Towards a covariant formulation of electromagnetic wave polarization

Chapter 2
Background
2.1 Covariant formulation of Maxwell equations
In this Section we introduce some basic concepts from the theory of relativity
and a covariant formulation of Maxwell equations. In the process we derive
the field strength tensor, which is a fundamental object in electromagnetism,
that will be used later in this paper. With the field strength tensor we express
Maxwell’s equations as covariant 1 tensor equations. We will use Lorentz
notation with the proper four-vector x µ = (ct, x, y, z). This notation together
with the Lorentz scalar product, Equation (2.11), is often called Lorentz space
(Barut [1]).
2.1.1 Basic relativity and tensor operations
In relativity one considers the concept of Lorentz invariance. The Lorentz transformations
are the transformations that keep the expression
c 2 ∆t 2 − ∆x 2 − ∆y 2 − ∆z 2 , (2.1)
invariant. The ∆ refers to a difference between two vectors, which gives the
invariance under translations. These transformations form a group, which is
usually divided into different parts.
The transformations satisfying Equation (2.1) include translations and form
the inhomogeneous Lorentz group, which is also called the Poincaré group (see
Barut [1] or Tung [24]). The transformations which keep the expression
c 2 t 2 − x 2 − y 2 − z 2 , (2.2)
invariant form the inhomogeneous Lorentz group and does not include translations.
All homogeneous Lorentz transformations, denoted by L, have det L =
±1, see Barut [1]. Transformations with det L = 1 are the proper Lorentz transformations
and the transformations with det L = −1 are the improper Lorentz
transformations, which include reflections. The proper Lorentz transformations
1 The covariance refers to a specific group, in this case the Lorentz group (see Section
2.4). Lorentz covariant quantities and equations are often referred to as proper quantities and
equations.
3