Minimal coupling and the balance equations

In many interesting cases, there is a minimal coupling between geometry and matter, namely a decomposition of the kind

$$\lambda = \lambda^G + \lambda^M,$$ (4.55)

where $\lambda^G$ describes the geometry, namely the gravitation and other gauge fields and does not contain $\Psi$ and its derivatives, while $\lambda^M$ describes the matter and does not contain the structure coefficients or the two-forms $d \omega^{\alpha}$. We shall see in Section 5.1 that there are terms that depend only on $\omega^{\alpha}$ and can be attributed arbitrarily to $\lambda^G$ or to $\lambda^M$.

The conserved quantities can be decomposed in a similar way:

$$\tau(B) = \tau^G(B) + \tau^M(B),$$ (4.56)

$$\tau^G(B) = L(B) \omega^{\alpha} \wedge \sigma_{\alpha} - i(B) \lambda^G,$$ (4.57)

$$\tau^M(B) = B \Psi^U \pi_U - i(B) \lambda^M$$ (4.58)

and the tangential field equation (4.35) can be written in the form

$$d \sigma_{\alpha} + F_{\alpha \beta}^{\eta} \, \omega^{\beta... ...edge \sigma_{\eta} + i_{\alpha} \lambda^G = - \tau^M_{\alpha},$$ (4.59)

$$\tau^M_{\alpha} = \tau^M(- A_{\alpha}) = - A_{\alpha} \Psi^U \pi_U + i_{\alpha} \lambda^M.$$ (4.60)

In general $\tau^M(B)$ is not conserved separately and its conservation law has to be replaced by a balance equation that takes into account the forces exerted by the geometric fields. By means of the same procedure used to derive eq. (4.32) and by taking the tangential field equations into account, we obtain

$$\delta \lambda^M = d (\delta \Psi^U \pi_U) + \delta \omega^{\alpha} \wedge \tau^M_{\alpha}.$$ (4.61)

By introducing the variations (4.42), we have

$$i(B) d \lambda^M + d i(B) \lambda^M = L(B) \lambda^M = d (B \Psi^U \pi_U) + L(B) \omega^{\alpha} \wedge \tau^M_{\alpha}$$ (4.62)

and finally the balance equation

$$d \tau^M(B) = - L(B) \omega^{\alpha} \wedge \tau^M_{\alpha} + i(B) d \lambda^M.$$ (4.63)

In many cases the last term vanishes. In particular, if $\lambda^M$ is a homogeneous function of degree $m \neq 0$ of $\Psi$ and its derivatives, we can put

$$\delta \omega^{\alpha} = 0, \qquad \delta \Psi^U = \zeta \P... ...A_{\alpha} \Psi^U, \qquad \delta \lambda^M = m \zeta \lambda^M$$ (4.64)

and from eq. (4.61) we obtain

$$m \lambda^M = d (\Psi^U \pi_U), \qquad d \lambda^M = 0.$$ (4.65)

If $\mathcal{S}$ is the bundle of frames of the Minkowski spacetime, and we consider the isomorphism $g \to \hat s g$ between $\mathcal{S}$ and the Poincaré group, the vector fields $A^L_{\alpha}$, defined by eq. (1.66), commute with all the fundamental vector fields $A_{\alpha}$ and we have $L(A_{\alpha}) \omega^{\beta} = 0$. It follows that the quantities

$$\tau^M(- A^L_{\alpha}) = D^{\beta}{}_{\alpha}(g^{-1}) \tau^M_{\beta}$$ (4.66)

are conserved. They describe the energy-momentum and the relativistic total angular momentum of matter with respect to the fixed frame $\hat s$. No symmetry property of the Lagrangian form is required for these conservation laws.

For $B = - A_{\alpha}$ we have

$$d \tau^M_{\alpha} = - F_{\alpha \beta}^{\gamma} \omega^{\beta} \wedge \tau^M_{\gamma}.$$ (4.67)

We shall use this formula in Chapter 8 in order to describe the motion of a test particle in external geometric fields. The differential 3-forms $\tau^M_{\alpha}$ describe the density and the flow of the (10+n)-momentum of matter (see Section 1.9). The balance equations show that the change of a component of the (10+n)-momentum is given by the product of some other components of (10+n)-momentum and some geometric fields. For instance, a change of momentum is given by the product of the electric charge and the electric field. All the relations of this kind are contained in a compact way in eq. (4.67).

We see that the structure constants of the Poincaré group play a role similar to the role played by the electromagnetic field. This remark gives another support to the idea that all the structure coefficient should have a dynamical nature.

If we consider spatially localized quantities of the kind (4.4), or, more in general

$$\tau^M_{\alpha} = T^k_{\alpha} \eta_k,$$ (4.68)

we assume that $\mathcal{S}$ has the structure of principal bundle as in Chapter 1, and we use the formula

$$d \eta_i = - F_{ik}^k \eta + 2^{-1} F_{[kl]i}^j \omega^{[kl]} \wedge \eta_j,$$ (4.69)

from eq. (4.67) we obtain the Lorentz and gauge transformation properties

$$A_{[ik]} T^j_{\alpha} = F_{[ik] \alpha}^{\beta} T^j_{\beta} ... ...a}, \qquad A_a T^j_{\alpha} = F_{a \alpha}^{\beta} T^j_{\beta}$$ (4.70)

and the balance equations

$$A_i T^i_{\alpha} - F_{ik}^k T^i_{\alpha} = F_{i \alpha}^{\beta} T^i_{\beta}.$$ (4.71)

In particular we have the equation

$$A_j T^j_{[ik]} - F_{jk}^k T^j_{[ik]} = g_{ij} T^j_k - g_{kj} T^j_i,$$ (4.72)

showing that, even in the absence of external fields, the spin density is not conserved, unless the energy momentum tensor $T_{ik}$ is symmetric.