Z4

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Z4

original form

$$ \begin{aligned} \mathcal{L}m \gamma{i j}= & -2 \alpha K_{i j}, \ \mathcal{L}m K{i j}= & -D_i D_j \alpha+\alpha\left(R_{i j}+2 D_{(i} {Z}{j)}-2 K{i k} Kk{ }j+(K-2 \Theta) K{i j}\right) \ & +4 \pi \alpha\left(\gamma_{i j}(S-\rho)-2 S_{i j}\right)-\alpha \kappa_1\left(1+\kappa_2\right) \gamma_{i j} \Theta, \ \mathcal{L}_m \Theta= & \frac{\alpha}{2}\left(R+K2-K_{i j} K{i j}-16 \pi \rho-2 \Theta K+2 D_i {Z}i-2 {Z}i D_i \ln \alpha\right)-\alpha \kappa_1\left(2+\kappa_2\right) \Theta, \ \mathcal{L}_m {Z}_i= & \alpha\left(D_j Kj{ }_i-D_i K-8 \pi j_i+D_i \Theta-2 {Z}_j K^j{ }_i-\Theta D_i \ln \alpha-\kappa_1 {Z}_i\right) . \end{aligned} $$

Omiting non-principal items (for Z4c)

$$ \begin{aligned} \mathcal{L}m \gamma{i j}= & -2 \alpha K_{i j}, \ \mathcal{L}m K{i j}= & -D_i D_j \alpha+\alpha\left(R_{i j}+2 D_{(i} {Z}{j)}-2 K{i k} Kk{ }j+(K{\color{blue}-2 \Theta}) K{i j}\right) \ & +4 \pi \alpha\left(\gamma_{i j}(S-\rho)-2 S_{i j}\right)-\alpha \kappa_1\left(1+\kappa_2\right) \gamma_{i j} \Theta, \ \mathcal{L}_m \Theta= & \frac{\alpha}{2}\left(R+K2-K_{i j} K{i j}-16 \pi \rho+2 D_i {Z}i{\color{blue}-2 \Theta K-2 {Z}i D_i \ln \alpha}\right)-\alpha \kappa_1\left(2+\kappa_2\right) \Theta, \ \mathcal{L}_m {Z}_i= & \alpha\left(D_j Kj{ }_i-D_i K-8 \pi j_i+D_i \Theta{\color{blue}-2 {Z}_j K^j{ }_i-\Theta D_i \ln \alpha}-\kappa_1 {Z}_i\right) . \end{aligned} $$

Ref: CCZ4(Ried 2023), Z4c(Bernuzzi 2010）

Reid’s version

$$ \begin{aligned} \gamma_{i j} & =e{4 \chi} \hat{\gamma}{i j} \ K{i j} & =e{4 \chi}\left(\hat{A}{i j}-\frac{1}{3} \hat{\gamma}{i j} K\right) \end{aligned} $$

Reid(also NR Shapiro), 3+1 Gourgoulhon, CCZ4_Alic, Z4c_Bernuzzi

$$ e{4\chi }=\Psi{4}=\phi{-2} {\color{grey} =\chi{-1}} \to \chi = \ln \Psi =-\frac{1}{2} \ln \phi{\color{grey}=-\frac{1}{4}\ln \chi} $$

Comformal factor

$$ {\cal L}{m} \chi=-\frac{1} {6} \alpha K+\frac{1} {6} \hat{D}{k} \beta^{k}, $$

Metric

$$ \mathcal{L}{m} \hat{\gamma}{i j}=-2 \alpha\hat{A}{i j}-\frac{2} {3} \hat{\gamma}{i j} \hat{D}_{k} \beta^{k} $$

trace of K

(derivation)

$$ \mathcal{L}m K=\gamma^{i j} \mathcal{L}m K{i j}+K{i j} \mathcal{L}_m \gamma^{i j}, $$

符合！上面两项都是一样的

BSSN: from ADM eqs

$$ {\cal L}{m} K{i j}=-D_{i} D_{j} \alpha+\alpha\left( R_{i j}+K K_{i j}-2 K_{i k} K^{k} {}_{j} \right)+4 \pi\alpha\left[ \gamma_{i j} \left( S-\rho\right)-2 S_{i j} \right] $$

then

$$ {\gamma}{i j} {\cal L}{m} K{i j}=-D_{i} D{i} \alpha+\alpha\left( R+K{2}-2 K_{i j} K{i j} \right)+4 \pi\alpha\left( S-3 \rho\right) $$

$$ \begin{aligned} \mathcal{L}m K & =\mathcal{L}m\left(\gamma^{i j} K{i j}\right) = \gamma^{i j} \mathcal{L}m K{i j} + K{i j}\mathcal{L}m \gamma{i j} \ & =\gamma{i j} \mathcal{L}m K{i j}{\color{red}+}2 \alpha K{i j} K{i j} \ & =-D2 \alpha+\alpha\left(R+K^2\right)+4 \pi \alpha(S-3 \rho) \end{aligned} $$

补充第二行：

$$ \begin{aligned} \mathcal{L}m \gamma{i j}&=-2 \alpha K_{i j} \ \gamma{i k} \gamma{j l} \mathcal{L}m \gamma{k l}&=-2 \alpha K{i j} \ \gamma{i k}[\mathcal{L}m(\underbrace{\gamma^{j l} \gamma{k l}}{\delta^j_k})-\gamma{k l} \mathcal{L}_m \gamma^{j l}]&=-2 \alpha K{i j} \ -\deltai_l\mathcal{L}_m \gamma{j l}&=-2 \alpha K{i j}. \ \mathcal{L}_m \gamma{i j}&=+\ 2 \alpha K{i j} \end{aligned} $$

Using Hamiltonian constraint

with

$$ K_{i j} K{i j}=\left(A_{i j}+\frac{K}{3} \gamma_{i j}\right)\left(A{i j}+\frac{K}{3} \gamma{i j}\right)=A_{i j} A{i j}+\frac{K2}{3}=\tilde{A}_{i j} \tilde{A}{i j}+\frac{K^2}{3} $$

$$ H=\frac{1}{2}\left(R+K2-K_{i j} K{i j}-16 \pi \rho \right)=\frac{1}{2}\left(R+\frac{2}{3}K2-\hat{A}_{i j} \hat{A}{i j}-16 \pi \rho \right)=0 $$

finally

$$ \mathcal{L}{m} K=-D^{2} \alpha+\alpha\left( \hat{A}{i j} \hat{A}{i j}+\frac{1} {3} K{2} \right)+4 \pi\alpha\left( S+\rho\right) $$

符合！

CCZ4:from Z4

$$ \begin{aligned} \mathcal{L}m K{i j}= & -D_i D_j \alpha+\alpha\left(R_{i j}+2 D_{(i} {Z}{j)}-2 K{i k} K^k{ }j+(K-2 \Theta) K{i j}\right) \ & +4 \pi \alpha\left(\gamma_{i j}(S-\rho)-2 S_{i j}\right)-\alpha \kappa_1\left(1+\kappa_2\right) \gamma_{i j} \Theta \end{aligned} $$

$$ \mathcal{L}m K=\gamma^{i j} \mathcal{L}m K{i j}+K{i j} \mathcal{L}_m \gamma^{i j}, $$

$$ \begin{aligned} \mathcal{L}m K= & -D_i Di \alpha+\alpha\left(R+2 D_i {Z}i-2 K^{l m} K{l m}+(K-2 \Theta) K\right)-8 \pi \alpha\left(S-\frac{3}{2}(S-\rho)\right) \ & -3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta+2 \alpha K_{i j} K{i j}, \ = & -D_i Di \alpha+\alpha\left(R+2 D_i {Z}i+K2-2 \Theta K+4 \pi(S-3 \rho)\right)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta, \ = & -D_i Di \alpha+\alpha R+\alpha\left(K2-2 \Theta K\right)+2 \alpha D_i {Z}^i+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta . \end{aligned} $$

符合！

Z4c: from Z4_o:

$$ \begin{aligned} \mathcal{L}m K{i j}= & -D_i D_j \alpha+\alpha\left(R_{i j}+2 D_{(i} {Z}{j)}-2 K{i k} K^k{ }j+(K{\color{blue}-2 \Theta}) K{i j}\right) \ & +4 \pi \alpha\left(\gamma_{i j}(S-\rho)-2 S_{i j}\right)-\alpha \kappa_1\left(1+\kappa_2\right) \gamma_{i j} \Theta \end{aligned} $$

$$ \begin{aligned} \mathcal{L}m K= & -D_i Di \alpha+\alpha\left(R+2 D_i {Z}i-2 K^{l m} K{l m}+(K{\color{blue}-2 \Theta}) K\right)-8 \pi \alpha\left(S-\frac{3}{2}(S-\rho)\right) \ & -3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta+2 \alpha K_{i j} K{i j}, \ = & -D_i Di \alpha+\alpha\left(R+2 D_i {Z}i+K2{\color{blue}-2 \Theta K}+4 \pi(S-3 \rho)\right)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta, \ = & -D_i Di \alpha+\alpha R+\alpha\left(K2{\color{blue}-2 \Theta K}\right)+2 \alpha D_i {Z}^i+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta . \end{aligned} $$

进一步可以转换为Z4c_Bernuzzi 2010:

$$ \begin{aligned} \mathcal{L}m K= & -D_i Di \alpha+\alpha R+\alpha\left(K2{\color{blue}-2 \Theta K}\right)+2 \alpha D_i {Z}i+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta \ = & -D_i Di \alpha+\alpha (R+K2)+2 \alpha D_i {Z}i+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta \end{aligned} $$

with Hamiltonian constraint ($\mathcal{L}m \Theta$)

$$ \mathcal{L}m \Theta= \frac{\alpha}{2}\left(R+K^2-K{i j} K{i j}-16 \pi \rho+2 D_i {Z}i{\color{blue}-2 \Theta K-2 {Z}i D_i \ln \alpha}\right)-\alpha \kappa_1\left(2+\kappa_2\right) \Theta\ {\color{blue}=0} $$

then

$$ \begin{aligned} \mathcal{L}_m K= & -D_i Di \alpha+\alpha (R+K2)+2 \alpha D_i {Z}i+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta \ = & -D_i Di \alpha+\alpha ( \hat{A}{ij}\hat{A}{ij}+\frac{1}{3}K2+16\pi\rho+2\kappa_1(2+\kappa_2)\Theta)+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta\ = & -D_i Di \alpha+\alpha ( \hat{A}^{ij}\hat{A}{ij}+\frac{1}{3}K^2)+4 \pi \alpha(S+ \rho)+ \alpha \kappa_1\left(1-\kappa_2\right) \Theta \end{aligned} $$

符合！

同理可以接着改造CCZ4,这里似乎并不可以

$$ \mathcal{L}m \Theta= \frac{\alpha}{2}\left(R+K^2-K{i j} K{i j}-16 \pi \rho+2 D_i {Z}i{\color{blue}-2 \Theta K-2 {Z}^i D_i \ln \alpha}\right)-\alpha \kappa_1\left(2+\kappa_2\right) \Theta $$

$$ \begin{aligned} \mathcal{L}m K= & -D_i Di \alpha+\alpha R+\alpha\left(K2{\color{blue}-2 \Theta K}\right)+2 \alpha D_i {Z}i+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta \ = & -D_i Di \alpha+\alpha (K{ij}K{ij}+16\pi\rho+2 {Z}i D_i \ln \alpha+2\kappa_1(2+\kappa_2)\Theta))+4 \pi \alpha(S-3 \rho)-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta \ =& -D_i Di \alpha+\alpha ( \hat{A}{ij}\hat{A}_{ij}+\frac{1}{3}K2 {\color{navy}+2 {Z}i D_i \ln \alpha})+4 \pi \alpha(S+ \rho)+ \alpha \kappa_1\left(1-\kappa_2\right) \Theta \end{aligned} $$

$\Theta$ term

CCZ4, direct from Z4

$$ \begin{aligned} \mathcal{L}m \Theta= & \frac{\alpha}{2}\left(R+K^2-K{i j} K{i j}-16 \pi \rho+2 D_i {Z}i{\color{blue}-2 \Theta K-2 {Z}i D_i \ln \alpha}\right)-\alpha \kappa_1\left(2+\kappa_2\right) \Theta \ =& \frac{\alpha}{2}\left(R+2 D_i {Z}i-\hat{A}_{i j} \hat{A}{i j}+\frac{2}{3} K2 {\color{navy}-2 \Theta K-2 {Z}^i D_i \ln \alpha}-16 \pi \rho\right)-\alpha \kappa_1\left(2+\kappa_2\right) \Theta \end{aligned} $$

符合！

$A_{ij}$ term

BSSN: $K_{ij}$展开

$$ \begin{aligned} \mathcal{L}m K{i j}= & -D_i D_j \alpha+\alpha\left(R_{i j}+K K_{i j}-2 K_{i k} Kk{ }_j\right)+4 \pi \alpha\left[\gamma_{i j}(S-\rho)-2 S_{i j}\right], \ = & -D_i D_j \alpha+\alpha\left[R_{i j}+K\left(\hat{A}{i j} e^{4 \chi}+\frac{1}{3} \hat{\gamma}{i j} e^{4 \chi} K\right)\right]+4 \pi \alpha\left[\gamma_{i j}(S-\rho)-2 S_{i j}\right] \ & -2 \hat{\gamma}{k m} \alpha e{-4 \chi}\left(\hat{A}_{i k} e{4 \chi}+\frac{1}{3} \hat{\gamma}{i k} e^{4 \chi} K\right)\left(\hat{A}{m j} e{4 \chi}+\frac{1}{3} \hat{\gamma}_{m j} e{4 \chi} K\right) \ = & -D_i D_j \alpha+\alpha\left[R_{i j}+e{4 \chi}\left(-\frac{1}{3} K \hat{A}_{i j}+\frac{1}{9} K2 \hat{\gamma}{i j}-2 \hat{A}{i k} \hat{A}^k{ }_j\right)\right]+4 \pi \alpha\left[\gamma_{i j}(S-\rho)-2 S_{i j}\right] . \end{aligned} $$

with

$$ \begin{aligned} \mathcal{L}m K{i j} & =\mathcal{L}m\left(\hat{A}{i j} e{4 \chi}+\frac{1}{3} \hat{\gamma}_{i j} e{4 \chi} K\right) \ & =e{4 \chi}\left[\mathcal{L}m \hat{A}{i j}+\frac{1}{3} \hat{\gamma}_{i j} \mathcal{L}m K+\frac{1}{3} K \mathcal{L}m \hat{\gamma}{i j}+\mathcal{L}m \chi\left(4 \tilde{A}{i j}+\frac{4}{3} \hat{\gamma}{i j} K\right)\right] \end{aligned} $$

get

$$ \begin{aligned} \mathcal{L}m \hat{A}{i j}= & e{-4 \chi}\left[-D_i D_j \alpha+ \alpha R_{i j}+4 \pi \alpha\left(\gamma_{i j}[S-\rho]-2 S_{i j}\right)\right. \ & \left.-\frac{1}{3} \gamma_{i j}\left(-D2 \alpha+\alpha\left[\hat{A}_{i j} \hat{A}{i j}+\frac{1}{3} K2+4 \pi(S+\rho)-K^2\right]\right)\right] \ & +\alpha\left[K \hat{A}{i j}-2 \hat{A}{i k} \hat{A}^k{ }_j\right]-\frac{2}{3} \hat{A}_{i j} \hat{D}_m \betam . \end{aligned} $$

substitute the Hamiltonian constraint for R to find

$$ R_{i j}=R_{i j}{\mathrm{T F}}+\frac{1} {3} \gamma_{i j} \left( 1 6 \pi\rho-\frac{2} {3} K{2}+\hat{A}{i j} \hat{A}^{i j} \right) $$

simplifying:

$$ {\cal L}{m} \hat{A}{i j}=e^{-4 \chi} \left[-D{i} D_{j} \alpha+\alpha R_{i j}-8 \pi\alpha S_{i j} \right]{\mathrm{T F}}-\frac{2} {3} \hat{A}{i j} \hat{D}{k} \beta{k}+\alpha\left( K \hat{A}{i j}-2 \hat{A}{i k} \hat{A}^{k} {}_{j} \right) $$

CCZ4 case

compare BSSN：

$$ \mathcal{L}m K{i j}= -D_i D_j \alpha+\alpha\left(R_{i j}+K K_{i j}-2 K_{i k} Kk{ }j\right)+4 \pi \alpha\left[\gamma_{i j}(S-\rho)-2 S_{i j}\right] $$

and z4:

$$ \mathcal{L}m K{i j}= -D_i D_j \alpha+\alpha\left(R{i j} {\color{green}+2 D_{(i} {Z}{j)}}-2 K{i k} Kk{ }j+(K{\color{blue}-2 \Theta}) K{i j}\right)\ +4 \pi \alpha\left(\gamma_{i j}(S-\rho)-2 S_{i j}\right) {\color{green} -\alpha \kappa_1\left(1+\kappa_2\right) \gamma_{i j} \Theta} $$

redeine

$$ \bar{R}{i j}=R{i j}+2 D_{( i} {Z}{j )}{\color{navy}-2 \Theta K{i j}} $$

then as following bssn case:

$$ \begin{aligned} \mathcal{L}m \hat{A}{i j}= & e{-4 \chi}\left[-D_i D_j \alpha+\alpha \bar{R}{i j}-8 \pi \alpha S{i j}\right]{\mathrm{TF}}-\frac{2}{3} \hat{A}{i j} \hat{D}k \beta^k+\alpha\left(K \hat{A}{i j}-2 \hat{A}{i k} \hat{A}k{ }j\right), \ = & -\frac{2}{3} \hat{A}{i j} \hat{D}_k \betak-2 \alpha \hat{A}{i k} \hat{A}k{ }j+\alpha \hat{A}{i j}(K {\color{navy}-2 \Theta}) \ & +e{-4 \chi}\left[-D_i D_j \alpha+\alpha\left(\hat{R}{i j}+\stackrel{\chi}{R}{i j}+2 D{(i} {Z}{j)}-8 \pi S{i j}\right)\right]^{\mathrm{TF}} \end{aligned} $$

Same with Z4c

the navy term come from:

$$ e{-4 \chi}\left[-2 \alpha \Theta K_{i j}\right]{\mathrm{TF}} = e^{-4 \chi}\left[-2 \alpha \Theta A_{i j}\right] = -2 \alpha \Theta \hat{A}_{i j} $$

note: the disapper of the last term in $\mathcal{L}m K{i j}$

recall $\mathcal{L}m K$ form CCZ4：

$$ \begin{aligned} \mathcal{L}m K= & -D_i Di \alpha+\alpha\left(R+2 D_i {Z}i-2 K^{l m} K{l m}+(K-2 \Theta) K\right)-8 \pi \alpha\left(S-\frac{3}{2}(S-\rho)\right) \ & {\color{green}-3 \alpha \kappa_1\left(1+\kappa_2\right) \Theta} +2 \alpha K{i j} K^{i j}, \ \end{aligned} $$

然后过程中会消去，注意是用的此式子进行的消去，不是改版

last term

CCZ4

$$ \begin{aligned} \mathcal{L}_m \hat{\Lambda}i & =\mathcal{L}_m \hat{\Delta}i+2 \mathcal{L}_m\left(\hat{\gamma}{i j} \bar{Z}_j\right) \ & =\mathcal{L}_m \hat{\Delta}i+2 \mathcal{L}_m\left(e{{\color{red}+4\chi}}\bar{Z}i\right)\ & =\mathcal{L}_m \hat{\Delta}i+8 e{4 \chi} \bar{Z}i \mathcal{L}_m \chi+2 e{4 \chi} \mathcal{L}_m \bar{Z}^i \end{aligned} $$

recall

$$ \mathcal{L}_m \bar{Z}_i=\alpha\left(D_j Kj{ }_i-D_i K-8 \pi j_i {\color{green}+D_i \Theta} {\color{blue}-2 \bar{Z}_j Kj{ }_i-\Theta D_i \ln \alpha}{\color{green}-\kappa_1 \bar{Z}_i}\right) $$

蓝标第一项直接消掉了

$$ $$

$$ $$

{\cal L}{m} \chi=-\frac{1} {6} \alpha K+\frac{1} {6} \hat{D}{k} \beta^{k} $$

C.98

$$ \mathcal{L}_m \hat{\Delta}i=\hat{\gamma}{m n} \stackrel{\circ}{D}_m \stackrel{\circ}{D}n \betai-2 \hat{D}_j\left(\alpha \hat{A}{i j}\right)+2 \alpha \hat{A}{m n} \hat{\Delta}i{ }{m n}+\frac{1}{3} \hat{\gamma}{m i} \hat{D}_m \hat{D}_n \betan+\frac{2}{3} \hat{\Delta}i \hat{D}_n \betan $$

$$ \begin{aligned} \mathcal{L}_m \hat{\Lambda}i = &\mathcal{L}_m \hat{\Delta}i + 8 e{4 \chi} \bar{Z}i \mathcal{L}m \chi+2 e{4 \chi} \mathcal{L}_m \bar{Z}i \ = &\hat{\gamma}{m n} \stackrel{\circ}{D}_m \stackrel{\circ}{D}_n \betai-2 \hat{D}j\left(\alpha \hat{A}{i j}\right)+2 \alpha \hat{A}{m n} \hat{\Delta}^i{ }{m n}+\frac{1}{3} \hat{\gamma}{m i} \hat{D}_m \hat{D}_n \betan+\frac{2}{3} \hat{\Delta}i \hat{D}_n \betan \ & +8 e{4 \chi} \bar{Z}i \left(-\frac{1} {6} \alpha K+\frac{1} {6} \hat{D}{k} \beta{k}\right) \ & +2 e{4 \chi}\alpha\left[ e{-4 x}\left(\hat{D}_l \hat{A}{li}+6 \hat{A}{li} \hat{D}_{l}\chi -\frac{2}{3} \hat{D}i k-8\pi \hat{j}i+\hat{D}i \Theta{\color{blue}-\Theta \hat{D}i \ln \alpha} -\kappa_1 \hat\gamma{ij} Z_j \right)\right] \ = &\hat{\gamma}{m n} \stackrel{\circ}{D}_m \stackrel{\circ}{D}_n \betai-2 \hat{D}j\left(\alpha \hat{A}{i j}\right)+2 \alpha \hat{A}{m n} \hat{\Delta}^i{ }{m n}+\frac{1}{3} \hat{\gamma}{m i} \hat{D}_m \hat{D}_n \betan+\frac{2}{3} \hat{\Delta}i \hat{D}_n \betan \ & +8 \hat{\gamma}{ij} {Z}{j} \left(-\frac{1} {6} \alpha K+\frac{1} {6} \hat{D}{k} \beta{k}\right) \ & +2 \alpha \left(\hat{D}l \hat{A}{li}+6 \hat{A}{li} \hat{D}{l}\chi -\frac{2}{3} \hat{D}i k-8\pi \hat{j}i+\hat{D}i \Theta{\color{blue}-\Theta \hat{D}i \ln \alpha} -\kappa_1 \hat\gamma^{ij} Z_j \right) \end{aligned} $$ final $$ \begin{aligned} \mathcal{L}_m \hat{\Lambda}i= & \hat{\gamma}{m n} \stackrel{\circ}{D}_m \stackrel{\circ}{D}_n \betai+{\color{grey}\frac{2}{3} \hat{\Lambda}i \hat{D}_n \betan}+\frac{1}{3} \hat{D}i \hat{D}_n \betan-2 \hat{A}{i k}\left(\hat{D}_k \alpha-6 \alpha \hat{D}k \chi\right)+2 \alpha \hat{A}{j k} \hat{\Delta}i{ }{j k} \ & -\frac{4}{3} \alpha \hat{D}i K+2 \hat{\gamma}{i k}\left({\color{green}\alpha \hat{D}_k \Theta} {\color{blue}-\Theta \hat{D}_k \alpha} {\color{grey}-\frac{2}{3} \alpha K \bar{Z}_k}\right)-16 \pi \alpha \hat{\gamma}{i j} j_j {\color{green}-2 \alpha \kappa_1 \hat{\gamma}{i j} \bar{Z}_j} \end{aligned} $$

退回BSSN：$2Z_j \to 0, \Lambdai \to \Deltai$

$$ \begin{aligned} \mathcal{L}_m \hat{\Delta}i= & \hat{\gamma}{m n} \stackrel{\circ}{D}_m \stackrel{\circ}{D}_n \betai-2 \hat{A}{i m} \hat{D}m \alpha+2 \alpha \hat{A}{m n} \hat{\Delta}i{ }{m n}+2 \alpha\left(6 \hat{A}{i j} \hat{D}_j \chi-\frac{2}{3} \hat{\gamma}{i j} \hat{D}_j K-8 \pi \hat{j}^i\right) \ & +\frac{1}{3}\left[\hat{D}i\left(\hat{D}_n \betan\right)+2 \hat{\Delta}i \hat{D}_n \betan\right] . \end{aligned} $$

Z4c case:

少的项：