Towards a covariant formulation of electromagnetic wave polarization

2.1. COVARIANT FORMULATION OF MAXWELL EQUATIONS 5
The metric, g µν , is a symmetric tensor, which describes the curvature of space,
and is used to ’raise’ and ’lower’ indices. This metric is constant under covariant
differentiation. The metric we use is the flat metric 6
⎛
⎞
1 0 0 0
g µν = g µν = ⎜0 −1 0 0
⎟
⎝0 0 −1 0 ⎠ . (2.6)
0 0 0 −1
We are also allowed to use the four dimensional Levi-Civita symbol or the
totally anti-symmetric tensor 7 (see e.g. Goldstein [11] or Misner, Thorne &
Wheeler [7]). With the flat metric, g µν , it is defined by
ɛ αβγδ = −ɛ αβγδ
ɛ 0123 = 1
ɛ αβγδ = [αβγδ] =
{
1 αβγδ even permutation of 0,1,2,3,
−1 αβγδ odd permutation of 0,1,2,3,
(2.7)
where [αβγδ] is the permutation symbol. Generalizing the Levi-Civita tensor
to an arbitrary metric it is defined as
ɛ αβγδ = √ −g[αβγδ], (2.8)
where
g = det g µν . (2.9)
The contravariant and mixed Levi-Civita tensors are then formed by raising the
indices. Note that the sign convention of ɛ 0123 varies and some authors define
it as -1, which just changes the sign of ɛ αβγδ and ɛ αβγδ .
The differential invariant distance can, in our flat space be written
ds 2 = g µν dx µ dx ν = c 2 dt 2 − dx 2 − dy 2 − dz 2 = c 2 dτ 2 . (2.10)
In the last step here we have introduced the proper time, τ, a term used in
relativity and it is the time experienced by the particle in question and not
by an observer of the particle. Note that the coordinate time, t, is the time
experienced by the observer defining the coordinate system and also defining
the metric.
The scalar product in Lorentz space is defined according to contraction. The
procedure is the following: a vector a µ , multiplied with a vector b µ , gives a rank
two tensor, which via contraction gives a scalar. This is also the covariant trace
of a tensor, the formula is
a µ b ν = c µν , g µν c µν = c µ µ = a 0 b 0 − a 1 b 1 − a 2 b 2 − a 3 b 3 . (2.11)
6 The strong equivalence principle (see Schutz [21]) states that ’any physical law which can
be expressed in tensor notation in special relativity has exactly the same form in a locally
inertial frame of curved space-time’, i.e. if we find a tensor equation using the flat metric, it
will be the same in a curved space time but with another metric.
7 The totally anti-symmetric tensor will be further discussed in Section 2.4, when the group
theory of the Lorentz group is considered.