Transformation of fields

The electromagnetic field tensor transforms according to the standard rule

$$F^{\mu'\nu'} = F^{\mu\nu} p_\mu^{\mu'} p_{\nu}^{\nu'}.$$ (1511)

This easily yields the celebrated rules for transforming electromagnetic fields:

$$E_\parallel' = E_\parallel,$$ (1512)

$$B_\parallel' = B_\parallel,$$ (1513)

$${\bf E}_\perp' = \gamma ({\bf E}_\perp +{\bf v}\times{\bf B}),$$ (1514)

$${\bf B}_\perp' = \gamma ({\bf B}_\perp-{\bf v}\times{\bf E}/c^2),$$ (1515)

where ${\bf v}$ is the relative velocity between the primed and unprimed frames, and the perpendicular and parallel directions are, respectively, perpendicular and parallel to ${\bf v}$.

At this stage, we may conveniently note two important invariants of the electromagnetic field. They are

$$\frac{1}{2} F_{\mu\nu} F^{\mu\nu} = c^2 B^2 - E^2,$$ (1516)

and

$$\frac{1}{4} G_{\mu\nu} F^{\mu\nu} = c {\bf E}\!\cdot \! {\bf B}.$$ (1517)

The first of these quantities is a proper-scalar, and the second a pseudo-scalar.