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The electromagnetic field tensor transforms according to the standard
rule
![$\displaystyle F^{\,\mu'\nu'} = F^{\,\mu\nu} \,p_\mu^{\,\mu'} \,p_{\nu}^{\,\nu'}.$](img3850.png) |
(1806) |
This easily yields the celebrated rules for transforming electromagnetic
fields:
where
is the relative velocity between the primed and unprimed
frames, and the perpendicular and parallel directions are, respectively,
perpendicular and parallel to
.
At this stage, we may conveniently note two important invariants of the
electromagnetic field. They are
![$\displaystyle \frac{1}{2}\, F_{\mu\nu} \,F^{\,\mu\nu} = c^{\,2} \,B^{\,2} - E^{\,2},$](img3859.png) |
(1811) |
and
![$\displaystyle \frac{1}{4} \,G_{\mu\nu} \,F^{\,\mu\nu} = c\,{\bf E} \cdot {\bf B}.$](img3860.png) |
(1812) |
The first of these quantities is a proper-scalar, and the second
a pseudo-scalar.
Next: Potential Due to a
Up: Relativity and Electromagnetism
Previous: Dual Electromagnetic Field Tensor
Richard Fitzpatrick
2014-06-27