Copyright © 1999 David E. Rutherford
All Rights Reserved

• The Electric Field Equations
• The Force Equations
• The Energy-Momentum Equations

1. The Electric Field Equations

We would like now to write out, in full, the analogs of the equations mentioned in Eqs. (3) of "Additions: July 13, 1999" on this website

First we will expand the analogs of Maxwell's field equations

Remembering that , where is the proper magnitude of the test charge and is its four-velocity, and , where is the potential four-vector at the position of due to all charges other than , we expand the first set of equations, ,

The second set of equations, , obtained by setting in Eqs. (3), is

The terms on the right-hand side of the last equality in Eqs. (4) can be simplified by noting that

and

so that two of the terms cancel. The remaining term we interpret as

and the final relations for the analogs of Maxwell's inhomogeneous equations become

where the are the components of the current density four-vector, and there is an implied sum over the repeated indices. In addition to the usual terms on the left-hand side of Maxwell's equations we have, in Eqs. (8), the terms not found in Maxwell's equations.

2. The Force Equations

The analogs of the Lorentz four-force equations are

The first set of equations, , is

The second set of equations, , obtained by setting in Eqs. (10), is

Since

and similarly

we see that two of the terms on the right-hand side of the last equality in Eqs. (11) cancel. The remaining term we identify as

where the are the components of the momentum four-vector. We can now write the complete analogs of the Lorentz four-force equations as

where there is an additional implied sum over the repeated indices. Here, again, there is an additional term, , which does not appear in the Lorentz equations.

3. The Energy-Momentum Equations

The energy-momentum equations are

where the last index in the first equation is the greek letter sigma, not the number zero. Expanding the first set of equations, , we get

Combining terms we have

or

The second set of equations, , obtained by setting in Eqs. (17), is

combining terms, again, we get

Separating terms that cancel onto opposite sides of Eqs. (21), we end up with the same terms on each side. Identifying the terms on the left-hand side with the components of the energy-momentum tensor, , we have

and finally, we have the equations for the energy-momentum tensor

where there is, as before, an additional implied sum over the repeated indices.

Copyright © 1999 David E. Rutherford
All Rights Reserved