The divergence and conserved quantities

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

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

Note that both this metric-dependent expression and the expression \({\nabla_{a}u^{a}}\) (sometimes called the covariant divergence) in terms of the Levi-Civita connection are coordinate-independent and equal to \({\partial_{a}u^{a}}\) in Riemann normal coordinates, confirming our expectation that for zero torsion we have

\(\require{cancel}\displaystyle \mathrm{div}(u)=\overline{\nabla}_{a}u^{a}. \)

Recall however that the connection coefficients do not vanish in Riemann normal coordinates for non-zero torsion; in this case we can use the contorsion tensor contraction \({K^{a}{}_{ba}=T^{a}{}_{ab}}\) to relate the pseudo-Riemannian divergence to the covariant divergence by

\(\displaystyle \mathrm{div}(u) =\nabla_{a}u^{a}-T^{a}{}_{ab}u^{b}. \)

Δ Note that the different symbols and names given here for the pseudo-Riemannian divergence versus the covariant divergence are oftentimes not distinguished, since they are the same for zero torsion. The distinction also vanishes if the torsion is completely anti-symmetric, i.e. if it leaves geodesics unchanged.

The divergence measures the change in volume due to the flow. Here we assume the vector field \({u}\) has unit length at point \({p}\), and choose an orthonormal frame which aligns \({e_{2}}\) with \({u}\). Each covariant derivative extends a face of the volume, with their sum being proportional to the total change in volume. Note that the upper right corner is of order \({\varepsilon^{4}}\) and so can be neglected, and e.g. any component of \({\nabla_{1}u}\) orthogonal to \({e_{1}}\) leaves the volume unchanged, since a more accurate depiction would include the volume with edge \({-\varepsilon e_{1}}\), where by linearity this component would be in the opposite direction and thus cancel the volume change. Also note that non-zero torsion would reduce the top edge \({\parallel_{\varepsilon e_{2}}\varepsilon e_{1}}\) by \({\varepsilon^{2}T^{1}{}_{1b}u^{b}}\), which must be added back by subtracting this component, matching the algebraic result.

Using the relation \({\mathrm{div}(u)\Omega=\mathrm{d}(i_{u}\Omega)}\) above, along with Stokes’ theorem, we again 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.”

In physics, one considers the divergence of the current vector (AKA current density, flux, flux density) \({j\equiv\rho u}\) of a physical flow in space at a moment in time, 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 \({Q/(\mathrm{length})^{2}(\mathrm{time})}\). For a flat Riemannian metric on the manifold representing space, the continuity equation (AKA equation of continuity) is

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

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 four-current

\(\displaystyle J\equiv(\rho,j^{\mu}), \)

and express the continuity equation with \({\sigma=0}\) as

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

whereupon \({J}\) is called a conserved current. Note that 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

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

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. A conserved current nevertheless means that the quantity is conserved in finite volumes of spacetime, in the sense that \({\int_{\partial V}\left\langle J,\hat{n}\right\rangle \mathrm{d}S=0}\) over any spacetime volume \({V}\), and the continuity equation holds for the components of the coordinate-dependent quantity \({\mathfrak{J}\equiv J\sqrt{\left|\mathrm{det}(g)\right|}}\), since

\begin{aligned}\partial_{\mu}\mathfrak{J}^{\mu} & =\partial_{t}\mathfrak{J}^{t}+\partial_{i}\mathfrak{J}^{i}\\
& =\partial_{t}\mathfrak{J}^{t}+\overline{\nabla}_{i}\mathfrak{J}^{i}=0.

Noether’s theorem derives conserved currents from transformations (“symmetries”) on the variables of an expression called the action that leave it unchanged.

More details on the divergence, currents, and conservation laws may be found in an Appendix.

An Illustrated Handbook