Continuing off of the BSSNOK worksheet.
Following the $\chi = exp(-4\phi)$ option
Next the whole thing over with $\chi = exp(-4 \phi)$
$\chi_{,\mu}$
$= exp(-4 \phi)_{,\mu}$
$= -4 exp(-4 \phi) \phi_{,\mu}$
$= -4 \chi \phi_{,\mu}$
so $\phi_{,\mu} = -\frac{1}{4 \chi} \chi_{,\mu}$
$\chi_{,t}$
$= -4 \chi \phi_{,t}$
$= -4 \chi ( -\frac{1}{6} \alpha K
+ \phi_{,k} \beta^k
+ \frac{1}{6} {\beta^k}_{,k} )$
$= \chi (
\frac{2}{3} \alpha K
- 4 (
-\frac{1}{4 \chi} \chi_{,k}
)\beta^k
- \frac{2}{3} {\beta^k}_{,k}
)$
$= \frac{2}{3} \alpha \chi K
+ \chi_{,k} \beta^k
- \frac{2}{3} \chi {\beta^k}_{,k}$
$D_i D_j \alpha = \alpha_{,ij}
- {\bar{\Gamma}^k}_{ij} \alpha_{,k}
- 2 \phi_{,j} \alpha_{,i}
- 2 \phi_{,i} \alpha_{,j}
+ 2 \gamma_{ij} \gamma^{kl} \phi_{,l} \alpha_{,k}$
$= \alpha_{,ij}
- {\bar{\Gamma}^k}_{ij} \alpha_{,k}
- 2 \alpha_{,i} (-\frac{1}{4 \chi} \chi_{,j})
- 2 \alpha_{,j} (-\frac{1}{4 \chi} \chi_{,i})
+ 2 \gamma_{ij} \gamma^{kl} (-\frac{1}{4 \chi} \chi_{,l}) \alpha_{,k}$
$= \alpha_{,ij}
- {\bar{\Gamma}^k}_{ij} \alpha_{,k}
+ \frac{1}{2 \chi} (
\alpha_{,i} \chi_{,j}
+ \alpha_{,j} \chi_{,i}
- \bar{\gamma}_{ij} \bar{\gamma}^{kl} \chi_{,l} \alpha_{,k}
)$
$\gamma^{ij} D_i D_j \alpha$
$= \gamma^{ij} \alpha_{,ij}
- exp(-4 \phi) \bar{\Gamma}^i \alpha_{,i}
+ 2 exp(-4 \phi) \bar{\gamma}^{ij} \phi_{,i} \alpha_{,j}$
$= \chi \bar{\gamma}^{ij} \alpha_{,ij}
- \chi \bar{\Gamma}^i \alpha_{,i}
- \frac{1}{2} \bar{\gamma}^{ij} \chi_{,i} \alpha_{,j}$
$(D_i D_j \alpha)^{TF}$
$= D_i D_j \alpha
- \frac{1}{3} \gamma_{ij} \gamma^{kl} D_k D_l \alpha$
$= D_i D_j \alpha
- \frac{1}{3} \gamma_{ij} (
\chi \bar{\gamma}^{kl} \alpha_{,kl}
- \chi \bar{\Gamma}^k \alpha_{,k}
- \frac{1}{2} \bar{\gamma}^{kl} \chi_{,k} \alpha_{,l}
)$
$= D_i D_j \alpha
- \frac{1}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \alpha_{,kl}
+ \frac{1}{3} \bar{\gamma}_{ij} \bar{\Gamma}^k \alpha_{,k}
+ \frac{1}{6 \chi} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \chi_{,k} \alpha_{,l}
$
Substitute $D_i D_j \alpha$
$= \alpha_{,ij}
- {\bar{\Gamma}^k}_{ij} \alpha_{,k}
+ \frac{1}{2 \chi} \alpha_{,i} \chi_{,j}
+ \frac{1}{2 \chi} \alpha_{,j} \chi_{,i}
- \frac{1}{2 \chi} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \chi_{,l} \alpha_{,k}
- \frac{1}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \alpha_{,kl}
+ \frac{1}{3} \bar{\gamma}_{ij} \bar{\Gamma}^k \alpha_{,k}
+ \frac{1}{6 \chi} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \chi_{,k} \alpha_{,l}
$
$= \alpha_{,ij}
- {\bar{\Gamma}^k}_{ij} \alpha_{,k}
+ \frac{1}{3} \bar{\gamma}_{ij} \bar{\Gamma}^k \alpha_{,k}
+ \frac{1}{2 \chi} \alpha_{,i} \chi_{,j}
+ \frac{1}{2 \chi} \alpha_{,j} \chi_{,i}
- \frac{1}{3 \chi} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \chi_{,l} \alpha_{,k}
- \frac{1}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \alpha_{,kl}
$
$R^\phi_{ij}$
$= -2 \bar{D}_i \bar{D}_j \phi
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{D}_k \bar{D}_l \phi
+ 4 \bar{D}_i \phi \bar{D}_j \phi
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{D}_k \phi \bar{D}_l \phi
$
$= -2 (\phi_{,ji} - {\bar{\Gamma}^k}_{ji} \phi_{,k})
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} (\phi_{,lk} - {\bar{\Gamma}^m}_{lk} \phi_{,m})
+ 4 \phi_{,i} \phi_{,j}
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
$= -2 ((-\frac{1}{4 \chi} \chi_{,j})_{,i} - {\bar{\Gamma}^k}_{ji} (-\frac{1}{4 \chi} \chi_{,k}))
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} ( (-\frac{1}{4 \chi} \chi_{,l})_{,k} - {\bar{\Gamma}^m}_{lk} (-\frac{1}{4 \chi} \chi_{,m}))
+ 4 (-\frac{1}{4 \chi} \chi_{,i})(-\frac{1}{4 \chi} \chi_{,j})
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{kl} (-\frac{1}{4 \chi} \chi_{,k})(-\frac{1}{4 \chi} \chi_{,l})
$
$= \frac{1}{4 \chi^2} (
- \chi_{,i} \chi_{,j}
+ 2 \chi \chi_{,ij}
- 2 \chi {\bar{\Gamma}^k}_{ji} \chi_{,k}
+ \bar{\gamma}_{ij} \bar{\gamma}^{kl} (
- 3 \chi_{,k} \chi_{,l}
+ 2 \chi \chi_{,kl}
- 2 \chi {\bar{\Gamma}^m}_{lk} \chi_{,m}
)
)
$
$(R^\phi_{ij})^{TF}$
$= (
\frac{1}{4 \chi^2} (
- \chi_{,i} \chi_{,j}
+ 2 \chi \chi_{,ij}
- 2 \chi {\bar{\Gamma}^k}_{ji} \chi_{,k}
+ \bar{\gamma}_{ij} \bar{\gamma}^{kl} (
- 3 \chi_{,k} \chi_{,l}
+ 2 \chi \chi_{,kl}
- 2 \chi {\bar{\Gamma}^m}_{lk} \chi_{,m}
)
)
)^{TF}
$
$= \frac{1}{2 \chi^2}
(- \chi_{,i} \chi_{,j}
+ 2 \chi \chi_{,ij}
- 2 \chi {\bar{\Gamma}^k}_{ji} \chi_{,k}
)^{TF}
$
Z4c formalism ... basically BSSN except with the added $\Theta$ from Z4, and replacing $K_{ij}$ with $\hat{K}_{ij}$
(Following 2011 Cao, Hilditch "Numerical stability of the Z4c formulation of general relativity")
TODO derive this from the Z4 equations: $G_{ab} = 8 \pi T_{ab} + \mathcal{L}_Z g_{ab}$
$\hat{K} = K - 2 \Theta$