time-like potentials

The 4-current density \(\jmath^\mu\) is a timelike vector, \(\eta_{\mu\nu}\jmath^\mu \jmath^\nu > 0\) as a consequence of the 4-velocity \(\upsilon^\mu\) being time-like. What can you say about the 4-potential \(A_\mu\): Is it time-like as well? Would it depend on the chosen gauge? Can you generalise the classification to the Faraday-tensor \(F_{\mu\nu}\) as well, i.e. in the sense that the value of \(\eta^{\alpha\mu}\eta^{\beta\nu}F_{\alpha\beta}F_{\mu\nu}\) is bounded from below?