Pre-Quantum Electrodynamics

We have seen that a four-vector transforms according to $${\overline{a}}^{\mu }={\mathrm{\Lambda }}^{\mu }{}_{\nu }{a}^{\nu }$$ in which $\mathrm{\Lambda }$ is a matrix representing the Lorentz transformation. The form this matrix takes depends on the actual transformation: for the specific case of motion in the $x$ direction with velocity $v$, $${\mathrm{\Lambda }}^{\mu }{}_{\nu }=\left(\begin{array}{cccc}\gamma & -\gamma \beta & 0& 0\\ -\gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ (in which $\mu$ is the "row" index and $\nu$ the "column" one).

A four-vector is synonymous to a rank-one tensor. Higher-rank tensors are simply objects carrying more indices. For example, a rank-two tensor transforms as $${\overline{t}}^{\mu \nu }={\mathrm{\Lambda }}^{\mu }{}_{\lambda }{\mathrm{\Lambda }}^{\nu }{}_{\sigma }{t}^{\lambda \sigma }.$$ Such a rank-two tensor can be represented similarly to a matrix: $$t^{\mu \nu }=\left(\begin{array}{cccc}{t}^{00}& t^{01}& t^{02}& t^{03}\\ t^{10}& t^{11}& t^{12}& t^{13}\\ t^{20}& t^{21}& t^{22}& t^{23}\\ t^{30}& t^{31}& t^{32}& t^{33}\end{array}\right).$$ Special cases include symmetric $t_s^{\mu \nu }=t_s^{\nu \mu }$ and antisymmetric $t_a^{\mu \nu }=-t_a^{\nu \mu }$ tensors. The latter contains six independent elements: $$t_a^{\mu \nu }=\left(\begin{array}{cccc}0& t_a^{01}& t_a^{02}& t_a^{03}\\ -t_a^{01}& 0& t_a^{12}& t_a^{13}\\ -t_a^{02}& -t_a^{12}& 0& t_a^{23}\\ -t_a^{03}& -t_a^{13}& -t_a^{23}& 0\end{array}\right)$$ Under the Lorentz transformation along $x$ defined above, we can work out how the nonvanishing elements of an antisymmetric tensor transform:

Comparing with the transformation rules for the electromagnetic field which we obtained in EMtr, we can define the

together with the handy

obtained from the field tensor by the substitution $\mathit{E}/c\to \mathit{B}$, $\mathit{B}\to -\mathit{E}/c$.

Our electromagnetic field transformation laws then become the simple