Kerr metric in Eddington–Finkelstein form

I am searching for a reference in which I can find out the Kerr metric in Eddington–Finkelstein form.
I have computed it by hand and I have obtained the following form but I am not sure above all of the red term. $$\begin{bmatrix} -1+\frac{2mr}{r^2+\alpha^2(\cos{\theta})^2} & 1 & 0 & \frac{2mr\alpha(\sin{\theta})^2}{r^2+\alpha^2(\cos{\theta})^2} \\ 1 & 0 & 0 & \alpha(\sin{\theta})^2\\ 0 & 0 & r^2+\alpha^2(\cos{\theta})^2 & 0\\ \frac{2mr\alpha(\sin{\theta})^2}{r^2+\alpha^2(\cos{\theta})^2} & \alpha(\sin{\theta})^2 & 0 & \color{red}{(r^2+\alpha^2)(\sin{\theta})^2+\frac{2mr\alpha^2(\sin{\theta})^4}{r^2+\alpha^2(\cos{\theta})^2}}\\ \end{bmatrix}$$ Can you help me giving me a reference where I can find this matrix or telling me if I am right?

• arxiv.org/pdf/0706.0622.pdf – G. Smith Apr 8 at 16:39
• Sorry where there is indicated this matrix? I have tried to search for it but I can't find out it. – willie Apr 8 at 16:42
• Compute the matrix for the line element (3). – G. Smith Apr 8 at 16:57