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update: auto commit

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Yingjie Wang
2026-04-10 01:45:39 -04:00
parent 93cd56980b
commit 32048b15db
1 changed files with 28 additions and 4 deletions
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@@ -785,13 +785,37 @@ $}
\[
\scalebox{0.8}{$\displaystyle
\begin{aligned}
\pt \constraintA{_i} &= \dots - c_1 \constraintA{_i},\\
\pt \constraintB{^i_j} &= \dots - c_2 \constraintB{^i_j},\\
\pt \constraintC*{_i} &= \dots - c_3 \constraintC*{_i},\\
\pt \constraintD*{_i_j_k} &= \dots - c_4 \constraintD*{_i_j_k},
\pt \constraintA{_i} &= - \pdv{\mu_L}{\lapse} \lapse^2 \hat{K} \constraintA{_i} - 2
\mu_L \lapse \hat{K} \constraintA{_i} + \left( \Partial{_i}
\shift{^j} \right) \constraintA{_j} + \dlapse{_j}
\constraintB{^j_i} + \shift{^j} \Partial{_i} \constraintA{_j},\\
\pt \constraintB{^i_j} &= \pdv{\mu_S}{\lapse} \lapse^2 \Gam*{^i} \constraintA{_j} + 2
\mu_S \lapse \Gam*{^i} \constraintA{_j} - \pdv{\eta}{\lapse}
\shift{^i} \constraintA{_j} - \eta \constraintB{^i_j} +
\dshift{^k_j} \constraintB{^i_k} + \left( \Partial{_k} \shift{^i}
\right) \constraintB{^k_j} + \shift{^k} \Partial{_k} \constraintB{^i_j},\\
\pt \constraintC*{_i} &= \frac{2}{3} \left[ \left( \hat{K} +
2\Theta \right) \constraintA{_i} - \left( \Gam*{^j_j_k} -
\frac{3}{2} \dchi*{_k} \right) \constraintB{^k_i} \right] +
\shift{^k} \Partial{_i} \constraintC*{_k} + \left( \Partial{_i}
\shift{^k} \right) \constraintC*{_k},\\
\pt \constraintD*{_i_j_k} &= -2 \Ktf*{_j_k} \constraintA{_i} +
\dsmetric*{_l_j_k} \constraintB{^l_i} + \shift{^l} \Partial{_l}
\constraintD*{_i_j_k} + 2 \constraintD*{_{il(j}}
\dshift{^l_{k)}} - \frac{2}{3} \dshift{^l_l} \constraintD*{_i_j_k},
\end{aligned}
$}
\]
% \[
% \scalebox{0.8}{$\displaystyle
% \begin{aligned}
% \pt \constraintA{_i} &= \dots - c_1 \constraintA{_i},\\
% \pt \constraintB{^i_j} &= \dots - c_2 \constraintB{^i_j},\\
% \pt \constraintC*{_i} &= \dots - c_3 \constraintC*{_i},\\
% \pt \constraintD*{_i_j_k} &= \dots - c_4 \constraintD*{_i_j_k},
% \end{aligned}
% $}
% \]
\end{frame}
\begin{frame}{References}