Conserved currents and quantities

We previously saw that a Lorentzian conserved current $${\mathrm{div}(J)=0}$$ does not imply a conserved quantity in the presence of curvature. If we are willing to consider coordinate-dependent currents, at any given point we can choose Riemann normal coordinates, which allows us to recover a conserved quantity at that point in those coordinates.

In the integral form, we may also identify a coordinate-dependent conserved quantity for a Lorentzian conserved current by integrating over a space-like volume $${S}$$ with coordinates such that $${t\equiv x^{0}}$$ is constant on $${S}$$ and normal to it, while $${x^{1}}$$ is constant on $${\partial S}$$ and normal to it:

\begin{aligned}0 & =\int_{S}\sqrt{g}\mathrm{div}(J)\mathrm{d}^{3}x\\
& =\int_{S}\partial_{\mu}\mathfrak{J}^{\mu}\mathrm{d}^{3}x\\
& =\partial_{t}\left(\int_{S}\mathfrak{J}^{t}\mathrm{d}^{3}x\right)+\int_{S}\partial_{i}\mathfrak{J}^{i}\mathrm{d}^{3}x\\
& =\partial_{t}\left(\int_{S}\mathfrak{J}^{t}\mathrm{d}^{3}x\right)+\int_{\partial S}\mathfrak{J}^{1}\mathrm{d}^{2}x
\end{aligned}

Note that the coordinate-dependent factor $${\sqrt{g}}$$ in $${\mathfrak{J}=\sqrt{g}J}$$ cannot be absorbed into either $${\mathrm{d}^{3}x}$$ or $${\mathrm{d}^{2}x}$$ to yield a coordinate-independent quantity. More specifically, if $${\mathfrak{J}}$$ is either also normal to $${S}$$ or vanishes on $${\partial S}$$, we have $${\partial_{t}\left(\int_{S}\mathfrak{J}^{t}\mathrm{d}^{3}x\right)=0}$$. This also holds if $${S}$$ is infinite and $${\mathfrak{J}}$$ vanishes rapidly enough at spatial infinity.

 Δ A conserved quantity as we have defined it is a quantity whose amount in a volume of space changes in time by the net amount that crosses the volume boundary. This concept is not valid when $${\mathrm{div}(J)=0}$$ in the presence of spacetime curvature, but it is important to remember that this still means that $${\int_{\partial V}\left\langle J,\hat{n}\right\rangle \mathrm{d}S=0}$$, so that the same amount of the quantity enters and exits any finite volume of spacetime; it is in this sense that the current is “conserved.”

With regard to tensors, we can conclude from our divergence theorem variants that in the case of an orthonormal coordinate frame under a flat metric and the Levi-Civita covariant derivative, we have a coordinate-dependent conserved quantity for each component of a tensor, corresponding to a coordinate-dependent conserved current:

\begin{aligned}\require{cancel}\overline{\nabla}_{\nu}T^{\mu\nu}&=0\\\Rightarrow\partial_{0}T^{\mu0}&\overset{\cancel{R}}{=}-\overline{\nabla}_{j}T^{\mu j},\\\int_{\partial V}T^{\mu}{}_{b}\hat{n}^{b}\mathrm{d}S&\overset{\cancel{R}}{=}0\end{aligned}

In the special case of an anti-symmetric tensor and the Levi-Civita covariant derivative we also have a divergence theorem, and therefore a coordinate-dependent conserved current for each component:

\begin{aligned}\overline{\nabla}_{\nu}F^{\mu\nu}&=0\\\Rightarrow\int_{\partial V}F^{\mu}{}_{b}\hat{n}^{b}\mathrm{d}S&=0\end{aligned}