Copyright © 2000 David E. Rutherford
All Rights Reserved

The Divergence of the Current Density

We would like, now, to find the four-divergence of the current density of Eq. (105). This takes the form

After eliminating terms that cancel from the left and right-hand sides of Eq. (110), we are left with

where is the time component of the generalized electric field four-vector, , from Eq. (102). Simplifying Eq. (111) further, we get , which is equivalent to Eq. (76). Since is the negative four-divergence of the potential four-vector, it is a scalar quantity. We can say then that , where is a scalar quantity with the units of force per unit charge. The term, , on the right-hand side of Eq. (111) is the four-divergence of the extra current density. It is, therefore, a scalar quantity with the units of charge per unit four-dimensional volume or four-dimensional charge density. To simplify this term, we say that , where is the four-dimensional charge density. In order to simplify Eq. (111) still further, we introduce the operator ,

which is the four-dimensional Laplacian operator. Putting all of this together, Eq. (111) becomes

where is a constant of proportionality. Since represents a matter density, we suspect that represents a scalar potential.

Now,  remembering that the proper mass is

we can replace in Eq. (114) with the four-dimensional mass density, , and with the four-dimensional charge density, , to get

Multiplying both sides of Eq. (113) by , we have

Inserting Eq. (115) into Eq. (116) and setting , Eq. (116) becomes

where is the gravitational potential.

Copyright © 2000 David E. Rutherford
All Rights Reserved