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This_began_as_some.tex  This_leads_directly_.tex  subsectionKerrSchild.tex  subsectionBowenYork_.tex  Bowen__York_citeBowe.tex  BowenYork_data_relat.tex  diff --git a/subsectionKerrSchild.tex b/subsectionKerrSchild.tex  deleted file mode 100644  index abfc804..0000000  --- a/subsectionKerrSchild.tex  +++ /dev/null 
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\subsection{Kerr-Schild Coordinates}   \label{sec:ks_general}     An important sub-class of black-hole space-times can be written in   {\it Kerr-Schild} form. The four-metric is written   %   \[   g_{\mu \nu} = \eta_{\mu \nu} + 2 H \ell_{\mu}\ell_{\nu} \Rightarrow g^{\mu \nu} = \eta^{\mu \nu} - 2 H \ell^{\mu} \ell^{\nu},   \]   %   where $\ell_{\mu}$ is a flat-space null vector:   %   \[   \ell_{\mu} \ell^{\mu} \equiv \eta^{\mu \nu} \ell_{\mu}\ell_{\nu} = 0.   \]   %   From this generic form, we can deduce something of the 3+1 decomposition:   %   \begin{array*}   \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},\\   \gamma_{i j} &= \eta_{i j} + 2H \ell_i \ell_j,\\   \gamma^{i j} &= \eta^{i j} - \frac{2H}{1 + 2 H \ell_0^2} \ell^i \ell^j,\\   \Rightarrow \gamma_{i j,k} &= 2 \left[ H_{,k} \ell_i \ell_j + H \ell_{i,k} \ell_j + H \ell_i \ell_{j,k} \right], \\   K_{i j} &= \alpha \left[ \ell_i H_{,j} + \ell_j H_{,i} + H \ell_{i,j} + H \ell_{j,i} + 2 H ^2 \left( \ell_i \ell_m \ell_{j,m} + \ell_j \ell_m \ell_{i,m} \right) + 2 H \ell_i \ell_j \ell_m H_{,m} \right].   \end{array*}       This agrees with the formula presented in eqn (35) of \citet{Yo:2002bm}.