# Coordinate transformation from Boyer-Lindquist to rotating frame

The Kerr-Newman metric in Boyer-Lindquist coordinates $x^{\mu} = (t,r,\theta,\phi)$ is of the form $$ds^2 = g_{tt}dt^2 + 2g_{t\phi}dt d\phi + g_{rr}dr^2 + g_{\theta \theta} d\theta^2 + g_{\phi \phi} d\phi^2$$ The vector potential of the electromagnetic field is $$A_{\mu} = -\frac{Qr}{\Sigma}\left[(dt)_{\mu} - a\sin^2 \theta(d\phi)_{\mu}\right]$$ where $\Sigma = r^2 +a^2 \cos^2 \theta$, and $Q , a,$ and $M$ are the parameters of the family. What is the transformation from the Boyer-Lindquist coordinate basis to the rotating basis $\{x^{\tilde{\mu}} \}$ $$dx^{\tilde{\mu}} = \Lambda^{\tilde{\mu}}_{\nu}x^{\nu}$$ so that the metric reduces to the form $$ds^2 = g_{\tilde{t} \tilde{t}} d\tilde{t}^2 + g_{\tilde{r}\tilde{r}}d\tilde{r}^2 + g_{\tilde{\theta} \tilde{\theta}} d \tilde{\theta}^2 ?$$ I am interested in the form of $\Lambda^{\tilde{\mu}}_{\nu}$ so that I can calculate how $A_{\mu}$ transform.