Let us suppose, following Lorentz, that all charge is made up of elementary particles, each carrying the invariant amount . Suppose that is the number density of such charges at some given point and time, moving with velocity , as observed in a frame . Let be the number density of charges in the frame in which the charges are momentarily at rest. As is well known, a volume of measure in has measure in (because of length contraction). Because observers in both frames must agree on how many particles are contained in the volume, and, hence, on how much charge it contains, it follows that . If and are the charge densities in and , respectively, then

(1726) |

The quantity is called the

Suppose that
are the coordinates of the moving charge in
.
The *current density 4-vector* is constructed as follows:

(1727) |

Thus,

where is the current density 3-vector. Clearly, charge density and current density transform as the time-like and space-like components of the same 4-vector.

Consider the invariant 4-divergence of :

(1729) |

We know that one of the caveats of Maxwell's equations is the charge conservation law

(1730) |

It is clear that this expression can be rewritten in the manifestly Lorentz invariant form

This equation tells us that there are no net sources or sinks of electric charge in nature: that is, electric charge is neither created nor destroyed.