From 3D to 4D: Lorentz Transformation, Maxwell Equations

Abstract

This chapter provides an outlook onto Special Relativity Theory and the four-dimensional formulation of the Maxwell equations of electrodynamics. Co- and contra-variant four-dimensional vectors and tensors are introduced, the Lorentz transformation is discussed, properties of the four-dimensional epsilon tensor are stated, some historical remarks are added. The formulation of the homogeneous Maxwell equations involves the field tensors derived from the four-dimensional electric potential. The inhomogeneous Maxwell equations, which can also be derived from a Lagrange density, contain the four-dimensional flux density as a source term. The transformation behavior of the electromagnetic fields is stated. A discussion of the four-dimensional force density and the Maxwell stress tensor conclude the final chapter. The Maxwell equations in four-dimensional form are closely linked with the Lorentz-invariance of these equations. Similarities and differences between the 3D and 4D formulation are discussed. First the Lorentz transformation as well as four-dimensional vectors and tensors are introduced.