Electromagnetic laws of motion

Maxwell's equations

The laws of motion of electromagnetic fields are governed by the equations,

\begin{align*}\partial_\mu F^{\mu \nu} &= j^\nu\\\partial_\mu \tilde{F}^{\mu \nu} &= 0\end{align*} where

• $j^\nu$ are the components of the four-current $$j^\mu \rightarrow (\rho, j_x, j_y, j_y)$$ comprising of the charge density $\rho$, and the current density $\vec{j}$
• $F^{\mu \nu}$ are the components of the Faraday tensor $$F^{\mu \nu} = \begin{bmatrix}0 & -E_x & -E_y & -E_z\\E_x & 0 & -B_z & B_y\\E_y & B_z & 0 & -B_x\\E_z & -B_y & B_x & 0\end{bmatrix}$$ and $\tilde{F}^{\mu \nu}$ are the components of the second Faraday tensor $$\tilde{F}^{\mu \nu} = \begin{bmatrix}0 & B_x & B_y & B_z\\-B_x & 0 & -E_z & E_y\\-B_y & E_z & 0 & -E_x\\-B_z & -E_y & E_x & 0\end{bmatrix}$$ comprising of the components of the electric and magnetic fields $(E_x, E_y, E_z), (B_x, B_y, B_z)$.

Lorentz force

The laws of motion of charged particles in electromagnetic fields are governed by the equation,

$$\frac{\mathrm{d} P^\mu}{\mathrm{d} \tau} = q F^{\mu \nu} U_\nu$$ where

• $P^\nu$ are the components of the four-momentum
• $q$ is the electric charge on the particle
• $F^{\mu \nu}$ are the components of the Faraday tensor defined above
• $U^\nu$ are the components of the four-velocity

Final word

Two goals have been met:

1. We have shown that the electromagnetic laws of motion are the same in every reference frame
2. The electromagnetic laws lead to a value for the speed of light

From this we can conclude that the speed of light is the same in every reference frame.