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update: auto commit

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Yingjie Wang
2026-04-10 01:21:13 -04:00
parent 85b57fb9ba
commit 13dca20c05
1 changed files with 22 additions and 1 deletions
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@@ -745,7 +745,28 @@ $}
\right] + \shift{^i} \Partial{_i} \Theta,\\
\pt \lapse ={}& - \mu_L \lapse^2 \hat{K} + \shift{^i} \Partial{_i} \lapse,\\
\pt \shift{^i} ={}& \mu_S \lapse^2 \Gam*{^i} - \eta \shift{^i} +
\shift{^j} \Partial{_j} \shift{^i},
\shift{^j} \Partial{_j} \shift{^i},\\
\pt \dlapse{_i} ={} & - \pdv{\mu_L}{\lapse} \lapse^2 \hat{K} \dlapse{_i} - 2 \mu_L
\lapse \hat{K} \dlapse{_i} - \mu_L \lapse^2 \Partial{_i} \hat{K} +
\dshift{^j_i} \dlapse{_j} + \shift{^j} \Partial{_j} \dlapse{_i},\\
\pt \dshift{^i_j} ={} & \pdv{\mu_S}{\lapse} \lapse^2 \Gam*{^i} \dlapse{_j} + 2 \mu_S
\lapse \Gam*{^i} \dlapse{_j} + \mu_S \lapse^2 \Partial{_j}
\Gam*{^i} - \pdv{\eta}{\lapse} \dlapse{_j} \shift{^i} - \eta
\dshift{^i_j} + \dshift{^k_j} \dshift{^i_k} + \shift{^k}
\Partial{_k} \dshift{^i_j},\\
\pt \dchi*{_i} ={} & \frac{2}{3} \left[ \dlapse{_i} \left( \hat{K} + 2\Theta \right)
+ \lapse \Partial{_i} \left( \hat{K} + 2\Theta \right) -
\Partial{_i} \dshift{^j_j} - \frac{1}{2} \shift{^k}
\dsmetric*{_k_l_j} \Partial{_i} \smetric*{^l^j} - \frac{1}{2}
\smetric*{^l^j} \shift{^k} \Partial{_i} \dsmetric*{_k_l_j} - \left(
\Gam*{^j_j_k} - \frac{3}{2} \dchi*{_k} \right) \dshift{^k_i}
\right] + \shift{^k} \Partial{_k} \dchi*{_i},\\
\pt \dsmetric*{_{ijk}} ={} & -2 \dlapse{_i} \Ktf*{_j_k} -2 \lapse \Partial{_i} \Ktf*{_j_k} +
\dshift{^l_i} \dsmetric*{_l_j_k} + \shift{^l} \Partial{_l}
\smetric*{_i_j_k} + 2 \dsmetric*{_{il(j}} \dshift{^l_{k)}} + 2
\smetric*{_{l(j}} \Partial{_{|i|}} \dshift{^l_{k)}} - \frac{2}{3}
\dsmetric*{_i_j_k} \dshift{^l_l} - \frac{2}{3} \smetric*{_j_k}
\Partial{_i} \dshift{^l_l},
\end{aligned}
$}
\]