Tensor densities

The current density \({\mathfrak{j}}\) defined in the previous section is an example of a tensor density, which in general takes the form

\(\displaystyle \mathfrak{T}\equiv\left(\sqrt{\left|\mathrm{det}(g)\right|}\right)^{W}T, \)

where \({T}\) is a tensor and \({W}\) is called the weight. Note that tensor densities are not coordinate-independent quantities, and \({\sqrt{g}}\) itself can thus be called a scalar density.

From the previous expressions we get

\begin{aligned}\partial_{\lambda}\left(\mathfrak{T}\right) & =\sqrt{g}^{W}\partial_{\lambda}T+W\left(\Gamma^{\mu}{}_{\lambda\mu}-T^{\mu}{}_{\mu\lambda}\right)\mathfrak{T}\\
& =\sqrt{g}^{W}\partial_{\lambda}T+W\ \overline{\Gamma}^{\mu}{}_{\lambda\mu}\mathfrak{T}\\
& =\sqrt{g}^{W}\partial_{\lambda}T+\frac{W}{2}g^{\mu\nu}\partial_{\lambda}g_{\mu\nu}\mathfrak{T},\\
L_{u}\left(\mathfrak{T}\right) & =\sqrt{g}^{W}L_{u}T+W\mathrm{div}\left(u\right)\mathfrak{T}\\
& =\sqrt{g}^{W}L_{u}T+\frac{W}{2}g^{\mu\nu}L_{u}g_{\mu\nu}\mathfrak{T},\\
\nabla_{\lambda}\left(\mathfrak{T}\right) & =\sqrt{g}^{W}\nabla_{\lambda}T,\end{aligned}

where the last is due to the covariant derivative of the metric vanishing. In particular, this means that for zero torsion the divergence of a vector density is

\begin{aligned}\overline{\nabla}_{\lambda}\mathfrak{J}^{\lambda}&=\sqrt{g}\ \overline{\nabla}_{\lambda}J^{\lambda}\\&=\sqrt{g}\ \mathrm{div}\left(J\right)\\&=\partial_{\lambda}\mathfrak{J}^{\lambda}.\end{aligned}

Δ A potential source of confusion is the use of the word “density” to indicate both an amount per unit area or volume and the presence of the coordinate-dependent factor \({\sqrt{g}}\). Also, while the current density is metric-independent, reflecting the amount per unit time crossing a unit coordinate area instead of metric area, tensor densities in general are not necessarily metric-independent.

An Illustrated Handbook