending on the 1-forms $\omega^{\alpha}$ and at most linearly on the 2-forms $d \omega^{\alpha}$, with the property that it is odd under space reflection, namely it contains the antisymmetric quantity $\epsilon_{ikjl}$. One can easily see that a 4-form with these properties depending only on the 1-forms $\omega^{[ik]}$ necessarily vanishes.

In refs. [51,52] the special case with $k_1 = k_2 = k_4 = 0$ was considered. The two terms proportional to $k_1$ and $k_2$ are exact forms, since they can be written as

$$- d \left(k_1 \epsilon_{ikjl} \, \omega^{[ik]} \wedge \omega... ...omega^{[ik]} \wedge \omega^{[jl]} \wedge \omega^{[mn]}\right).$$ (5.3)

They do not affect the field equations and the action integral, if $\delta \omega^{\alpha} = 0$ on the boundary $\partial S$. However, they influence the definition of the four-momentum and angular momentum of the geometric fields, which are known to be ambiguous in General Relativity [99,28]. They also play a role in the construction of the theories treated in Section 6.1 and in the Chapter 7. The term proportional to $k_4$ takes into account a cosmological constant, that has raised a considerable interest in the recent years [100,101].

The normal field equations are automatically satisfied and we have

$$\sigma_{[ik]} = 2 (k - k_1) \epsilon_{ikjl} \, \omega^j \wedge \omega^l - 2 k_2 \epsilon_{ikjn} g_{lm} \omega^{[jl]} \wedge \omega^{[mn]},$$

$$\sigma_i = 2k_1 \epsilon_{ikjl} \, \omega^{[kj]} \wedge \omega^l.$$ (5.4)

These quantities do not depend on the structure coefficients and we can write

$$\lambda^H = d \omega^{\alpha} \wedge \sigma_{\alpha}.$$ (5.5)

As a consequence, we obtain the following simplified formulas for the gravitational 10-momentum (4.57) and for the geometric tangential field equations (4.59):

$$\tau_{\alpha}^G = d \omega^{\beta} \wedge i_{\alpha} \sigma_{\beta} + i_{\alpha} \lambda^A,$$ (5.6)

$$d \sigma_{\alpha} + d \omega^{\beta} \wedge i_{\alpha} \sigma_{\beta} + i_{\alpha} \lambda^A = - \tau_{\alpha}^M.$$ (5.7)

Even in a flat spacetime, the forms $\tau_{\alpha}^G$ are not spatially localized, since they contain the forms $\omega^{[ik]}$. In a theory in which all the structure coefficients have a dynamical role, this flow of 10-momentum can be considered as the source of the structure constants of the Poincaré group. This is a relevant change of perspective, which can influence the construction of new modified classical theories of gravitation.

The tangential field equations have the explicit form

$$2 k \epsilon_{ikjl} F_{\alpha \beta}^j \omega^{\alpha} \wedge \omega^{\beta} \wedge \omega^l$$

$$- 2 k_3 (\epsilon_{injl} g_{km} - \epsilon_{knjl} g_{im}) \omega^{[mn]} \wedge \omega^j \wedge \omega^l = \tau^M_{[ik]},$$ (5.8)

$$k \epsilon_{ikjl} F_{\alpha \beta}^{[kj]} \omega^{\alpha} \wedge \omega^{\beta} \wedge \omega^l$$

$$- 2 k_3 \epsilon_{ijkl} g_{mn} \omega^{[jm]} \wedge \omega^{[nk]} \wedge \omega^l + k_4 \eta_i = \tau^M_i.$$ (5.9)

We call them the vertical and the horizontal tangential equations. As it was expected, the coefficients $k_1$ and $k_2$ have disappeared.

If the 10-momentum of matter has the spatially localized form (4.68), we obtain

$$F_{jk}^i + \delta^i_j F_{kl}^l - \delta^i_k F_{jl}^l = 8 \pi... ...uad F_{[jk]l}^i = \hat F_{[jk]l}^i, \qquad F_{[jk][mn]}^i = 0,$$ (5.10)

$$- F_{jk}^{[ik]} + 2^{-1} \delta^i_j F_{lk}^{[lk]} = 8 \pi G (T^i_j - k_4 \delta^i_j),$$

$$F_{[jl]m}^{[ik]} = 0, \qquad F_{[jl][mn]}^{[ik]} = \hat F_{[jl][mn]}^{[ik]},$$ (5.11)

namely the field equations of the Einstein-Cartan theory, together with the properties of the structure coefficients that are expected when $\mathcal{S}$ is the bundle of the Lorentz frames of a spacetime manifold. We stress that all these properties are consequences of the action principle.

Besides these equations, we have to take into account two Bianchi identities, which are special cases of eq. (1.55). Also the tensor transformation properties of curvature and torsion follow from eq. (1.55), as we have already observed in Section 1.7.