Towards a covariant formulation of electromagnetic wave polarization

Chapter 5
Covariant polarization
tensors in E,B-component
form
In Chapter 5 we saw that the S, V, T parameters can be found in Lorentz covariant
tensors. These tensors were constructed according to
where
is the metric,
and
1 X µν = F αµ (F αδ ) ∗ g δν , (5.1)
2 X µν = F αµ (F αδ ) ∗ g δν , (5.2)
3 X µν = F αµ (F αδ ) ∗ g δν , (5.3)
4 X µν = F αµ (F αδ ) ∗ g δν , (5.4)
⎛
⎞
1 0 0 0
g µν = g µν = ⎜0 −1 0 0
⎟
⎝0 0 −1 0 ⎠ , (5.5)
0 0 0 −1
⎛
⎞
0 −E 1 −E 2 −E3
F µν = ⎜E 1 0 −cB 3 cB 2
⎟
⎝E 2 cB 3 0 −cB 1
⎠ (5.6)
E 3 −cB 2 cB 1 0
⎛
⎞
0 E 1 E 2 E 3
F µν = ⎜−E 1 0 −cB 3 cB 2
⎟
⎝−E 2 cB 3 0 −cB 1
⎠ (5.7)
−E 3 −cB 2 cB 1 0
are the contravariant and covariant field strength tensors, respectively, and
⎛
⎞
0 −cB 1 −cB 2 −cB 3
F µν = ⎜cB 1 0 E 3 −E 2
⎟
⎝cB 2 −E 3 0 E 1
⎠ (5.8)
cB 3 E 2 −E 1 0
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