update: auto commit

report.tex

@@ -661,8 +661,36 @@
     \[
 \scalebox{0.5}{$\displaystyle
 \begin{aligned}
-  E &= mc^2 \\
+  \pt \chi ={}& \frac{2}{3} \chi \left[ \alpha \left( \hat{K} +
-  F &= \frac{a+b}{c+d}
+  2\Theta \right) - \spaceD{_i} \shift{^i} \right],\\
+  \pt \smetric*{_i_j} ={} & - 2 \lapse \Ktf*{_i_j} + \shift{^k}
+  \Partial{_k} \smetric*{_i_j} + 2 \smetric*{_{k(i}} \Partial{_{j)}}
+  \shift{^k} - \frac{2}{3} \smetric*{_i_j} \Partial{_k} \shift{^k},\\
+  \pt \hat{K} ={}& - \spaceD{^i} \spaceD{_i} \lapse + \lapse \left[
+    \Ktf*{_i_j} \Ktf*{^i^j} + \frac{1}{3} \left( \hat{K} + 2 \Theta
+  \right)^2 \right] + 4 \pi \lapse \left( S + \rho \right) + \lapse
+  \kappa_1 (1-\kappa_2) \Theta + \shift{^i} \Partial{_i} \hat{K},\\
+  \pt \Ktf*{_i_j} ={} & \chi \left[ - \spaceD{_i} \spaceD{_j} \lapse
+    + \lapse \left( \Rictensor{_i_j} - 8 \pi \tensor{S}{_i_j} \right)
+  \right]^{\text{tf}} + \lapse \left[ \left( \hat{K} + 2 \Theta
+  \right) \Ktf*{_i_j} - 2 \Ktf*{^k_i} \Ktf*{_k_j} \right] +
+  \shift{^k} \Partial{_k} \Ktf*{_i_j} + 2 \Ktf*{_{k(i}}
+  \Partial{_{j)}} \shift{^k} - \frac{2}{3} \Ktf*{_i_j} \Partial{_k}
+  \shift{^k},\\
+  \pt \Gam*{^i} ={} & - 2 \Ktf*{^i^j} \Partial{_j} \lapse + 2 \lapse
+  \left[ \Gam*{^i_j_k} \Ktf*{^j^k} - \frac{3}{2} \Ktf*{^i^j}
+    \Partial{_j} \ln(\chi) - \frac{1}{3} \smetric*{^i^j} \Partial{_j}
+    \left( 2 \hat{K} + \Theta \right) - 8 \pi \smetric*{^i^j}
+  \tensor{S}{_j} \right] + \smetric*{^j^k} \Partial{_j} \Partial{_k}
+  \shift{^i} + \frac{1}{3} \smetric*{^i^j} \Partial{_j} \Partial{_k}
+  \shift{^k} \notag\\
+  & {} + \shift{^j} \Partial{_j} \Gam*{^i} - \Gamd*{^j} \Partial{_j}
+  \shift{^i} + \frac{2}{3} \Gamd*{^i} \Partial{_j} \shift{^j} - 2
+  \lapse \kappa_1 \left[ \Gam*{^i} - \Gamd*{^i} \right], \\
+  \pt \Theta ={} & \frac{1}{2} \lapse \left[ \Rscalar - \Ktf*{_i_j}
+    \Ktf*{^i^j} + \frac{2}{3} \left( \hat{K} + 2 \Theta \right)^2
+  \right] - \lapse \left[ 8 \pi \rho + \kappa_1 (2+\kappa_2) \Theta
+  \right] + \shift{^i} \Partial{_i} \Theta,
 \end{aligned}
 $}
 \]