Copyright © 1999 David E. Rutherford
All Rights Reserved

• The Current Density Four-vector
• The Trace of the Electric Field Tensor

1. The Current Density Four-vector

The current density four-vector is

or

with

where is the invariant charge density, the are the components of the velocity four-vector of the charge distribution, and is the trace of the electric field tensor. Eqs. (2) are the analogs of Maxwell's inhomogeneous equations. The analogs of Maxwell's homogeneous equations are

2. The Trace of the Electric Field Tensor

The equations for the force four-vector, Eqs. (4) in "Additions: July 8, 1999" on this website with sign corrections,

can also be written

We see an interesting relationship between Eqs. (3), above, and Eqs. (5) of "Additions: July 8, 1999" on this website

Each of these two sets of equations includes the trace of the electric field tensor, F, which must be equal in both cases, since it is an invariant, therefore, we must show that

To do this, we substitute , , and from Eq. (3) of "Additions: July 4, 1999" with and , into Eqs. (8) and get

which shows that the two sides of Eqs. (8) are, indeed, equal as required.

Copyright © 1999 David E. Rutherford
All Rights Reserved