This one is going by 2005 Bona, Ledvinka, Palenzuela-Luque, Zacek, "Constraint-preserving boundary conditions in the Z4 Numerical Relativity formalism".
or maybe better 2004 Bona, Palenzuela "Dynamical shift conditions for the Z4 and BSSN formalisms".
or maybe even better 2005 Bona, Luher, Palenzuela-Luque "Geometrically motivated hyperbolic coordinate condions for numerical relativity- Analysis, issues and implementation".
or maybe even better 2009 Alic, Bona, Bona-Casas "Towards a Gauge-Polyvariant Numerical Relativity Code".
Start with the EFE:
$G_{ab} = 8 \pi T_{ab}$
$G_{ab} = 8 \pi T_{ab}$
$R_{ab} - \frac{1}{2} R g_{ab} = 8 \pi T_{ab}$
contract:
$R = -8 \pi T$
substitute:
$R_{ab} = 8 \pi (T_{ab} - \frac{1}{2} T g_{ab})$
For Killing vector $Z_u$ we have $\nabla_u Z_v + \nabla_v Z_u = 0$
add this onto the EFE:
$R_{ab} + \nabla_a Z_b + \nabla_b Z_a = 8 \pi (T_{ab} - \frac{1}{2} T g_{ab})$
From there, run through the typical ADM stuff:
(Project twice, project once and contract once, contract twice)
Z4-specific variables:
$\Theta = -n_a Z^a = \alpha Z^0$
(2005 Bona defines $n_a$ as negative was 2008 Alcubierre and 2010 Baumgarte et al define $n_a$, so that's where the negative sign comes from.)
hyperbolic variables (as usual):
$a_i = \alpha_{,i} / \alpha = (ln \alpha)_{,i}$
${b^j}_i = {\beta^j}_{,i}$. Notice I'm changing the order of indexes from the typical Z4 group. I like suffix derivatives.
$d_{kij} = \frac{1}{2} \gamma_{ij,k}$
aux vars:
$d_i = d_{ijk} \gamma^{jk}$
$e_i = \gamma^{jk} d_{jki}$
parameters:
$m = $ coefficient of $(K - m \Theta)$ that goes with $\alpha_{,t}$. Set it to 2 to match the other $(K - 2 \Theta)$'s that are found in the Z4 formalism, and something about that equating to BSSN.
Z4
$\frac{d}{dt} \alpha = -f \alpha^2 (K - m \Theta)$
$\frac{d}{dt} \beta^i = -\alpha Q^i$
$\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
$\frac{d}{dt} K_{ij} = -\nabla_i \alpha_{,j} + \alpha (^3 R_{ij} + \nabla_i Z_j + \nabla_j Z_i - 2 K_{ik} {K^k}_j + (K - 2 \Theta) K_{ij} - 8 \pi S_{ij} + 4 \pi (S - \rho) \gamma_{ij} )$
$\frac{d}{dt} \Theta = \frac{1}{2} \alpha (^3 R + 2 \nabla_k Z^k + (K - 2 \Theta) K - K_{ij} K^{ij} - 16 \pi \rho) - Z^k \alpha_{,k}$
$\frac{d}{dt} Z_i = \alpha (\nabla_j ({K_i}^j - \delta_i^j K) + \Theta_{,i} - 2 {K_i}^j Z_j - 8 \pi S_i) - \Theta \alpha_{,i}$
expanded, in terms of $\partial_t$:
$\alpha_{,t} = \alpha_{,i} \beta^i - f \alpha^2 (K - m \Theta)$
${\beta^i}_{,t} = -\alpha Q^i$
$\gamma_{ij,t} = \gamma_{ij,k} \beta^k + \gamma_{kj} {\beta^k}_{,i} + \gamma_{ik} {\beta^k}_{,j} - 2 \alpha K_{ij}$
$K_{ij,t} = K_{ij,k} \beta^k + K_{kj} {\beta^k}_{,i} + K_{ik} {\beta^k}_{,j} - \nabla_i \alpha_{,j} + \alpha (^3 R_{ij} + \nabla_i Z_j + \nabla_j Z_i - 2 K_{ik} {K^k}_j + (K - 2 \Theta) K_{ij} - 8 \pi S_{ij} + 4 \pi (S - \rho) \gamma_{ij} )$
$\Theta_{,t} = \Theta_{,i} \beta^i + \frac{1}{2} \alpha (^3 R + 2 \nabla_k Z^k + (K - 2 \Theta) K - K_{ij} K^{ij} - 16 \pi \rho) - Z^k \alpha_{,k}$
$Z_{i,t} = Z_{i,j} \beta^j + Z_j {\beta^j}_{,i} + \alpha (\nabla_j ({K_i}^j - \delta_i^j K) + \Theta_{,i} - 2 {K_i}^j Z_j - 8 \pi S_i) - \Theta \alpha_{,i}$
with hyperbolic vars substituted for partials:
$\alpha_{,t} = \alpha a_i \beta^i - f \alpha^2 (K - m \Theta)$
$\gamma_{ij,t} = 2 d_{kij} \beta^k + \gamma_{ki} {b^k}_j + \gamma_{kj} {b^k}_i - 2 \alpha K_{ij}$
hyperbolic variable time derivatives in terms of flux:
$a_{k,t} = (ln \alpha)_{,kt}$
$= ((ln \alpha)_{,t})_{,k}$
$= ((ln \alpha)_{,i} \beta^i - f \alpha (K - m \Theta))_{,k}$
$= (a_i \beta^i - f \alpha (K - m \Theta))_{,k}$
shift-less:
$= (f \alpha (K - m \Theta))_{,k}$
expanded:
$= a_{i,k} \beta^i + a_i {\beta^i}_{,k} - f_{,k} \alpha (K - m \Theta) - f \alpha_{,k} (K - m \Theta) - f \alpha ((\gamma^{jl} K_{jl})_{,k} - m \Theta_{,k})$
$= a_{i,k} \beta^i + a_i {b^i}_k - f' \alpha_{,k} \alpha (K - m \Theta) - f \alpha_{,k} (K - m \Theta) - f \alpha ({\gamma^{jl}}_{,k} K_{jl} + \gamma^{jl} K_{jl,k} - m \Theta_{,k})$
$= a_{i,k} \beta^i - f \alpha \gamma^{jl} K_{jl,k} + f \alpha m \Theta_{,k} + a_i {b^i}_k - \alpha a_k (f' \alpha + f) (K - m \Theta) + 2 f \alpha {d_k}^{jl} K_{jl}$
shift-less expanded:
$= -f \alpha \gamma^{jl} K_{jl,k} + f \alpha m \Theta_{,k} - \alpha a_k (f' \alpha + f) (K - m \Theta) + 2 f \alpha {d_k}^{jl} K_{jl}$
${b^i}_{j,t} = {\beta^i}_{,jt} = {\beta^i}_{,tj}$
$ = (-\alpha Q^i)_{,t} = -\alpha_{,t} Q^i - \alpha {Q^i}_{,t}$
TODO rewrite in terms of spatial derivatives of state variables.
$d_{kij,t} = (\frac{1}{2} \gamma_{ij,k})_{,t}$
$= (\frac{1}{2} \gamma_{ij,t})_{,k}$
$= (d_{lij} \beta^l + \frac{1}{2} \gamma_{li} {b^l}_j + \frac{1}{2} \gamma_{lj} {b^l}_i - \alpha K_{ij})_{,k}$
shift-less:
$= (-\alpha K_{ij})_{,k}$
expanded:
$= d_{lij,k} \beta^l
+ d_{lij} {\beta^l}_{,k}
+ \frac{1}{2} \gamma_{li,k} {b^l}_j
+ \frac{1}{2} \gamma_{li} {b^l}_{j,k}
+ \frac{1}{2} \gamma_{lj,k} {b^l}_i
+ \frac{1}{2} \gamma_{lj} {b^l}_{i,k}
- \alpha_{,k} K_{ij}
- \alpha K_{ij,k}
$
$= d_{lij,k} \beta^l
+ \frac{1}{2} \gamma_{li} {b^l}_{j,k}
+ \frac{1}{2} \gamma_{lj} {b^l}_{i,k}
- \alpha K_{ij,k}
+ d_{lij} {b^l}_k
+ d_{kli} {b^l}_j
+ d_{klj} {b^l}_i
- \alpha a_k K_{ij}
$
shift-less expanded:
$= -\alpha K_{ij,k} - \alpha a_k K_{ij}$
hyperbolic variable time derivatives fully expanded in terms of other hyperbolic variable first order derivatives:
$K_{ij,t} = $ the same as ADM's $K_{ij,t}$ except with an additional $\alpha (\nabla_i Z_j + \nabla_j Z_i - 2 \Theta K_{ij})$
$K_{ij,t} =
\beta^k K_{(ij),k}
+ 2 K_{k(i} {b^k}_j
- \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r}
+ \alpha (
2 \gamma^{pr} \delta^m_{(i} \delta^q_{j)}
- \gamma^{pq} \delta^m_{(i} \delta^r_{j)}
- \gamma^{mr} \delta^p_{(i} \delta^q_{j)}
) d_{mpq,r}
+ \alpha (Z_{i,j} + Z_{j,i})
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + {d_{kl}}^l - 2 {d^l}_{lk} - Z_k)
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ {d_{(i}}^{kl} d_{j)kl}
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
- \alpha ({\Gamma^k}_{ij} Z_k + 2 \Theta K_{ij})
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
$\Theta_{,t} = \Theta_{,i} \beta^i
+ \frac{1}{2} \alpha (^3 R
+ 2 \nabla_k Z^k
+ (K - 2 \Theta) K
- K_{ij} K^{ij}
- 16 \pi \rho)
- Z^k \alpha_{,k}
$
$\Theta_{,t} =
\Theta_{,i} \beta^i
+ \frac{1}{2} \alpha \gamma^{kl} (d_{ilj,k} - d_{ikl,j} - d_{kij,l} + d_{jik,l})
+ \frac{1}{2} \alpha \gamma^{km} Z^j \gamma_{km,j}
+ \alpha {Z^k}_{,k}
+ \frac{1}{2} \alpha (d_{ijl} + d_{jil} - d_{lij}) (d^l - 2 e^l)
+ \alpha ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ \frac{1}{2} \alpha d_{ikl} {d_j}^{kl}
+ \frac{1}{2} \alpha (K - 2 \Theta) K
- \frac{1}{2} \alpha K_{ij} K^{ij}
- \alpha Z^k a_k
- 8 \pi \alpha \rho
$
$\Theta_{,t} =
\Theta_{,i} \beta^i
+ \frac{1}{2} \alpha \gamma^{kl} (d_{ilj,k} - d_{ikl,j} - d_{kij,l} + d_{jik,l})
+ \alpha {Z^k}_{,k}
+ \frac{1}{2} \alpha (d_{ijl} + d_{jil} - d_{lij}) (d^l - 2 e^l)
+ \alpha ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ \frac{1}{2} \alpha d_{ikl} {d_j}^{kl}
+ \frac{1}{2} \alpha (K - 2 \Theta) K
- \frac{1}{2} \alpha K_{ij} K^{ij}
+ \alpha Z_j d^j
- \alpha Z^k a_k
- 8 \pi \alpha \rho
$
shift-less expanded:
$\Theta_{,t} =
\frac{1}{2} \alpha \gamma^{kl} (d_{ilj,k} - d_{ikl,j} - d_{kij,l} + d_{jik,l})
+ \alpha {Z^k}_{,k}
+ \alpha Z_j d^j
+ \frac{1}{2} \alpha (d_{ijl} + d_{jil} - d_{lij}) (d^l - 2 e^l)
+ \alpha ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ \frac{1}{2} \alpha d_{ikl} {d_j}^{kl}
+ \frac{1}{2} \alpha (K - 2 \Theta) K
- \frac{1}{2} \alpha K_{ij} K^{ij}
- \alpha Z^k a_k
- 8 \pi \alpha \rho
$
$Z_{k,t} =
\alpha \gamma^{ij} K_{ki,j}
- \alpha \gamma^{ji} K_{ji,k}
- \alpha K_{ki} \gamma^{im} \gamma^{nj} \gamma_{mn,j}
+ \alpha K_{ji} \gamma^{jm} \gamma^{ni} \gamma_{mn,k}
+ \alpha \Theta_{,k}
+ Z_{k,j} \beta^j
+ Z_j {b^j}_k
- \alpha {\Gamma^i}_{kj} {K_i}^j
+ \alpha {\Gamma^j}_{ij} {K_k}^i
- 2 \alpha {K_k}^j Z_j
- 8 \pi \alpha S_k
- \alpha \Theta a_k
$
$Z_{k,t} =
\alpha \gamma^{ij} K_{ki,j}
- \alpha \gamma^{ji} K_{ji,k}
+ \alpha \Theta_{,k}
+ Z_{k,j} \beta^j
- 2 \alpha K_{ki} e^i
+ 2 \alpha K_{ij} {d_k}^{ij}
+ Z_j {b^j}_k
- \alpha {\Gamma^i}_{kj} {K_i}^j
+ \alpha {\Gamma^j}_{ij} {K_k}^i
- 2 \alpha {K_k}^j Z_j
- 8 \pi \alpha S_k
- \alpha \Theta a_k
$
shift-less expanded:
$Z_{k,t} =
\alpha \gamma^{ij} K_{ki,j}
- \alpha \gamma^{ji} K_{ji,k}
- \alpha K_{ki} \gamma^{im} \gamma^{nj} \gamma_{mn,j}
+ \alpha K_{ji} \gamma^{jm} \gamma^{ni} \gamma_{mn,k}
+ \alpha \Theta_{,k}
- \alpha {\Gamma^i}_{kj} {K_i}^j
+ \alpha {\Gamma^j}_{ij} {K_k}^i
- 2 \alpha {K_k}^j Z_j
- 8 \pi \alpha S_k
- \alpha \Theta a_k
$
matrix form, favoring source terms:
$\left[\matrix{
a_k \\ d_{kij} \\ K_{ij} \\ \Theta \\ Z_k \\ {b^l}_k
}\right]_{,t}
+ \left[\matrix{
-\beta^r \delta_k^m &
0 &
f \gamma^{pq} \delta^r_k &
- f \alpha m \delta^r_k &
0 &
0 \\
0 &
- \delta^m_k \delta^p_i \delta^q_j \beta^r &
\alpha \delta^p_i \delta^q_j \delta^r_k &
0 &
0 &
-\gamma_{n(i} \delta^m_{j)} \delta^r_k
\\
\alpha \delta^{(m}_i \delta^{r)}_j &
- \alpha (
\gamma^{pr} \delta_i^m \delta_j^q
- \gamma^{pq} \delta_i^m \delta_j^r
- \gamma^{mr} \delta_i^p \delta_j^q
+ \gamma^{qr} \delta_j^m \delta_i^p
) &
- \beta^r \delta^p_i \delta^q_j &
0 &
-2 \alpha \delta^m_{(i} \delta^r_{j)} &
0 \\
0 &
- \frac{1}{2} \alpha (
\gamma^{pr} \delta_i^m \delta_j^q
- \gamma^{pq} \delta_i^m \delta_j^r
- \gamma^{mr} \delta_i^p \delta_j^q
+ \gamma^{qr} \delta_j^m \delta_i^p
) &
0 &
- \beta^r &
- \alpha \delta^m_r &
0 \\
0 &
0 &
- \alpha (\gamma^{qr} \delta^p_k - \gamma^{qp} \delta^r_k) &
- \alpha \delta^r_k &
- \beta^r \delta^m_k &
0 \\
\alpha \frac{\partial Q^l}{\partial a_m} &
\alpha \frac{\partial Q^l}{\partial d_{mpq}} &
\alpha \frac{\partial Q^l}{\partial K_{pq}} &
\alpha \frac{\partial Q^l}{\partial \Theta} &
\alpha \frac{\partial Q^l}{\partial Z_m} &
\alpha \frac{\partial Q^l}{\partial {b^n}_m}
}\right]
\left[\matrix{
a_m \\ d_{mpq} \\ K_{pq} \\ \Theta \\ Z_m \\ {b^n}_m
}\right]_{,r}
= \left[\matrix{
a_i {b^i}_k
+ 2 f \alpha K_{ij} {d_k}^{ij}
- a_k \alpha (f' \alpha + f) (K - m \Theta) \\
d_{lij} {b^l}_k
+ d_{kli} {b^l}_j
+ d_{klj} {b^l}_i
- \alpha a_k K_{ij} \\
K_{kj} {b^k}_i
+ K_{ik} {b^k}_j
- \alpha a_i a_j
+ ({d_{ij}}^k + {d_{ji}}^k - {d^k}_{ij}) \alpha a_k
+ \alpha d_{ijl} ({d^{lk}}_k - 2 {d_k}^{kl})
+ \alpha d_{jil} ({d^{lk}}_k - 2 {d_k}^{kl})
- \alpha d_{lij} ({d^{lk}}_k - 2 {d_k}^{kl})
+ 2 \alpha ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ \alpha d_{ikl} {d_j}^{kl}
- 2 \alpha ({d_{ij}}^k + {d_{ji}}^k - {d^k}_{ij}) Z_k
- 2 \alpha K_{ik} {K^k}_j
+ \alpha (K - 2 \Theta) K_{ij}
- 8 \pi \alpha S_{ij}
+ 4 \pi \alpha (S - \rho) \gamma_{ij} \\
\alpha Z_j d^j
+ \frac{1}{2} \alpha d_{ijl} (d^l - 2 e^l)
+ \frac{1}{2} \alpha d_{jil} (d^l - 2 e^l)
- \frac{1}{2} \alpha d_{lij} (d^l - 2 e^l)
+ \alpha ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ \frac{1}{2} \alpha d_{ikl} {d_j}^{kl}
+ \frac{1}{2} \alpha (K - 2 \Theta) K
- \frac{1}{2} \alpha K_{ij} K^{ij}
- 8 \pi \alpha \rho
- \alpha Z^k a_k \\
Z_j {b^j}_k
- 2 \alpha K_{ki} e^i
+ 2 \alpha K_{ji} {d_k}^{ij}
- \alpha {\Gamma^i}_{kj} {K_i}^j
+ \alpha {\Gamma^j}_{ij} {K_k}^i
- 2 \alpha {K_k}^j Z_j
- 8 \pi \alpha S_k
- \alpha \Theta a_k \\
...
}\right]$
One possible shift condition, from 2005 Bona, Lehner, Palenzuela-Luque, eqn 10:
$Q^i = -\frac{1}{\alpha} \beta^k {\beta^i}_{,k} - \alpha \gamma^{ki} ( {\gamma_{jk}}^{,j} - (ln \sqrt\gamma)_{,k} - (ln \alpha)_{,k})$
$\frac{d}{dt} \beta^i = \beta^k {\beta^i}_{,k} + \alpha^2 \gamma^{ki} ({\gamma_{jk}}^{,j} - (ln \sqrt\gamma)_{,k} - (ln \alpha)_{,k})$
${\beta^i}_{,t} = \beta^k {b^i}_k + \alpha^2 \gamma^{ki} (2 \gamma^{jl} d_{ljk} - \frac{1}{2 \gamma} \gamma_{,k} - a_k)$
${\beta^i}_{,t} = \beta^k {b^i}_k + \alpha^2 \gamma^{ki} (2 \gamma^{jl} d_{ljk} - \frac{1}{2 \gamma} \gamma \gamma^{jl} \gamma_{jl,k} - a_k)$
${\beta^i}_{,t} = \beta^k {b^i}_k + \alpha^2 (2 e^i - d^i - a^i)$
${b^i}_{j,t} = {\beta^i}_{,jt} = {\beta^i}_{,tj}$
$ = ( \beta^k {b^i}_k + \alpha^2 (2 e^i - d^i - a^i) )_{,j}$
$ = {\beta^k}_{,j} {b^i}_k + \beta^k {b^i}_{k,j} + 2 \alpha \alpha_{,j} (2 e^i - d^i - a^i) + \alpha^2 (2 {e^i}_{,j} - {d^i}_{,j} - {a^i}_{,j})$
$ = {\beta^k}_{,j} {b^i}_k + \beta^k {b^i}_{k,j} + 2 \alpha^2 a_j (2 e^i - d^i - a^i) + \alpha^2 (2 {e^i}_{,j} - {d^i}_{,j} - {a^i}_{,j})$
Also, 2009 Alic, Bona, Bona-Casas, eqn 36:
I think this is the same as above.
$(\frac{\sqrt{\gamma}}{\alpha} \beta^i)_{,t} - (\frac{\sqrt{\gamma}}{\alpha} \beta^k \beta^i)_{,k} + (\alpha \sqrt{\gamma} \gamma^{ik})_{,k} = 0$