chapter  Chapter 8
18 Pages

Covariant electrodynamics

ByJames H. Luscombe

This chapter focuses on the material that is often referred to as “relativistic electrodynamics.” The electromagnetic field is described by some interrelated equations known collectively as Maxwell's equations. The Lorenz condition can always be satisfied through a gauge transformation. Conserved quantities satisfy continuity equations. Charge is conserved and thus it obeys the continuity equation. A hypersurface is an (n - 1)-dimensional surface embedded in an n-dimensional space. Spacelike hypersurfaces (SH) have timelike normals, and timelike hypersurfaces have spacelike normals. A continuity equation implies the existence of a fixed quantity contained in a SH. The field tensor is the generalization of the curl to four dimensions applied to the four-potential: It's a four-dimensional covariant bivector. The “relativism” between electric and magnetic fields can be illustrated by showing that the force experienced by a charge in motion relative to a current-carrying wire arises from an electric field in the rest frame of the charge.