New Transformation Equations and the Electric Field Four-vector

We would like, now, to find the four-divergence of the current density of (12.2). This takes the form

After eliminating terms that cancel from the left and right-hand sides of (15.1), we are left with

where is an extra current density associated with the terms by way of

where is the time component, , of the electric field four-vector from (11.1).