New Transformation Equations and the Electric Field Four-vector

17. Wave Equations

We see that (16.2) and (16.9) 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 (16.2) and (16.9) and, therefore, (17.1) and (17.2) are similar in form to some of our previous equations. For a single stationary charge, (12.3) reduces to

Setting in (17.1), we get the four-dimensional wave equation

Interestingly, we see that (17.4) and (17.5) are equivalent, since from (10.7),

If we, now, multiply in (17.5) by we get the four-dimensional wave equations associated with the potential four-vector field

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