Are the electric and magnetic fields functions on $\mathbb{R}^4$?

Question

Are the electric and magnetic fields functions from $\mathbb{R}^4$ to $\mathbb{R}^4$ (where $\mathbb{R}^4$ is then interpreted as space-time) or do we consider them to be functions from $\mathbb{R}^3$ to $\mathbb{R}^3$ with the parametrization $r(t) = x(t)i + y(t)j + z(t)k$? Is there a difference mathematically? And if it can be done in $\mathbb{R}^4$ and E&M is the basis of relativity, why do physicists study relativity in Minkowski space rather than $\mathbb{R}^4$?

Answer 1

In the relativistic formulation, the "electric and magnetic field functions" are merged into a single tensor field in $\mathbb{R}^4$ (with $x,y,z,t$ coordinates), called the "Faraday field" $\bf{F}$ also known as the "electromagnetic field tensor".

The four Maxwell equations, which can be thought of as time dependent equations in $x,y,z$, are then merged into two equations in $x,y,z,t$. Specifically, the electrostatic equation $$\nabla \cdot E = 4 \pi \rho $$ and the electrodynamic equation $$\partial E / \partial t - \nabla \times B = - 4 \pi J $$ (in which I have suppressed $x,y,z$) are merged into a single $\mathbb{R}^4$ electrodynamic law $$\bf{\nabla} \cdot \bf{F} = 4 \pi \bf{J} $$ (in which I have suppressed $x,y,z,t$).

Similarly the magnetostatic and magnetodynamic equations, which are time dependent equations in $x,y,z$, are merged into a single $x,y,z,t$ equation.

This is all laid out quite explicitly in Section 3.4 of the book "Gravitation" by Misner-Thorne-Wheeler.

The significance of $\mathbb{R}^4$ versus Minkowsi space is that the $x,y,z,t$ versions of these equations are invariant under Lorentz transformations.