# How is the idea of the static electric potential $\phi$ extended relativistically?

In classical electrodynamics there's the electrostatic potential $\phi$ which can be differentiated wrt space to give the static electric field $\vec E$. Is this idea perhaps extended relativistically by instead differentiating the potential wrt a space-time interval to give the electromagnetic fields?

• See e.g. Jackson: Classical Electrodynamics and come back for any more precise questions? Dec 11, 2016 at 20:52
• Any book about classical field theory treats this subject. So does wikipedia and many other sites. Dec 11, 2016 at 21:03
• I've deleted an number of comments that were developing into a rather personal argument. Keep it civil, please. Dec 11, 2016 at 21:17

In relativistic classical electrodynamics, the electric and magnetic fields are deduced from the potentials $\phi$ and $\vec{A}$ just as you know, i.e. :
$$\left\{ \begin{array}{c} \vec{E} =-\vec{\nabla}\phi - \dfrac{\partial\vec{A}}{\partial t} \\ \vec{B} = \vec{\nabla}\times\vec{A} \end{array} \right.$$
Then, $A^\mu \equiv (\phi/c, \vec{A})$ is a four-vector, i.e. it transforms from one inertial reference frame to another according to the Lorentz transformation. Defining $F^{\mu\nu}\equiv \partial^\mu A^\nu - \partial^\nu A^\mu$, the equation of motion for a test charged particle is : $$m\dfrac{dv^\mu}{ds} = qv_\nu F^{\mu\nu}$$