deletions | additions       diff --git a/The_projected_fourme.tex b/The_projected_fourme.tex  deleted file mode 100644  index 110cced..0000000  --- a/The_projected_fourme.tex  +++ /dev/null 
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The ``projected'' four-metric is $h_{\mu \nu} \equiv g_{\mu \nu} +  (\cite{York_1979} eq. (19),(35))\footnote{This agrees with \cite{MTW}  and Cook. Wald \cite{Wald} uses the opposite sign (eq  (10.2.13),(E.2.30)) but is self-consistent.}:  K_{i j} & \equiv & - \frac{1}{2} \mathcal{L}_{\nh} h_{i j} \\  & = & \frac{1}{2\alpha} \left( \beta_{i|j} + \beta_{j|i} - \partial _t  & = & \frac{1}{2\alpha} \left( \beta_{i,j} + \beta_{j,i} - \partial _t \gamma_{i j} - 2 \Gamma^p_{i j} \beta_p \right) \\  & = & - \alpha \,^4\Gamma^0_{i j}.  diff --git a/layout.md b/layout.md  index b973acb..5825a39 100644  --- a/layout.md  +++ b/layout.md 
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brace_g_mu_nu_.tex  This_leads_directly_.tex  To_get_the_extrinsic.tex  The_projected_fourme.tex  Smarr_citeSmarr_1977.tex  The_Riemann_and_Ricc.tex  subsectionMaple_and_.tex