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I am trying to put the Kerr metric into the famous Bondi gauge, which is given for instance by the formula (6.2.10) at page 154 of the following paper: https://arxiv.org/abs/1801.01714. Now, Barnich and Troessaert in their paper BMS charge algebra (https://arxiv.org/abs/1106.0213) did the calculations, which can be found in Appendix D and F of the paper. My questions are: 1) In the original metric the determinant of the angular part is different from the unit sphere metric determinant, hence they redefine the radial coordinate. I managed to work out all the new metric coefficients except for $g_{u\theta}$; in particular, I don't understand from where the new leading order coefficient $\frac{a\cos{\theta}}{2\sin^2{\theta}}$ comes from. Any help to understand this transformation would be very appreciated. 2) As one can see in Appendix F (it is pretty straightforward to calculate thiz), $C^{AB}C_{AB}=\frac{2a^2}{\sin^2{\theta}}$. However, in the Bondi gauge one has that the coefficient of the power $r^{-2}$ in $g_{ur}$ should be $\frac{C^{AB}C_{AB}}{16}=\frac{a^2}{8\sin^2{\theta}}$, and in the metric found by Barnich the same coefficient is actually $a^2(\frac{1}{2}-\cos^2{\theta})$. These are very different, unless I am missing some property of sinusoidal functions. Can anyone help me to solve this incongruence?

Thanks for any help or hints!

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