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Bernard Kelly edited subsectionKSGeneral_.tex
about 10 years ago
Commit id: 8dff06fe86eb8edfd05dd4e5b9313d73b2c8ff24
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diff --git a/subsectionKSGeneral_.tex b/subsectionKSGeneral_.tex
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From this generic form, we can deduce something of the 3+1 decomposition:
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\begin{array*} \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} - \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*} \end{array}
This agrees with the formula presented in eqn (35) of \citet{Yo:2002bm}.