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.


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