deletions | additions       diff --git a/This_leads_directly_.tex b/This_leads_directly_.tex  index 840a5a3..ee03a19 100644  --- a/This_leads_directly_.tex  +++ b/This_leads_directly_.tex 
...
\section{3+1 Decomposition and Conventions}  The standard reference for the 3+1 decomposition is \cite{York_1979}. \citet{York_1979}.  I'll repeat only the most important resulting equations here. The line element $ ds $ is given by:  % 
...
\Rightarrow \nh^{\mu} &= g^{\mu \alpha} \nh_{\alpha} = \left[ \frac{1}{\alpha}, - \frac{\beta^i}{\alpha} \right].  \end{align*}  The ``projected'' four-metric is $h_{\mu \nu} \equiv g_{\mu \nu} + \nh_{\mu} \nh_{\nu}$. Then we define the extrinsic curvature (\cite{York_1979} (\citet{York_1979}  eq. (19),(35))\footnote{This agrees with \cite{MTW} \citet{MTW}  and Cook. Wald \cite{Wald} \citet{Wald}  uses the opposite sign (eq (10.2.13),(E.2.30)) but is self-consistent.}: %  \begin{align*}  K_{i j} & \equiv - \frac{1}{2} \mathcal{L}_{\nh} h_{i j} \\ 
...
\Gfour^0_{i j} &= \frac{1}{2} \gamma_{i j, 0} = - K_{ij}, &\Gfour^i_{j k} &= \frac{1}{2} \gamma^{i m} \left[ \gamma_{j m, k} + \gamma_{m k, j} - \gamma_{j k, m} \right] = \Gamma^i_{jk},  \end{align*}  Smarr \cite{Smarr_1977} \citet{Smarr_1977}  defines the {\em electric} and {\em magnetic} parts of the Weyl curvature in the 3+1 split. These are spatial tensors (that is, they are orthogonal to the unit normal to the hypersurface), with components given by %  \begin{align}  E_{i j} &= - R_{i j} - K K_{i j} + K_{m i} K_j^{\;m}, \nonumber \\ 
...
R_{a b} &\equiv R^{c}_{\; a c b},  \end{align*}  %  the former following the Landau-Lifshitz Spacelike Convention (LLSC), as with \cite{MTW}. \citet{MTW}.