From be2a7bc0072e60d1ba22c9a7d89afc37c9c88262 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:17:27 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 52 +++++++++++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 47 insertions(+), 5 deletions(-)

diff --git a/report.tex b/report.tex
index bf33ff8..f4defc5 100644
--- a/report.tex
+++ b/report.tex
@@ -703,11 +703,53 @@ $}
   \begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
     We introduce the following auxiliary variables:
     \begin{equation}
-  \dlapse{_i} \definedby \Partial{_i} \lapse \qc
-  \dshift{^i_j} \definedby \Partial{_j} \shift{^i} \qc
-  \dchi*{_i} \definedby \spaceD*{_i} \ln(\chi) \qc
-  \dsmetric*{_i_j_k} \definedby \Partial{_i} \smetric*{_j_k},
-\end{equation}
+      \dlapse{_i} \definedby \Partial{_i} \lapse \qc
+      \dshift{^i_j} \definedby \Partial{_j} \shift{^i} \qc
+      \dchi*{_i} \definedby \spaceD*{_i} \ln(\chi) \qc
+      \dsmetric*{_i_j_k} \definedby \Partial{_i} \smetric*{_j_k},
+    \end{equation}
+    and replace all the spatical derivatives of the original variables with the new auxiliary variables, we got this
+    \[
+    \scalebox{0.6}{$\displaystyle
+    \begin{aligned}
+      \pt \chi ={}& \frac{2}{3} \chi \left[ \alpha \left( \hat{K} +
+    2\Theta \right) - \dshift{^i_i} - \left( \Gam*{^j_j_k} -
+  \frac{3}{2} \dchi*{_k} \right) \shift{^k} \right] \notag \\
+  ={}&\frac{2}{3} \chi \left[ \alpha \left( \hat{K} +
+  2\Theta \right) - \dshift{^i_i} - \Gam*{^j_j_k} \shift{^k} \right]
+  + \shift{^k} \Partial{_k} \chi,\\
+  \pt \smetric*{_i_j} ={} & - 2 \lapse \Ktf*{_i_j} + \shift{^k}
+  \Partial{_k} \smetric*{_i_j} + 2 \smetric*{_{k(i}} \dshift{^k_{j)}} -
+  \frac{2}{3} \smetric*{_i_j} \dshift{^k_k},\\
+  \pt \hat{K} ={}& - \spaceD{^i} \dlapse{_i} + \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}} \dlapse{_{j)}} +
+    \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}}
+  \dshift{^k_{j)}} - \frac{2}{3} \Ktf*{_i_j} \dshift{^k_k},\\
+  \pt \Gam*{^i} ={} & - 2 \Ktf*{^i^j} \dlapse{_j} + 2 \lapse \left[
+    \Gam*{^i_j_k} \Ktf*{^j^k} - \frac{3}{2} \Ktf*{^i^j} \dchi*{_j} -
+    \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} \dshift{^i_k} + \frac{1}{3}
+  \smetric*{^i^j} \Partial{_j} \dshift{^k_k} \notag\\
+  & {} + \shift{^j} \Partial{_j} \Gam*{^i} - \Gamd*{^j} \dshift{^i_j}
+  + \frac{2}{3} \Gamd*{^i} \dshift{^j_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,\\
+  \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},
+    \end{aligned}
+    $}
+    \]
 
   \end{frame}