New Transformation Equations and the Electric Field Four-vector

The Generalized Electric Field

Our intention, now, is to present the electric field in its most general form and to express the field and force equations and energy-momentum tensor in terms of the generalized electric field. First, we find the general form of the components of the electric field four-vector, . Since the electric field four-vector is a four-gradient it must transform as a covariant vector, therefore

where the are the components of the stationary electric field of Eq. (51). Remembering now that, for a single moving point charge, the components of the potential four-vector, , in the unprimed frame are given by Eq. (87), we can write the general expressions for the components of the electric field four-vector in terms of the potential field four-vector as

At this point, we would like to introduce a new operator; the four-dimensional analog of the three-dimensional curl of a vector field. This operator we call the "turn" of a four-vector field, which we define as

where , here, can be thought of as an arbitrary four-vector field and i, j, k, and l are unit vectors in the x, y, z, and t directions, respectively. Another way of writing is , where the operator, , is not to be mistaken as the D'Alembertian operator. Using this shorthand, we can express Eqs. (102) as a single four-vector equation by writing either or, equivalently, .

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