New Transformation Equations and the Electric Field Four-vector

Wave Equations

We see that Eqs. (127) and (134) can easily be put in the form of four-dimensional wave equations by subtracting the terms on the right-hand side of each equation from both sides, obtaining

and

respectively, where c is the speed of propogation of the waves. These waves propogate in four-dimensional spacetime rather than three-dimensional space.

We notice that Eqs. (127) and (134) and, therefore, Eqs. (135) and (136) are similar in form to some of our previous equations, for example, the differential equation for the scalar electric potential, ,

Setting in Eq. (135), we get

Interestingly, we see that Eqs. (138) and (139) are equivalent, providing

so that Eq. (137), in the form of Eq. (139) after inserting Eq. (140), is actually a four-dimensional wave equation. Similarly, we see that Eqs. (113) and (117) can be written in the form of the wave equations

and

respectively. In general, any equation which has the same form as Eqs. (113) and (117) can be written as wave equations. That is, any equation, in this theory, whose left-hand side is the four-dimensional Laplacian, Eq. (112), of a scalar potential function of the coordinates and whose right-hand side is a source term, can be written as a four-dimensional wave equation.

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