Four-current

In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the current density, with the dimension of electric charge per time per area. Also known as vector current, it is used in the context of four-dimensional spacetime, rather than separating time from three-dimensional space. It is a four-vector and is Lorentz covariant. 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 translate between them.

In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the current density, with the dimension of electric charge per time per area. Also known as vector current, it is used in the context of four-dimensional spacetime, rather than separating time from three-dimensional space. It is a four-vector and is Lorentz covariant. 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 translate between them. J^\alpha = \left(c \rho, j^1 , j^2 , j^3 \right) = \left(c \rho, \mathbf{j} \right) where: is the speed of light; is the volume charge density; is the conventional current density; the dummy index labels the spacetime dimensions. J^\alpha = \rho_0 U^\alpha , where: \rho_0 is "the rest charge density", i.e., the charge density in the rest frame of the charge (as seen by an observer moving along with the local charge). Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to Lorentz contraction. \dfrac{\partial J^\alpha}{\partial x^\alpha} = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0\,, where \partial/\partial x^\alpha is the four-gradient. This is the continuity equation. In general relativity, the continuity equation is written as: \nabla_\alpha J^\alpha =0\,, where ∇ α is the covariant derivative. \Box A^\alpha = \mu_0 J^\alpha where \Box is the D'Alembert operator, or the electromagnetic field tensor: \nabla_\alpha F^{\alpha\beta} = \mu_0 J^\beta where μ 0 is the permeability of free space and ∇ α is the covariant derivative.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1350575773 under CC BY-SA 4.0; modifications have been made.