New Transformation Equations and the Electric Field Four-vector

The Potential Field Four-vector

The scalar electric potential , or for short, at an arbitrary field event , or for short, due to a stationary point charge at an event , or for short, is

where is the spacetime interval between the events and .

To find the scalar potential due to a distribution of stationary charges, we need to sum the contributions from each of the individual elements of charge. For the contribution of an element of charge 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 equation for the scalar potential at an arbitrary field event due to a distribution of stationary charges is

The components of the potential field four-vector , not to be confused with the transformation matrix , at the field event due to a distribution of moving charges at the field event are,

where the are the components of the four-velocity of the element of charge , and the spacetime interval , in this case, is

where

and

are the components of the spacetime interval between events and . The components of Matrix (41) were obtained by transforming the coordinates of the primed frame with , since the primed observer's measurement of the interval between the events is made instantaneously in his frame.

Noting that the components of the current density four-vector, , at the event are

we can write the general relation for the components of the potential field four-vector at an arbitrary field event due to a distribution of moving charges. By inserting Eqs. (44) into Eqs. (39), we get

It may help, when determining the spacetime interval, of Eq. (40), to remember that

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