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}