New Transformation Equations and the Electric Field Four-vector

10. 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, in the unprimed frame is

where

and is the spacetime interval between events and in the unprimed frame from (3.2). Notice that the spatial part of , can be zero, while the time part remains non-zero. In this case, the potential (10.1) does not become infinite at r = 0.

If we transform the components of the primed spacetime interval, , using (6.2) with , since the primed observer's measurement of the interval between the events is made instantaneously in his frame, the spacetime interval becomes

where

and

are the components of the spacetime interval between 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 P(2) are

where the are the components of the four-velocity of the element of charge .

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 (10.9) into (10.8), we get

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