# Electromagnetic four-potential for Kerr-Newman solution

Any textbook or paper on the Kerr-Newman metric I found contains the solution for the electromagnetic tensor $F^{\mu}_{\phantom{\mu} \nu}$.

Can you provide a (reliable) reference for the solution of electromagnetic four-potential $A_{\mu}$ (the derivation is not necessary)?

The expression (in natural units, Boyer-Lindquist coordinates) is - $$\boxed{\vec{A}={rQ \over r^2+a^2\cos^2{\theta}}\left(-1,0,0,a\sin^2{\theta}\right)}$$ where $a$ is angular momentum per unit mass and $Q$ is the charge of the BH.
• sorry for the late comment but I don't get the dimensions of the various components. How can $a sin^2\theta$ be dimensionless while $a^2 cos^2\theta$ has dimension of $r^2$? Here I am naively thinking that the components must have all the same dimensions but maybe this is wrong since $d\varphi$ does not have the same dimensions than $dr$... – AoZora Jun 24 '19 at 18:04