# The divergence and conserved quantities

Recall that the divergence of a vector field $${u}$$ can be generalized to a pseudo-Riemannian manifold of signature $${\left(r,s\right)}$$ (sometimes called the covariant divergence) by defining $${\mathrm{div}(u)\equiv(-1)^{s}*\mathrm{d}(*(u^{\flat}))}$$. Using the previously stated relations $${i_{u}\Omega=(-1)^{s}*(u^{\flat})}$$ and $${A=(*A)\Omega}$$ for $${A\in\Lambda^{n}M^{n}}$$, we have $${\mathrm{d}(i_{u}\Omega)=(-1)^{s}\mathrm{d}(*(u^{\flat}))=(-1)^{s}*\mathrm{d}(*(u^{\flat}))\Omega=\mathrm{div}(u)\Omega}$$. 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:

QuantityDefinitionMeaning
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.