Building off that last one, what if I were to rewrite the metric, partial, and connection variables
in terms of differences with some background metric? Like how Baumgarte did wtih BSSN shortly after his 2010 book was published?
$
\def \devgamma {{ \overset{{}_\Delta}{\gamma} }}
\def \devd {{ \overset{{}_\Delta}{d} }}
\def \devGamma {{ \overset{{}_\Delta}{\Gamma} }}
\def \hatgamma {{ \hat{\gamma} }}
\def \hatGamma {{ \hat{\Gamma} }}
$
$\hatgamma_{ij} = $ background metric..
$\devgamma_{ij} = $ deviation of metric from the background metric..
$\gamma_{ij} = \devgamma_{ij} + \hatgamma_{ij} = $ metric..
$\hatgamma^{ij} = $ background metric inverse, such that $\hatgamma_{ik} \hatgamma^{kj} = \delta_i^j$.
$\gamma^{ij} = $ metric inverse, such that $\gamma_{ik} \gamma^{kj} = \delta_i^j$.
see here at the bottom:
Let ${r^a}_b = \hatgamma^{ac} \devgamma_{cb}$.
Let $r = \hatgamma^{uv} \devgamma_{uv} = {r^a}_a$.
$\gamma^{ab} = \hatgamma^{ab} - (\hatgamma^{ac} \devgamma_{cd} \hatgamma^{db}) \frac{1}{1 + r}$
$\hat{d}_{kij} = \frac{1}{2} \hatgamma_{ij,k} = $ partial derivative of background metric..
$\devd_{kij} = \frac{1}{2} \devgamma_{ij,k} = $ partial derivative of deviation of metric from background metric..
$d_{kij} = \devd_{kij} + \hat{d}_{kij} = $ partial of metric.
lower Levi-Civita zero-torsion connection associated with the background metric:
$\hatGamma_{ijk} = \frac{1}{2} ( \gamma_{ij,k} + \gamma_{ik,j} - \gamma_{jk,i} )$
$\hatGamma_{ijk} = \hat{d}_{kij} + \hat{d}_{jik} - \hat{d}_{ijk}$
lower Levi-Civita zero-torsion connection associated with the metric:
$\Gamma_{ijk} = d_{kij} + d_{jik} - d_{ijk}$
$\Gamma_{ijk} =
\devd_{kij} + \devd_{jik} - \devd_{ijk}
+ \hat{d}_{kij} + \hat{d}_{jik} - \hat{d}_{ijk}
$
$\Gamma_{ijk} = \devGamma_{ijk} + \hatGamma_{ijk}$
lower Levi-Civita zero-torsion connection deviation from the background metric:
$\devGamma_{ijk} = \devd_{kij} + \devd_{jik} - \devd_{ijk}$
upper Levi-Civita connection associated with the background metric:
${\hatGamma^i}_{jk} = \hatgamma^{il} \hatGamma_{ljk}$
Notice that this pseudo-tensor raises/lowers wrt the background metric, not the metric metric. Hopefully it's the only case.
upper Levi-Civita connection associated with the metric:
${\Gamma^i}_{jk} = \gamma^{il} \Gamma_{ljk}$
${\Gamma^i}_{jk} = \frac{1}{2} \gamma^{il} (
\gamma_{lj,k} + \gamma_{lk,j} - \gamma_{jk,l}
)$
${\Gamma^i}_{jk} = \gamma^{il} (d_{klj} + d_{jlk} - d_{ljk} )$
${\Gamma^i}_{jk} = {d_{jk}}^i + {d_{kj}}^i - {d^i}_{jk}$
${\Gamma^i}_{jk} = \gamma^{il} (
\devd_{klj} + \devd_{jlk} - \devd_{ljk}
+ \hat{d}_{klj} + \hat{d}_{jlk} - \hat{d}_{ljk}
)$
${\Gamma^i}_{jk} = \gamma^{il} (\devGamma_{ljk} + \hatGamma_{ljk})$
${\Gamma^i}_{jk} = {d_{kj}}^i + {d_{jk}}^i - {d^i}_{jk}$
$\Gamma^i = {\Gamma^{ij}}_j = 2 {d_j}^{ji} - {d^{ij}}_j$
Now, should ${\devGamma^i}_{jk}$ be the difference of upper connections, or should it be the raised difference of lower connections?
And if it is the raised difference of lower connections then what metric raises and lowers it?
I'll just say it is the difference of upper connections.
${\Gamma^i}_{jk} = {\devGamma^i}_{jk} + {\hatGamma^i}_{jk}$
So remember, don't try to use metric transforms to go between $\devGamma_{ijk}$ and ${\devGamma^i}_{jk}$.
Alright, now Z4 but using hats and deltas instead of metrics.
At least in the partials / flux terms. Don't mind the source terms.
$\alpha_{,t}$:
favoring source:
$\alpha_{,t}
=
\alpha a_i \beta^i
- f \alpha^2 (K - m \Theta)
$
favoring flux:
$\alpha_{,t}
- \beta^r \alpha_{,r}
=
- f \alpha^2 (K - m \Theta)
$
$\gamma_{ij,t}$:
favoring source:
$\gamma_{ij,t}
=
2 d_{kij} \beta^k
+ \gamma_{ki} {b^k}_j
+ \gamma_{kj} {b^k}_i
- 2 \alpha K_{ij}
$
favoring flux:
$\gamma_{ij,t}
- \delta^p_i \delta^q_j \beta^r \gamma_{pq,r}
- (
\gamma_{ni} \delta^r_j
+ \gamma_{nj} \delta^r_i
) {\beta^n}_{,r}
=
- 2 \alpha K_{ij}
$
$\devgamma_{ij,t}$:
favoring source:
$\devgamma_{ij,t}
=
- \hatgamma_{ij,t}
+ 2 d_{kij} \beta^k
+ \gamma_{ki} {b^k}_j
+ \gamma_{kj} {b^k}_i
- 2 \alpha K_{ij}
$
favoring flux:
$\devgamma_{ij,t}
- \beta^r \delta^p_i \delta^q_j \devgamma_{pq,r}
- (
\gamma_{ni} \delta^r_j
+ \gamma_{nj} \delta^r_i
) {\beta^n}_{,r}
=
- \hatgamma_{ij,t}
+ \beta^k \hatgamma_{ij,k}
- 2 \alpha K_{ij}
$
$a_{k,t}$:
favoring source:
$a_{k,t}
- \beta^r \delta^m_k a_{m,r}
+ f \alpha \gamma^{pq} \delta^r_k K_{pq,r}
- f \alpha m \delta^r_k \Theta_{,r}
=
+ a_i {b^i}_k
- \alpha a_k (f' \alpha + f) (K - m \Theta)
+ 2 f \alpha {d_k}^{jl} K_{jl}
$
favoring flux:
$a_{k,t}
+ (
- \frac{1}{2 \alpha} {b^r}_k
+ (f' \alpha + f) (K - m \Theta) \delta^r_k
) \alpha_{,r}
- f \alpha K^{pq} \delta^r_k \devgamma_{pq,r}
- \beta^r \delta^m_k a_{m,r}
+ f \alpha \gamma^{pq} \delta^r_k K_{pq,r}
- f \alpha m \delta^r_k \Theta_{,r}
- \frac{1}{2} a_n \delta^r_k {\beta^n}_{,r}
=
+ f \alpha K^{pq} \hatgamma_{pq,k}
$
$d_{kij,t}$:
favoring source;
$d_{kij,t}
- \beta^r \delta^m_k \delta^p_i \delta^q_j \devd_{mpq,r}
+ \alpha \delta^p_i \delta^q_j \delta^r_k K_{pq,r}
- \frac{1}{2} \delta^r_k (
\gamma_{ni} \delta^m_j
+ \gamma_{nj} \delta^m_i
) {b^n}_{m,r}
=
+ \hat{d}_{lij,k} \beta^l
+ d_{lij} {b^l}_k
+ d_{kli} {b^l}_j
+ d_{klj} {b^l}_i
- \alpha a_k K_{ij}
$
favoring flux:
$d_{kij,t}
+ K_{ij} \delta^r_k \alpha_{,r}
- \frac{1}{4} (
{b^r}_k \delta^p_i \delta^q_j
+ {b^q}_j \delta^p_i \delta^r_k
+ {b^p}_i \delta^q_j \delta^r_k
) \devgamma_{pq,r}
- \beta^r \delta^m_k \delta^p_i \delta^q_j \devd_{mpq,r}
+ \alpha \delta^p_i \delta^q_j \delta^r_k K_{pq,r}
- \frac{1}{2} (
d_{nij} \delta^r_k
+ d_{knj} \delta^r_i
+ d_{kin} \delta^r_j
) {\beta^n}_{,r}
- \frac{1}{2} \delta^r_k (
\gamma_{ni} \delta^m_j
+ \gamma_{nj} \delta^m_i
) {b^n}_{m,r}
=
\hat{d}_{lij,k} \beta^l
+ \frac{1}{4} \hatgamma_{ij,l} {b^l}_k
+ \frac{1}{4} \hatgamma_{li,k} {b^l}_j
+ \frac{1}{4} \hatgamma_{lj,k} {b^l}_i
$
$\devd_{kij,t}$:
favoring source;
$\devd_{kij,t}
- \beta^r \delta^m_k \delta^p_i \delta^q_j \devd_{mpq,r}
+ \alpha \delta^p_i \delta^q_j \delta^r_k K_{pq,r}
- \frac{1}{2} (
\gamma_{in} \delta^m_j \delta^r_k
+ \gamma_{nj} \delta^m_i \delta^r_k
) {b^n}_{m,k}
=
- \hat{d}_{kij,t}
+ \hat{d}_{lij,k} \beta^l
+ d_{lij} {b^l}_k
+ d_{kli} {b^l}_j
+ d_{klj} {b^l}_i
- \alpha a_k K_{ij}
$
favoring flux:
$\devd_{kij,t}
+ K_{ij} \delta^r_k \alpha_{,r}
- \frac{1}{4} (
{b^r}_k \delta^p_i \delta^q_j
+ {b^q}_j \delta^p_i \delta^r_k
+ {b^p}_i \delta^q_j \delta^r_k
) \devgamma_{pq,r}
- \beta^r \delta^m_k \delta^p_i \delta^q_j \devd_{mpq,r}
+ \alpha \delta^p_i \delta^q_j \delta^r_k K_{pq,r}
- \frac{1}{2} (
d_{nij} \delta^r_k
+ d_{knj} \delta^r_i
+ d_{kin} \delta^r_j
) {\beta^n}_{,r}
- \frac{1}{2} \delta^r_k (
\gamma_{in} \delta^m_j
+ \gamma_{nj} \delta^m_i
) {b^n}_{m,r}
=
- \hat{d}_{kij,t}
+ \hat{d}_{lij,k} \beta^l
+ \frac{1}{4} \hatgamma_{ij,l} {b^l}_k
+ \frac{1}{4} \hatgamma_{li,k} {b^l}_j
+ \frac{1}{4} \hatgamma_{lj,k} {b^l}_i
$
$K_{ij,t}$:
favoring source:
$K_{ij,t}
+ \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)}
) \devd_{mpq,r}
- \beta^r \delta^p_{(i} \delta^q_{j)} K_{pq,r}
- \alpha (
\delta^m_i \delta^r_j
+ \delta^m_j \delta^r_i
) Z_{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)}
) \hat{d}_{mpq,r}
+ 2 K_{k(i} {b^k}_{j)}
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + d_k - 2 e_k - 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})
$
favoring flux:
$K_{ij,t}
+ \frac{1}{2} (
a_i \delta^r_j
+ a_j \delta^r_i
- {\Gamma^r}_{ij}
) \alpha_{,r}
+ \alpha Z_k (
\gamma^{kp} \delta^q_j \delta^r_i
+ \gamma^{kq} \delta^p_i \delta^r_j
- \gamma^{kr} \delta^p_i \delta^q_j
) \devgamma_{pq,r}
- \frac{1}{2} \alpha (
+ \frac{1}{2} {d_i}^{pq} \delta^r_j
+ \frac{1}{2} {d_j}^{pq} \delta^r_i
+ {d^{rq}}_j \delta^p_i
- {d^{qr}}_j \delta^p_i
+ {d^{rp}}_i \delta^q_j
- {d^{pr}}_i \delta^q_j
- \frac{1}{2} (
{\Gamma^p}_{ij} \gamma^{qr}
+ {\Gamma^q}_{ij} \gamma^{pr}
- {\Gamma^r}_{ij} \gamma^{pq}
)
+ \frac{1}{2} (
\delta^p_m \delta^q_j \delta^r_i
+ \delta^p_i \delta^q_m \delta^r_j
- \delta^p_i \delta^q_j \delta^r_m
) (
a^m
+ d^m
- 2 e^m
+ 2 Z^m
)
) (\delgamma_{pq,r} + \hatgamma_{pq,r})
+ \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)}
) \devd_{mpq,r}
- \beta^r \delta^p_{(i} \delta^q_{j)} K_{pq,r}
- \alpha (
\delta^m_i \delta^r_j
+ \delta^m_j \delta^r_i
) Z_{m,r}
- (
K_{ni} \delta^r_j
+ K_{nj} \delta^r_i
) {\beta^n}_{,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)}
) \hat{d}_{mpq,r}
- \alpha Z^k \hatGamma_{kij}
+ \alpha (
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
- 2 \alpha \Theta K_{ij}
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
$\Theta_{,t}$:
favoring source:
$\Theta_{,t}
- \Theta_{,r} \beta^r
- \frac{1}{2} \alpha \gamma^{kl} (d_{ilj,k} - d_{ikl,j} - d_{kij,l} + d_{jik,l})
- \alpha \gamma^{kl} Z_{l,k}
=
- 2 \alpha e^l Z_l
+ \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
$
favoring flux: