New Transformation Equations and the Electric Field Four-vector

The Force Equations

The analogs of the Lorentz four-force equations are

Remembering that , where , is the proper magnitude of the test charge, is its four-velocity, and , where is the potential field 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. (78), is

Since

and similarly

we see that two of the terms on the right-hand side of the last equality in Eqs. (80) 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 the are the components of the force four-vector. Writing out the components of , we get

and, as before, there is an additional implied sum over the repeated index, . Here, again, there are the additional terms, , which do not appear in the Lorentz equations. These terms imply the existence of an extra force

We would like to look further into the relationship between the terms in Eq. (83), in hopes of finding a connection between mass, charge, and the scalar electric potential. In the case where the potential is created by a single point charge, we can write the components of the potential four-vector, , as

where the are the components of the four-velocity of the point charge. Noting that and starting with Eq. (83), we get

We conclude from Eq. (89) that the proper mass of a particle is proportional to its proper charge and the potential of the field at its location. The improper mass, , of a particle varies with the magnitude of the charge, therefore, the improper mass is

This states that mass is velocity dependent, however, its magnitude due to the particle's velocity is decreased, rather than increased, as it is in special relativity, therefore,

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