The divergence and conserved quantities

Recall that the divergence of a vector field \({u}\) can be generalized to a pseudo-Riemannian manifold (sometimes called the covariant divergence) by defining \({\mathrm{div}(u)\equiv*\mathrm{d}(*(u^{\flat}))}\). Using the previously stated relations \({i_{u}\Omega=*(u^{\flat})}\) and \({A=(*A)\Omega}\) for \({A\in\Lambda^{n}M^{n}}\), we have \({\mathrm{d}(i_{u}\Omega)=\mathrm{d}(*(u^{\flat}))=*\mathrm{d}(*(u^{\flat}))\Omega=\mathrm{div}(u)\Omega}\). Using \({i_{u}\mathrm{d}+\mathrm{d}i_{u}=L_{u}}\) we then arrive at \({\mathrm{div}(u)\Omega=L_{u}\Omega}\), or as it is more commonly written

\(\displaystyle \mathrm{div}(u)\mathrm{d}V=L_{u}\mathrm{d}V. \)

Thus we can say that \({\mathrm{div}(u)}\) is “the fraction by which a unit volume changes when transported by the flow of \({u}\),” and if \({\mathrm{div}(u)=0}\) then we can say that “the flow of \({u}\) leaves volumes unchanged.” Expanding the volume element in coordinates \({x^{\lambda}}\) we can obtain an expression for the divergence in terms of these coordinates,

\(\displaystyle \mathrm{div}(u)=\frac{1}{\sqrt{\left|\mathrm{det}(g)\right|}}\partial_{\lambda}\left(u^{\lambda}\sqrt{\left|\mathrm{det}(g)\right|}\right). \)

Note that both this expression and \({\nabla_{a}u^{a}}\) are coordinate-independent and equal to \({\partial_{a}u^{a}}\) in Riemann normal coordinates, confirming our expectation that in general we have

\(\displaystyle \mathrm{div}(u)=\nabla_{a}u^{a}. \)

Using the relation \({\mathrm{div}(u)\Omega=\mathrm{d}(i_{u}\Omega)}\) above, along with Stokes’ theorem, we recover the classical divergence theorem

\(\displaystyle \begin{aligned}\int_{V}\mathrm{div}(u)\mathrm{d}V & =\int_{\partial V}i_{u}\mathrm{d}V\\ & =\int_{\partial V}\left\langle u,\hat{n}\right\rangle \mathrm{d}S, \end{aligned} \)

where \({V}\) is an \({n}\)-dimensional compact submanifold of \({M^{n}}\), \({\hat{n}}\) is the unit normal vector to \({\partial V}\), and \({\mathrm{d}S\equiv i_{\hat{n}}\mathrm{d}V}\) is the induced volume element (“surface element”) for \({\partial V}\). In the case of a Riemannian metric, this can be thought of as reflecting the intuitive fact that “the change in a volume due to the flow of \({u}\) is equal to the net flow across that volume’s boundary.” If \({\mathrm{div}(u)=0}\) then we can say that “the net flow of \({u}\) across the boundary of a volume is zero.” We can also consider an infinitesimal \({V}\), so that the divergence at a point measures “the net flow of \({u}\) across the boundary of an infinitesimal volume.” As usual, for a pseudo-Riemannian metric these geometric intuitions have less meaning.

The divergence can be extended to contravariant tensors \({T}\) by defining \({\mathrm{div}(T)\equiv\nabla_{a}T^{ab}}\), although other conventions are in use. Since \({\mathrm{div}(T)}\) is vector-valued and the parallel transport of vectors is path-dependent, we cannot in general integrate to get a divergence theorem for tensors. In the case of a flat metric however, we are able to integrate to get a divergence theorem for each component

\(\displaystyle \begin{aligned}\int_{V}\nabla_{a}T^{ab}\mathrm{d}V & =\int_{\partial V}T_{a}{}^{b}\hat{n}^{a}\mathrm{d}S.\end{aligned} \)

In physics, the vector field \({u}\) often represents the current vector (AKA current density, flux, flux density) \({j\equiv\rho u}\) of an actual physical flow, where \({\rho}\) is the density of the physical quantity \({Q}\) and \({u}\) is thus a velocity field; e.g. in \({\mathbb{R}^{3}}\), \({j}\) has units \({Q/(\mathrm{length})^{2}(\mathrm{time})}\). There are several quantities that can be defined around this concept:

Quantity Definition Meaning
Current vector \({j\equiv\rho u}\) The vector whose length is the amount of \({Q}\) per unit time crossing a unit area perpendicular to \({j}\)
Current form
\begin{aligned}\zeta & \equiv i_{j}\mathrm{d}V\\
& =\left\langle j,\hat{n}\right\rangle \mathrm{d}S
\end{aligned}
The \({(n-1)}\)-form which gives the amount of \({Q}\) per unit time crossing the area defined by the argument vectors
Current density
\begin{aligned}\mathfrak{j} & \equiv\sqrt{\left|\mathrm{det}(g)\right|}\, j\\
\Rightarrow\zeta & =\left\langle \mathfrak{j},\hat{n}\right\rangle \mathrm{d}x^{\lambda_{1}}\wedge\cdots\wedge\mathrm{d}x^{\lambda_{n-1}}
\end{aligned}
The vector whose length is the amount of \({Q}\) per unit time crossing a unit coordinate area perpendicular to \({j}\)
Current
\begin{aligned}I & \equiv\int_{S}\zeta\\
& =\int_{S}\left\langle j,\hat{n}\right\rangle \mathrm{d}S\\
& =\int_{S(x^{\lambda})}\left\langle \mathfrak{j},\hat{n}\right\rangle \mathrm{d}x^{\lambda_{1}}\cdots\mathrm{d}x^{\lambda_{n-1}}
\end{aligned}
The amount of \({Q}\) per unit time crossing \({S}\)
Current 4-vector \({J\equiv(\rho,j^{\mu})}\) Current vector on the spacetime manifold

Notes: \({\rho}\) is the density of the physical quantity \({Q}\), \({u}\) is a velocity field, \({\hat{n}}\) is the unit normal to a surface \({S}\), and \({x^{\lambda}}\) are coordinates on the submanifold \({S}\). The current 4-vector can be generalized to other Lorentzian manifolds, and can also be turned into a form or a density.

Δ Note that the terms flux and current (as well as flux density and current density) are not used consistently in the literature.

The current density \({\mathfrak{j}}\) is an example of a tensor density, which in general takes the form \({\mathfrak{T}\equiv\left(\sqrt{\left|\mathrm{det}(g)\right|}\right)^{W}T}\), where \({T}\) is a tensor and \({W}\) is called the weight. Note that tensor densities are not coordinate-independent quantities.

For a Riemannian metric we now define the continuity equation (AKA equation of continuity)

\(\displaystyle \frac{\mathrm{d}q}{\mathrm{d}t}=\Sigma-\int_{\partial V}\left\langle j,\hat{n}\right\rangle \mathrm{d}S, \)

where \({q}\) is the amount of \({Q}\) contained in \({V}\), \({t}\) is time, and \({\Sigma}\) is the rate of \({Q}\) being created within \({V}\). The continuity equation thus states the intuitive fact that the change of \({Q}\) within \({V}\) equals the amount generated less the amount which passes through \({\partial V}\). Using the divergence theorem, we can then obtain the differential form of the continuity equation

\(\displaystyle \frac{\partial\rho}{\partial t}=\sigma-\mathrm{div}(j), \)

where \({\sigma}\) is the amount of \({Q}\) generated per unit volume per unit time. This equation then states the intuitive fact that at a point, the change in density of \({Q}\) equals the amount generated less the amount that moves away. Positive \({\sigma}\) is referred to as a source of \({Q}\), and negative \({\sigma}\) a sink. If \({\sigma=0}\) then we say that \({Q}\) is a conserved quantity and refer to the continuity equation as a (local) conservation law.

Under a flat Lorentzian metric, we can combine \({\rho}\) and \({j}\) into the current 4-vector \({J}\) and express the continuity equation with \({\sigma=0}\) as

\(\displaystyle \mathrm{div}(J)=0, \)

whereupon \({J}\) is called a conserved current. Note that in this approach we lose the intuitive meaning of the divergence under a Riemannian metric. If any curvature is present, when we split out the time component we recover a Riemannian divergence but introduce a source due to the non-zero Christoffel symbols

\(\displaystyle \begin{aligned}\nabla_{\mu}J^{\mu} & =\partial_{\mu}J^{\mu}+\Gamma^{\mu}{}_{\nu\mu}J^{\nu}\\ & =\partial_{t}\rho+\nabla_{i}j^{i}+\left(\Gamma^{\mu}{}_{t\mu}\rho+\Gamma^{t}{}_{it}j^{i}\right), \end{aligned} \)

where \({t}\) is the negative signature component and the index \({i}\) goes over the remaining positive signature components. Thus, since the Christoffel symbols are coordinate-dependent, in the presence of curvature there is in general no coordinate-independent conserved quantity associated with a vanishing Lorentzian divergence.

Several methodologies can be used to derive conserved quantities and currents from an expression that in some way describes a physical system (and is often call simply “the system”); in particular, Noether’s theorem derives conserved currents from transformations (“symmetries”) on the variables of an expression called the action that leave it unchanged.

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