Chapter 18
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 elec-
tric 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.
18.1 Lorentz Transformation
18.1.1 Invariance Condition
The Maxwell equations imply that the speed of lightc, in vacuum, observed in
a coordinate system which moves with a constant velocityvwith respect to the
original coordinate system, is the same as in the original system. And this is in
accord with experiments. Consequently, the rule for the transformation of coordinates
between these two systems must be supplemented by a transformation of the time,
as formulated by the Lorentz transformation.
Letr,tbe the position vector and the time in the original coordinate system,
r′,t′the corresponding variables in the system moving with the constant velocity.
The Maxwell equations enforce an invariance condition, viz. the square of the line
element or “length”s,viz.
s^2 =c^2 t^2 −r^2© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_18
369