New Transformation Equations and the Electric Field Four-vector

1. The Potentials of a Moving Charge Distribution

The scalar potential , or for short, at an arbitrary field event , or for short, due to a point charge moving with uniform four-velocity at an event , or for short, is

where is the time component of the four-velocity of the charge, and is the spacetime interval

between the events and , where

and

are the components of the spacetime vector between events and .

To find the scalar potential due to a distribution of moving charges, we need to sum the contributions from each of the individual elements of charge. For the contribution of an element of charge in uniform motion at an event , we make use of the fact that , where is the charge density at the event , and is the element of volume containing . The general relation for the scalar potential at an arbitrary field event due to a distribution of moving charges is

The components of the potential four-vector at the field event are

where is the scalar potential of Eq. (5), and the are the components of the velocity four-vector of the charge element . Noting that the components of the current density four-vector at the event are

and by inserting Eqs. (7) into Eqs. (6), we can write the general relation for the components of the potential four-vector at an arbitrary field event due to a distribution of moving charges as

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