D2 - relativistic electromagnetic forces

\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})

F^{α} = (\frac{γ}{c} \vec{v} \cdot \vec{F}, γ \vec{F})

\begin{aligned} F^{0} & = \frac{γ}{c} q (\vec{v} \cdot E + \underset{0}{\underset{⏟}{\vec{v} \cdot (\vec{v} \times \vec{B})}}) \\ = \frac{γ}{c} q \vec{v} \cdot \vec{E} = \frac{q}{c} γ (v_{i} E_{i}) \\ = \frac{q}{c} (u^{1} E_{x} + u^{2} E_{y} + u^{3} E_{z}) \end{aligned}

\begin{aligned} F^{1} & = γ F_{x} = q γ (\frac{c E_{x}}{c} + v_{y} B_{z} - v_{z} B_{y}) \\ = q (\frac{u^{0} E_{x}}{c} + u^{2} B_{z} - u^{3} B_{y}) \\ F^{2} & = q (\frac{u^{0} E_{y}}{c} + u^{3} B_{z} - u^{1} B_{z}) \\ F^{3} & = q (\frac{u^{0} E_{z}}{c} + u^{1} B_{y} - u^{2} B_{x}) \end{aligned}

F^{α} = q F^{α}_{β} u^{β}

F^{α}_{β} = (\begin{matrix} 0 & \frac{E_{x}}{c} & \frac{E_{y}}{c} & \frac{E_{z}}{z} \\ \frac{E_{x}}{c} & 0 & B_{z} & - B_{y} \\ \frac{E_{y}}{c} & - B_{z} & 0 & B_{x} \\ \frac{E_{z}}{c} & B_{y} & - B_{x} & 0 \end{matrix})

F^{α β} = η^{β ν} F^{α}_{ν} F_{α β} = η_{α μ} F^{μ}_{β}