From 32048b15db2dd25d0cfcaf0483f5831a2183954f Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:45:39 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 32 ++++++++++++++++++++++++++++----
 1 file changed, 28 insertions(+), 4 deletions(-)

diff --git a/report.tex b/report.tex
index 3f74f09..31760c2 100644
--- a/report.tex
+++ b/report.tex
@@ -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}