In the traditional 3+1 spacetime, Maxwell's equations také the following form:
$\nabla × \mathbf E = -\frac 1 c\frac {\partial \mathbf B} {\partial t}$
$\nabla \cdot \mathbf B=0$
$\nabla \cdot \mathbf E=\rho$
$\nabla × \mathbf B = \frac 1 c \mathbf j + \frac 1 c \frac {\partial \mathbf E} {\partial t}$
But to make the relativistic formulation, we use a vector potential $\mathbf A$ and a scaler potential $\phi$, such that $\mathbf B=\nabla × \mathbf A$ and $\mathbf E = -\frac 1 c \frac {\partial \mathbf A} {\partial t} - \nabla \phi$. We than combine these quantities into the four vector $A^\mu = (\phi, A^1, A^2, A^3)$ and the electromagnetic field strenght $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$.
It is then easy to see that $E_i=F^{0i}$. But what are the remaining components? In a 3+1 spacetime, there are 3 components left over, which makes for the (3D) magnetic field vector. But in 4+1, there are 6 components left, which doesn't fit into a 4-D vector, as it has only 4 components. What are then the remaining components? Or is magnetic field a tensor in 4+1D?
