I have been trying to follow some of the computations of this paper: http://arxiv.org/abs/hep-th/0408217 and particularly I couldn't derive the asymptotic form of the Kerr-AdS background (3.27) using the coordinates that they introduced. Could someone please help me with any useful comment about the coordinate transformation and the limit. I got the contribution of the mass of the black hole, but not the factors in $g_{tt}$ and $g_{yy}$.

The general metric for the five-dimensional Kerr-AdS black hole with two independent rotation parameters is given by

\begin{align*} ds^{2} &= -\dfrac{\Delta_{r}}{\rho^{2}}\left(dt-\dfrac{a\sin^{2}\theta}{\Xi_{a}}d\phi-\dfrac{b\cos^{2}\theta}{\Xi_{b}}d\psi\right)^{2}+\dfrac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\left( adt-\dfrac{(r^{2}+a^{2})}{\Xi_{a}}d\phi \right)^{2} \nonumber \\ &+ \dfrac{1+r^{2}/l^{2}}{r^{2}\rho^{2}}\left( abdt - \dfrac{b(r^{2}+a^{2})\sin^{2}\theta}{\Xi_{a}}d\phi - \dfrac{a(r^{2}+b^{2})\cos^{2}\theta}{\Xi_{b}}d\psi \right)^{2} \nonumber \\ &+ \dfrac{\Delta_{\theta}\cos^{2}\theta}{\rho^{2}}\left( bdt- \dfrac{(r^{2}+b^{2})}{\Xi_{b}}d\psi \right)^{2} +\dfrac{\rho^{2}}{\Delta_{r}}dr^{2}+\dfrac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}, \end{align*}

where \begin{align*} \Delta_{r} &= \dfrac{1}{r^{2}}(r^{2}+a^{2})(r^{2}+b^{2})\left(1+\dfrac{r^{2}}{l^{2}}\right)-2M, \nonumber \\ \Delta_{\theta} &= 1- \dfrac{a^{2}}{l^{2}}\cos^{2}\theta - \dfrac{b^{2}}{l^{2}}\sin^{2}\theta, \nonumber \\ \rho^{2} &= r^{2} + a^{2}\cos^{2}\theta + b^{2}\sin^{2}\theta, \nonumber \\ \Xi_{a} &= 1 -\dfrac{a^{2}}{l^{2}}, \qquad \Xi_{b} = 1 - \dfrac{b^{2}}{l^{2}} \end{align*}

and $a$ and $b$ are two independent rotation parameters.

The asymptotic structure is

$$ds^{2} = -\left(1+\dfrac{y^{2}}{l^{2}}\right)dt^{2}+\dfrac{dy^{2}}{1+\dfrac{y^{2}}{l^{2}}-\dfrac{2M}{y^{2}\Delta^{2}_{\hat{\theta}}}}+y^{2}d\hat{\Omega}^{2}_{3} + \dfrac{2M}{y^{2}\Delta^{3}_{\hat{\theta}}}(dt-a\sin^{2}\hat{\theta} d\hat{\phi} -b\cos^{2}\hat{\theta} d\hat{\psi})^{2} + \dots$$

after applying the coordinate transformation



$\phi=\hat{\phi} + \dfrac{a}{l^{2}}t$,

$\psi=\hat{\psi} + \dfrac{b}{l^{2}}t$.

  • 3
    $\begingroup$ Please include all relevant information into the question so that it is answerable without reading the paper to find out what eq. (3.27) is. $\endgroup$ – ACuriousMind Mar 22 '16 at 22:55

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