Fourcurrent

Fourcurrent is a term used in relativistic electrodynamics to describe the density and flow of electric charge in a single relativistic object. It is commonly denoted as J^μ and combines the charge density with the current density into a four-vector: J^μ = (cρ, J_x, J_y, J_z), where ρ is the electric charge density and J is the three-dimensional current density. The time component cρ reflects the charge per unit volume in a frame, while the spatial components J^i describe how that charge moves in space.

As a four-vector, the fourcurrent transforms under Lorentz transformations in the same way as other four-vectors. A central property is charge conservation, expressed in covariant form as ∂_μ J^μ = 0. In ordinary three-dimensional terms, this is the continuity equation ∂ρ/∂t + ∇·J = 0, which holds in all inertial frames.

In covariant electromagnetism, the fourcurrent serves as the source for the electromagnetic field. Maxwell’s equations can be written in a manifestly Lorentz-covariant form as ∂_ν F^{μν} = μ0 J^μ (in SI units), where F^{μν} is the electromagnetic field tensor. The fourcurrent thus provides a compact description of how charges and currents generate electromagnetic fields across different reference frames.

The concept is foundational in both special and general relativity and is widespread in field theory and particle physics. While the standard notation uses J^μ with the four-current, some texts may spell it as four-current or, less commonly, fourcurrent in plain text.