Four-current


Four-current

In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density, or indeed any conventional charge current density. Its four components are given by:

J^a = \left(c \rho, \mathbf{j} \right)

where

c is the speed of light
ρ the charge density
j the conventional current density.
a labels the space-time dimensions

This can also be expressed in terms of four-velocity as [1] [2]

J^a = \rho_0 u^a \,

where

\rho = \frac{\rho_0}{\sqrt{1-\frac{v^2}{c^2}}}

In this article the metric ημν = diag[1, − 1, − 1, − 1] is used.

Contents

Continuity equation

In special relativity, the statement of charge conservation (also called the continuity equation) is that the Lorentz invariant divergence of J is zero [3]:

D \cdot J = \partial_a J^a = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0

where D is an operator called the four-gradient and given by (1/c ∂/∂t, ∇). The summation convention has been used, so that the space-time dimensions are implicitly summed over. i.e.

 \partial_a J^a =  \sum_{i=0}^{3} \partial_i J^i

Sometimes, the above relation is written as

J^a{}_{,a}=0\,

In general relativity, the continuity equation is written as:

J^a{}_{;a}=0\,

where the semi-colon represents a covariant derivative.

Other uses

This four-vector is related to four-potential as [ref. 1, p519]

\Box A^a = \mu_0 J^a

where \Box is the D'Alembert operator, defined as

\Box = \partial_{\mu} \partial^{\mu} = -\frac{{1}}{{c^2}} \frac{{\partial}}{{\partial t^2}} + \nabla^2

Also, two of the Maxwell's Equations can be written in terms of the electromagnetic tensor as [ref. 1, p520]

\partial_b F^{ab} = \mu_0 J^a

Euclidean metric

Although Minkowski's space is not euclidean, some authors prefer to use a euclidean metric, where \eta_{\mu\nu} = diag[1,1,1,1] \,, and

J^a = J_a \,

Using this metric, the four-current is rewritten as

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

where i is the imaginary unit.

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} \,

Thus,

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

Physical interpretation

This four-vector unifies charge density (related to electricity) and current density (related to magnetism) in one electromagnetic entity. As studied in electrostatics, charges or distributions of charge moving only through time, thus having no velocity, have only charge density, while if they are moving through space too, thus having some velocity v, they have current density too. This means that charge density is related to time, while current density is related to space.

See also

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

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