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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)?

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I found the potential at the following paper -
Furuhashi, Hironobu, and Yasusada Nambu. "Instability of massive scalar fields in Kerr-Newman spacetime." Progress of theoretical physics 112.6 (2004): 983-995.‏
Also appear in "Black Hole Physics" by Frolov & Novikov (1997), Appendix D.

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.

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  • $\begingroup$ 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 $dr$... $\endgroup$ – AoZora Jun 24 '19 at 18:04

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