= Solution
The official notes state only $\mathcal F'=\Lambda\mathcal F\Lambda^T$ (PDF p. 15, handwritten p. 10); they do not evaluate the requested coefficients. For a boost along positive $x$, use $\Lambda^0{}_0=\Lambda^1{}_1=\gamma$, $\Lambda^0{}_1=\Lambda^1{}_0=-\gamma v/c$, with the transverse diagonal entries equal to one. Multiplying the specified <electromagnetic field tensor> gives
$$
\mathcal F'^{01}=\mathcal F^{01},\qquad
\mathcal F'^{02}=\gamma(\mathcal F^{02}-\beta\mathcal F^{12}),\qquad
\mathcal F'^{23}=\mathcal F^{23},\qquad
\mathcal F'^{31}=\gamma(\mathcal F^{31}-\beta\mathcal F^{30}).
$$
The corresponding <Lorentz transformation of electromagnetic fields> is
$$
\boxed{E_x'=E_x,\qquad E_y'=\gamma(E_y-vB_z),\qquad
B_x'=B_x,\qquad B_y'=\gamma\left(B_y+\frac{vE_z}{c^2}\right).}
$$
Consequently
$$
\boxed{a_1=\gamma,\qquad a_2=0,\qquad a_3=-\gamma v.}
$$
These coefficients depend on the frame speed, but are constant for the fixed boost.
Back to article page
