deletions | additions       diff --git a/subsectionKerrSchild.tex b/subsectionKerrSchild.tex  index 5bce06e..abfc804 100644  --- a/subsectionKerrSchild.tex  +++ b/subsectionKerrSchild.tex 
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An important sub-class of black-hole space-times can be written in  {\it Kerr-Schild} form. The four-metric is written  %  \begin{array} \[  g_{\mu \nu} &= =  \eta_{\mu \nu} + 2H\ell_{\mu}\ell_{\nu},\\ 2 H \ell_{\mu}\ell_{\nu}  \Rightarrow g^{\mu \nu} &= =  \eta^{\mu \nu} - 2 H \ell^{\mu} \ell^{\nu}   \end{array} \ell^{\nu},   \]  %  where $\ell_{\mu}$ is a flat-space null vector:  % 
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%  From this generic form, we can deduce something of the 3+1 decomposition:  %  \[   g_{\mu \nu} = \eta_{\mu \nu} + 2 H \ell_{\mu} \ell_{\nu},   \]   %   \begin{array*}  g_{\mu \nu} &= \eta_{\mu \nu} + 2 H \ell_{\mu} \ell_{\nu},\\   \Rightarrow g^{\mu \nu} &= \eta^{\mu \nu} - 2 H \ell^{\mu} \ell^{\nu},\\   \alpha &= \frac{1}{\sqrt{1 + 2 H \ell_0^2}},\\  \beta_i &= 2 H \ell_0 \ell_i,\\  \beta^i &= \frac{2 H \ell_0 \ell^i}{1 + 2 H \ell_0^2},\\