# Current forms and densities

We previously defined the current vector (AKA flux) $${j\equiv\rho u}$$, where $${\rho}$$ is the density of the physical quantity $${Q}$$ and $${u}$$ is a velocity field, and then combined them into the four-current $${J\equiv(\rho,j^{\mu})}$$. There are a number 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{g}\,j\\
\Rightarrow\zeta & =\mathfrak{j}^{1}\mathrm{d}^{2}x
\end{aligned}
The vector whose coordinate 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}\mathfrak{j}^{1}\mathrm{d}^{2}x
\end{aligned}
The amount of $${Q}$$ per unit time crossing $${S}$$
Four-current$${J\equiv(\rho,j^{\mu})}$$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 $${\mathrm{d}^{3}x}$$ are coordinates with $${x^{1}}$$ constant on $${S}$$ and normal to it. The four-current can be generalized to other Lorentzian manifolds, and can also be turned into a form $${\xi\equiv i_{J}\mathrm{d}V}$$ or a density $${\mathfrak{J}\equiv\sqrt{g}\,J}$$.

 Δ Note that the terms flux and current (as well as flux density and current density) are not used consistently in the literature.