Conserved forms

A scalar conserved quantity, for instance the electric charge, is described in the spacetime formalism by a four-vector field $J^{\mu}(x)$, called a current. The charge contained in a space region $\Sigma$ belonging to a spacelike surface $x^0 = t$ is given by

$$q = \int_{\Sigma} J^0(t, \mathbf{x}) (-\det g)^{1/2} \, d^3 \mathbf{x} = \int_{\Sigma} \hat \tau,$$ (4.1)

where we have introduced the differential 3-form

$$\hat \tau = 6^{-1} J^{\lambda}(x) (-\det g)^{1/2} \epsilon_{... ...\nu \sigma} \, d x^{\mu} \wedge d x^{\nu} \wedge d x^{\sigma}.$$ (4.2)

The conservation law

$$\frac{\partial}{\partial x^{\lambda}} \left((-\det g)^{1/2} J^{\lambda} \right) = 0$$ (4.3)

can be written in the simple form $d \hat \tau = 0$.

We consider the pull-back of $\hat \tau$ on $\mathcal{S}$, given by

$$\tau_{\bullet} = \pi_* \hat \tau = J^i(s) \, \eta_i = T_{\bullet}^i \, \eta_i,$$ (4.4)

where the forms $\eta_i$ are defined in Section 0.3. The quantities $J^i$, that we also indicate by $T_{\bullet}^i$, are the anholonomic components of the current and we assume that the determinant of the matrix $e^i_{\lambda}$ is positive. We have used the formula

$$\pi_* d x^{\mu} = e^{\mu}_k \omega^k.$$ (4.5)

The conservation law takes the form

$$d \tau_{\bullet} = 0$$ (4.6)

and the charge is given by an expression of the kind

$$q = \int_{\Sigma} \tau_{\bullet},$$ (4.7)

where $\Sigma$ is a three-dimensional submanifold of $\mathcal{S}$ which has a spacelike projection on $\mathcal{M}$. Note that, in order to obtain the correct sign, one has to choose the orientation of the submanifold $\Sigma$ in such a way that the 3-form $\omega^1 \wedge \omega^2 \wedge \omega^3$ defines a positive density.

If $\tau_{\bullet}$ has the form (4.4), we say that it describes a spatially localized quantity, but $\tau_{\bullet}$ may have a more general form, depending on all the one-forms $\omega^{\alpha}$. In this way, as we shall see in Section 5.1, one can describe the energy-momentum the gravitational field, that is known to be ``nonlocalized'' in the sense that its spatial density depends on the choice of the coordinates or of a tetrad field [28,31,32]. The description of conserved quantities by means of 3-forms is valid also when $\mathcal{S}$ is not a fibre bundle and a spacetime manifold cannot be defined. In this case, the choice of the submanifold $\Sigma$ and of its orientation becomes a delicate problem that we discuss in Capter 7.

In Maxwell's theory the current is given by the formula

$$(-\det g)^{1/2} J^{\mu} = \frac{\partial}{\partial x^{\nu}} \left( (-\det g)^{1/2} F^{\nu \mu} \right).$$ (4.8)

There is no agreement between various textbooks on the sign in this formula, namely on the definition of the electromagnetic field tensor $F_{\mu \nu}$. We indicate by $F_{ik}$ the anholonomic components of the electromagnetic field and we identify them with the structure coefficients $F^{\bullet}_{ik}$. With our conventions, the connection with the electric and magnetic 3-dimensional vectors is

$$\mathbf{E} = (F_{01}, F_{02}, F_{03}), \qquad \mathbf{B} = (F_{32}, F_{13}, F_{21}).$$ (4.9)

It follows that

$$\hat \tau = 6^{-1} \frac{\partial}{\partial x^{\tau}} \left... ...{\mu} \wedge d x^{\nu} \wedge d x^{\sigma} = - d \hat \sigma,$$ (4.10)

where

$$\hat \sigma = 2^{-2} (-\det g)^{1/2} F^{\lambda \tau} \epsilon_{\lambda \tau \mu \nu} d x^{\mu} \wedge d x^{\nu}.$$ (4.11)

The 2-form $- \hat \sigma$ is called sometimes the Maxwell 2-form [28] and it is related to the dual electromagnetic tensor.

In this case too, we consider the pull-back of $\hat \sigma$ on $\mathcal{S}$, given by

$$\sigma_{\bullet} = \pi_* \hat \sigma = 2^{-2} F^{\bullet}_{ik}(s) \epsilon^{ik}{}_{jl} \, \omega^j \wedge \omega^l$$ (4.12)

and we have

$$\tau_{\bullet} = - d \sigma_{\bullet}, \qquad q = \int_{\Sigma} \tau_{\bullet} = - \int_{\partial \Sigma} \sigma_{\bullet}.$$ (4.13)

This is a generalized form of the Gauss law, that gives the charge as a surface integral of the electric field. We shall see in the following that a similar generalized Gauss law holds for other conserved quantities, related to gauge symmetry properties.