Four-current

This article uses the summation convention for indices. See covariance and contravariance of vectors for background on raised and lowered indices, and raising and lowering indices on how to switch between them.

Definition

Using the Minkowski metric ${\displaystyle {\eta }_{\mu \nu }}$ of metric signature (+ − − −), the four-current components are given by:

${\displaystyle J^{\alpha }=\left(c\rho ,j^1,j^2,j^3\right)=\left(c\rho ,\mathbf{j}\right)}$

where:

• c is the speed of light;
• ρ is the volume charge density;
• j is the conventional current density;
• The dummy index α labels the spacetime dimensions.

Motion of charges in spacetime

This can also be expressed in terms of the four-velocity by the equation:

${\displaystyle J^{\alpha }={\rho }_0{U}^{\alpha }={\rho }_u\sqrt{1-\frac{u^2}{c^2}}{U}^{\alpha }}$

where:

• ${\displaystyle {\rho }_u}$ is the charge density measured by an inertial observer O who sees the electric current moving at speed u (the magnitude of the 3-velocity);
• ${\displaystyle {\rho }_0}$ is “the rest charge density”, i.e., the charge density for a comoving observer (an observer moving at the speed u - with respect to the inertial observer O - along with the charges).

Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to Lorentz contraction.

Physical interpretation

Charges (free or as a distribution) at rest will appear to remain at the same spatial position for some interval of time (as long as they're stationary). When they do move, this corresponds to changes in position, therefore the charges have velocity, and the motion of charge constitutes an electric current. This means that charge density is related to time, while current density is related to space.

The four-current unifies charge density (related to electricity) and current density (related to magnetism) in one electromagnetic entity.