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).
Since 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

(1431) |

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

(1432) |

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 :

(1434) |

(1435) |

This equation tells us that there are no net sources or sinks of electric charge in nature: