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We can also define a more normal isotropic radial coordinate $r$ as:  r \equiv \frac{\sqrt{M^2 - a^2}}{2} e^{\eta}.  This is related to the original Boyer-Lindquist radial coordinate $R$  through:  R &=& r \left( 1 + \frac{M + a}{2 r} \right) \left( 1 + \frac{M - a}{2 r} \right) = r + M + \frac{M^2 - a^2}{4 r} \nonumber \\  where the $+$ sign is appropriate for $R > 2 M$. Then the three-metric  can be written:  where the new conformal factor is $\Phi^4_0 \equiv \rho^2 / r^2$, and  $\xi^2 \equiv \Omega/\rho^4$. Now the horizon is at  r_+ = \frac{1}{2} \left( R_+ - M \right) = \frac{1}{2} \sqrt{M^2 - a^2}.  Note that in the limit $a \rightarrow  0$, $\Psi^4_0 \rightarrow ( 1 + M / 2 r )^4$, the correct factor for  isotropic Schwarzschild. The extrinsic curvature components are now:  diff --git a/layout.md b/layout.md  index 9fa2035..a230fb8 100644  --- a/layout.md  +++ b/layout.md 
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subsectionQuasiIsotr.tex  In_the_mid1990s_Stev.tex  brace_K_r_phi_.tex  We_can_also_define.tex  which_leads_to_the.tex  We_can_push_the.tex  eqnarray*_psi_04r___frac710560.tex