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:
| Quantity | Definition | Meaning |
|---|---|---|
| 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. |