Four-current

In special and general relativity, the four-current is the 4-dimensional analogue of the electric current density, which is used in the geometric context of 4d spacetime, rather than 3d space and time separately. Mathematically it is a four-vector, and is Lorentz covariant.

Analogously, it's possible to have any form of "current density", meaning the flow of a quantity per unit time per unit area, see current density for more on this quantity.

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

Minkowski metric

Using the Minkowski metric $\eta_{\mu\nu}$ of metric signature (+−−−), its four components are given by:

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

where c is the speed of light, ρ is the charge density and j the conventional current density. The dummy index α labels the spacetime dimensions.

Euclidean metric

Some authorswho? prefer to work under a Euclidean spacetime rather than Minkowski's, in which case the Euclidean metric of signature (++++) is used, so that:

$J^\alpha = J_\alpha$

so the four-current is rewritten as

$J^\alpha = \left(ic \rho, \mathbf{j} \right)$

where i is the imaginary unit.

Motion of charges in spacetime

This can also be expressed in terms of the four-velocity by the equation:12

$J^\alpha = \rho_0 U^\alpha = \rho\sqrt{1-\frac{u^2}{c^2}} U^\alpha$

where ρ is the charge density measured by an observer at rest observing the electric current, and ρ0 the charge density for an observer moving at the speed u (the magnitude of the 3-velocity) 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 electricity magnetism) in one electromagnetic entity.

Continuity equation

In special relativity, the statement of charge conservation is that the Lorentz invariant divergence of J is zero:3

$\dfrac{\partial J^\alpha}{\partial x^\alpha} = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0$

where $\partial/\partial x^\alpha$ is the 4-gradient. This is the continuity equation.

In general relativity, the continuity equation is written as:

$J^\alpha{}_{;\alpha}=0\,$

where the semi-colon represents a covariant derivative.

Maxwell's equations

The four-current appears in two equivalent formulations of Maxwell's equations, in terms of the four-potential:4

$\Box A^\alpha = \mu_0 J^\alpha$

where $\Box$ is the D'Alembert operator, or the electromagnetic field tensor:

$\partial_\beta F^{\alpha\beta} = \mu_0 J^\alpha$

where μ0 is the permeability of free space.

General Relativity

In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as

$\mathcal{D}^{\mu \nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu \alpha} \, F_{\alpha \beta} \, g^{\beta \nu} \, \sqrt{-g} \,$

then

$J^\mu = \partial_\nu \mathcal{D}^{\mu \nu}$

References

1. ^ Roald K. Wangsness, Electromagnetic Fields, 2nd edition (1986), p. 518, 519
2. ^ Melvin Schwartz, Principles of Electrodynamics, Dover edition (1987), p. 122, 123
3. ^ J. D. Jackson, Classical Electrodynamics, 3rd Edition (1999), p. 554
4. ^ as [ref. 1, p519]

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