Another, hyperbolic gamma driver (2008 Alcubierre, 4.3.33 & 34):
${\beta^i}_{,t} = B^i + \beta^j {\beta^i}_{,j}$
${\beta^i}_{,t} = B^i + \beta^j {b^i}_j$
${B^i}_{,t} = \alpha^2 \zeta (
\partial_t - \beta^k \partial_k
) \tilde{\Gamma}^i - \eta B^i + \beta^j {B^i}_{,j}$
${B^i}_{,t} =
\alpha^2 \zeta (\partial_t - \beta^l \partial_l) (
\gamma^{1/3} {\Gamma^i}_{jk} \gamma^{jk}
+ \frac{1}{6} \gamma^{-2/3} \gamma^{ij} \gamma_{,j}
)
- \eta B^i
+ \beta^j {B^i}_{,j}
$
${B^i}_{,t} =
\alpha^2 \zeta (
+ (\gamma^{1/3})_{,t} {\Gamma^i}_{jk} \gamma^{jk}
+ \gamma^{1/3} {\Gamma^i}_{jk,t} \gamma^{jk}
+ \gamma^{1/3} {\Gamma^i}_{jk} {\gamma^{jk}}_{,t}
+ \frac{1}{6} {\gamma^{-2/3}}_{,t} \gamma^{ij} \gamma_{,j}
+ \frac{1}{6} \gamma^{-2/3} {\gamma^{ij}}_{,t} \gamma_{,j}
+ \frac{1}{6} \gamma^{-2/3} \gamma^{ij} \gamma_{,jt}
- (\gamma^{1/3})_{,l} {\Gamma^i}_{jk} \gamma^{jk} \beta^l
- \gamma^{1/3} {\Gamma^i}_{jk,l} \gamma^{jk} \beta^l
- \gamma^{1/3} {\Gamma^i}_{jk} {\gamma^{jk}}_{,l} \beta^l
- \frac{1}{6} {\gamma^{-2/3}}_{,l} \gamma^{ij} \gamma_{,j} \beta^l
- \frac{1}{6} \gamma^{-2/3} {\gamma^{ij}}_{,l} \gamma_{,j} \beta^l
- \frac{1}{6} \gamma^{-2/3} \gamma^{ij} \gamma_{,jl} \beta^l
)
- \eta B^i
+ \beta^j {B^i}_{,j}
$
${B^i}_{,t} =
\alpha^2 \zeta (
+ \frac{1}{3} \gamma^{1/3} \gamma^{mn} \gamma_{mn,t} {\Gamma^i}_{jk} \gamma^{jk}
+ \gamma^{1/3} {\Gamma^i}_{jk,t} \gamma^{jk}
- \gamma^{1/3} \Gamma^{ijk} \gamma_{jk,t}
- \frac{1}{9} \gamma^{1/3} \gamma^{mn} \gamma_{mn,t} \gamma^{ij} \gamma^{pq} \gamma_{pq,j}
- \frac{1}{6} \gamma^{1/3} \gamma^{im} \gamma_{mn,t} \gamma^{nj} \gamma^{pq} \gamma_{pq,j}
+ \frac{1}{6} \gamma^{1/3} \gamma^{ij} (
+ \gamma^{pq} \gamma_{pq,j} \gamma^{mn} \gamma_{mn,t}
- \gamma^{pm} \gamma^{qn} \gamma_{pq,j} \gamma_{mn,t}
+ \gamma^{mn} \gamma_{mn,tj}
)
- \frac{1}{3} \gamma^{1/3} \gamma^{mn} \gamma_{mn,l} {\Gamma^i}_{jk} \gamma^{jk} \beta^l
- \gamma^{1/3} {\Gamma^i}_{jk,l} \gamma^{jk} \beta^l
+ \gamma^{1/3} \Gamma^{ijk} \gamma_{jk,l} \beta^l
+ \frac{1}{9} \gamma^{1/3} \gamma^{mn} \gamma_{mn,l} \gamma^{ij} \gamma^{pq} \gamma_{pq,j} \beta^l
- \frac{1}{6} \gamma^{1/3} \gamma^{im} \gamma^{jn} \gamma_{mn,l} \gamma^{pq} \gamma_{pq,j} \beta^l
- \frac{1}{6} \gamma^{1/3} \gamma^{ij} (
+ \gamma^{pq} \gamma_{pq,j} \gamma^{mn} \gamma_{mn,l}
- \gamma^{mp} \gamma^{nq} \gamma_{pq,j} \gamma_{mn,l}
+ \gamma^{mn} \gamma_{mn,lj}
) \beta^l
)
- \eta B^i
+ \beta^j {B^i}_{,j}
$
${B^i}_{,t} =
\alpha^2 \zeta \gamma^{1/3} (
+ {\Gamma^i}_{jk,t} \gamma^{jk}
- \Gamma^{ijk} \gamma_{jk,t}
+ \frac{1}{3} {\Gamma^i}_{jk} \gamma^{jk} \gamma^{mn} \gamma_{mn,t}
+ \frac{1}{18} \gamma^{ij} \gamma^{pq} \gamma^{mn} \gamma_{pq,j} \gamma_{mn,t}
- \frac{1}{6} \gamma^{im} \gamma^{nj} \gamma^{pq} \gamma_{pq,j} \gamma_{mn,t}
- \frac{1}{6} \gamma^{ij} \gamma^{pm} \gamma^{qn} \gamma_{pq,j} \gamma_{mn,t}
+ \frac{1}{18} \gamma^{ij} \gamma^{pq} \gamma^{mn} \gamma_{pq,j} \gamma_{mn,l} \beta^l
- \frac{1}{3} \gamma^{ij} \gamma^{mp} \gamma^{nq} \gamma_{pq,j} \gamma_{mn,l} \beta^l
+ \frac{1}{2} \gamma^{ip} \gamma^{mq} \gamma^{jn} \gamma_{pq,j} \gamma_{mn,l} \beta^l
+ \frac{1}{2} \gamma^{ip} \gamma^{mj} \gamma^{nq} \gamma_{pq,j} \gamma_{mn,l} \beta^l
- \frac{1}{3} \gamma^{ip} \gamma^{jq} \gamma^{mn} \gamma_{pq,j} \gamma_{mn,l} \beta^l
+ \frac{1}{6} \gamma^{ij} \gamma^{mn} \gamma^{pq} \gamma_{pq,j} \gamma_{mn,l} \beta^l
- \frac{2}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} \gamma_{pq,j} \gamma_{mn,l} \beta^l
+ \gamma^{im} \gamma^{np} \gamma^{jq} \gamma_{pq,j} \gamma_{mn,l} \beta^l
+ \frac{1}{3} \gamma^{ij} \gamma^{mn} \gamma_{mn,jl} \beta^l
- \gamma^{in} \gamma^{jk} \gamma_{nj,kl} \beta^l
+ \frac{1}{6} \gamma^{ij} \gamma^{mn} \gamma_{mn,jt}
)
- \eta B^i
+ \beta^j {B^i}_{,j}
$
${B^i}_{,t} =
\alpha^2 \zeta \gamma^{1/3} (
+ {\Gamma^i}_{jk,t} \gamma^{jk}
- \Gamma^{ijk} \gamma_{jk,t}
+ \frac{1}{3} \gamma^{im} \gamma^{jq} \gamma^{pn} d_{jpq} (2 d_{rmn} \beta^r + \gamma_{rm} {b^r}_n + \gamma_{rn} {b^r}_m - 2 \alpha K_{mn})
- \frac{1}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} d_{jpq} (2 d_{rmn} \beta^r + \gamma_{rm} {b^r}_n + \gamma_{rn} {b^r}_m - 2 \alpha K_{mn})
- \frac{1}{18} \gamma^{ij} \gamma^{pq} \gamma^{mn} d_{jpq} (2 d_{rmn} \beta^r + \gamma_{rm} {b^r}_n + \gamma_{rn} {b^r}_m - 2 \alpha K_{mn})
- \frac{1}{3} \gamma^{ij} \gamma^{pm} \gamma^{qn} d_{jpq} (2 d_{rmn} \beta^r + \gamma_{rm} {b^r}_n + \gamma_{rn} {b^r}_m - 2 \alpha K_{mn})
+ \frac{1}{9} \gamma^{ij} \gamma^{pq} \gamma^{mn} d_{jpq} d_{lmn} \beta^l
- \frac{2}{3} \gamma^{ij} \gamma^{mp} \gamma^{nq} d_{jpq} d_{lmn} \beta^l
+ \gamma^{ip} \gamma^{mq} \gamma^{jn} d_{jpq} d_{lmn} \beta^l
+ \gamma^{ip} \gamma^{mj} \gamma^{nq} d_{jpq} d_{lmn} \beta^l
- \frac{2}{3} \gamma^{ip} \gamma^{jq} \gamma^{mn} d_{jpq} d_{lmn} \beta^l
+ \frac{1}{3} \gamma^{ij} \gamma^{mn} \gamma^{pq} d_{jpq} d_{lmn} \beta^l
- \frac{4}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} d_{jpq} d_{lmn} \beta^l
+ \gamma^{im} \gamma^{np} \gamma^{jq} d_{jpq} d_{lmn} \beta^l
+ \frac{1}{3} \gamma^{ij} \gamma^{mn} \gamma_{mn,jl} \beta^l
- \gamma^{in} \gamma^{jk} \gamma_{nj,kl} \beta^l
+ \frac{1}{6} \gamma^{ij} \gamma^{mn} \gamma_{mn,jt}
)
- \eta B^i
+ \beta^j {B^i}_{,j}
$
${B^i}_{,t} =
\alpha^2 \zeta \gamma^{1/3} (
+ {\Gamma^i}_{jk,t} \gamma^{jk}
- \Gamma^{ijk} \gamma_{jk,t}
+ \frac{1}{3} \gamma^{im} \gamma^{jq} \gamma^{pn} d_{jpq} 2 d_{rmn} \beta^r
- \frac{1}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} d_{jpq} 2 d_{rmn} \beta^r
- \frac{1}{18} \gamma^{ij} \gamma^{pq} \gamma^{mn} d_{jpq} 2 d_{rmn} \beta^r
- \frac{1}{3} \gamma^{ij} \gamma^{pm} \gamma^{qn} d_{jpq} 2 d_{rmn} \beta^r
- \frac{1}{3} \gamma^{im} \gamma^{jq} \gamma^{pn} d_{jpq} 2 \alpha K_{mn}
+ \frac{1}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} d_{jpq} 2 \alpha K_{mn}
+ \frac{1}{18} \gamma^{ij} \gamma^{pq} \gamma^{mn} d_{jpq} 2 \alpha K_{mn}
+ \frac{1}{3} \gamma^{ij} \gamma^{pm} \gamma^{qn} d_{jpq} 2 \alpha K_{mn}
+ \frac{1}{3} \gamma^{im} \gamma^{jq} \gamma^{pn} d_{jpq} \gamma_{rm} {b^r}_n
- \frac{1}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} d_{jpq} \gamma_{rm} {b^r}_n
- \frac{1}{18} \gamma^{ij} \gamma^{pq} \gamma^{mn} d_{jpq} \gamma_{rm} {b^r}_n
- \frac{1}{3} \gamma^{ij} \gamma^{pm} \gamma^{qn} d_{jpq} \gamma_{rm} {b^r}_n
+ \frac{1}{3} \gamma^{im} \gamma^{jq} \gamma^{pn} d_{jpq} \gamma_{rn} {b^r}_m
- \frac{1}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} d_{jpq} \gamma_{rn} {b^r}_m
- \frac{1}{18} \gamma^{ij} \gamma^{pq} \gamma^{mn} d_{jpq} \gamma_{rn} {b^r}_m
- \frac{1}{3} \gamma^{ij} \gamma^{pm} \gamma^{qn} d_{jpq} \gamma_{rn} {b^r}_m
+ \frac{1}{9} \gamma^{ij} \gamma^{pq} \gamma^{mn} d_{jpq} d_{lmn} \beta^l
- \frac{2}{3} \gamma^{ij} \gamma^{mp} \gamma^{nq} d_{jpq} d_{lmn} \beta^l
+ \gamma^{ip} \gamma^{mq} \gamma^{jn} d_{jpq} d_{lmn} \beta^l
+ \gamma^{ip} \gamma^{mj} \gamma^{nq} d_{jpq} d_{lmn} \beta^l
- \frac{2}{3} \gamma^{ip} \gamma^{jq} \gamma^{mn} d_{jpq} d_{lmn} \beta^l
+ \frac{1}{3} \gamma^{ij} \gamma^{mn} \gamma^{pq} d_{jpq} d_{lmn} \beta^l
- \frac{4}{3} \gamma^{im} \gamma^{jn} \gamma^{pq} d_{jpq} d_{lmn} \beta^l
+ \gamma^{im} \gamma^{np} \gamma^{jq} d_{jpq} d_{lmn} \beta^l
+ \frac{1}{3} \gamma^{ij} \gamma^{mn} \gamma_{mn,jl} \beta^l
- \gamma^{in} \gamma^{jk} \gamma_{nj,kl} \beta^l
+ \frac{1}{6} \gamma^{ij} \gamma^{mn} \gamma_{mn,jt}
)
- \eta B^i
+ \beta^j {B^i}_{,j}
$
... TODO FINISHME.
$\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}
$
$a_{k,t} =
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}
$
$d_{kij,t} = 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}
$
$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 \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
$
$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
$
${\beta^i}_{,t} = B^i$
${B^i}_{,t} = ...$
${b^i}_{j,t} = {\beta^i}_{,jt} = {B^i}_{,j}$
$\left[\matrix{
\alpha \\
\gamma_{ij} \\
a_k \\
d_{kij} \\
K_{ij} \\
\Theta \\
Z_k \\
\beta^l \\
{b^l}_k \\
B^l
}\right]_{,t}
+ \left[\matrix{
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 &
-\beta^r \delta_k^m &
0 &
f \gamma^{pq} \delta^r_k &
- f \alpha m \delta^r_k &
0 &
0 &
0 &
0 \\
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 &
0 &
-\gamma_{n(i} \delta^m_{j)} \delta^r_k &
0 \\
0 & 0 &
\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 &
0 \\
0 & 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 \\
0 & 0 &
0 &
0 &
- \alpha (\gamma^{qr} \delta^p_k - \gamma^{qp} \delta^r_k) &
- \alpha \delta^r_k &
- \beta^r \delta^m_k &
0 &
0 &
0 \\
0 & 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} &
0 &
\alpha \frac{\partial Q^l}{\partial {b^n}_m} &
0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\delta^l_n \delta^r_k \\
... & ... & ... & ... & ... & ... & ... & ... & ... & ...
}\right]
\left[\matrix{
\alpha \\
\gamma_{pq} \\
a_m \\
d_{mpq} \\
K_{pq} \\
\Theta \\
Z_m \\
\beta^n \\
{b^n}_m \\
B^n
}\right]_{,r}
= \left[\matrix{
\alpha a_i \beta^i - f \alpha^2 (K - m \Theta) \\
2 d_{kij} \beta^k
+ \gamma_{ki} {b^k}_j
+ \gamma_{kj} {b^k}_i
- 2 \alpha K_{ij} \\
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 \\
-\alpha Q^l \\
0 \\
...
}\right]$
favoring flux terms:
$\left[\matrix{
\alpha \\
\gamma_{ij} \\
a_k \\
d_{kij} \\
K_{ij} \\
\Theta \\
Z_k \\
\beta^l \\
{b^l}_k \\
B^l
}\right]_{,t}
+ \left[\matrix{
-\beta^r &
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 &
-\delta_i^p \delta_j^q \beta^r &
0 & 0 & 0 & 0 & 0 &
- \delta^r_i \gamma_{nj} - \delta^r_j \gamma_{ni} &
0 & 0 \\
\delta^r_k (f' \alpha + f) (K - m \Theta) &
- f \alpha K^{pq} \delta^r_k &
-\delta_k^m \beta^r &
0 &
f \gamma^{pq} \delta^r_k &
- f \alpha m \delta^r_k &
0 &
-a_n \delta^r_k
&
0 &
0 \\
K_{ij} \delta^r_k &
- \frac{1}{4} (
{b^r}_k \delta_i^p \delta_j^q
+ {b^p}_j \delta_i^q \delta_k^r
+ {b^p}_i \delta_j^q \delta_k^r
) &
0 &
- \delta^m_k \delta^p_i \delta^q_j \beta^r &
\alpha \delta^p_i \delta^q_j \delta^r_k &
0 &
0 &
- \frac{1}{2} (
d_{nij} \delta^r_k
+ d_{kni} \delta^r_j
+ d_{knj} \delta^r_i
) &
- \gamma_{n(i} \delta_{j)}^m \delta^r_k &
0 \\
\delta_{(i}^r a_{j)}
- {d_{(ij)}}^r
+ \frac{1}{2} {d^r}_{ij} &
- \frac{1}{2} \alpha (
a^q \delta_{(i}^p \delta_{j)}^r
- a^r \delta_i^p \delta_j^q
) &
\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
+ \delta_i^m \delta_j^p ({d^{qn}}_n - 2 {d_n}^{nq})
+ \delta_i^p \delta_j^m ({d^{qn}}_n - 2 {d_n}^{nq})
- \delta_i^p \delta_j^q ({d^{mn}}_n - 2 {d_n}^{nm})
+ 2 \delta_i^p {d^{mq}}_j
- 2 \delta_i^p {d^{qm}}_j
+ \delta_i^m {d_j}^{pq}
- 2 \delta_i^m \delta_j^p Z^q
- 2 \delta_j^m \delta_i^p Z^q
+ 2 \delta_i^p \delta_j^q Z^m
) &
- \beta^r \delta^p_i \delta^q_j &
0 &
-2 \alpha \delta^m_{(i} \delta^r_{j)} &
- K_{nj} \delta^r_i
- K_{ni} \delta^r_j &
0 &
0 \\
Z^r &
\alpha (
- \frac{1}{2} Z^r \gamma^{pq}
+ \frac{1}{8} d^r \delta^p_i \delta^q_j
+ \frac{1}{8} {d^r}_{ij} \gamma^{pq}
+ \frac{1}{2} {d_{(ij)}}^q \gamma^{pr}
+ \frac{1}{2} e^q \delta^p_{(i} \delta^r_{j)}
+ \frac{1}{4} {d^{pr}}_i \delta_j^q
+ \frac{1}{4} {d^{qr}}_j \delta_i^p
- \frac{1}{4} d^q \delta^p_{(i} \delta^r_{j)}
- \frac{1}{4} {d_{(ij)}}^r \gamma^{pq}
- \frac{1}{4} {d_{(i}}^{pq} \delta^r_{j)}
- \frac{1}{4} {d^q}_{ij} \gamma^{pr}
- \frac{1}{4} e^r \delta^p_i \delta^q_j
- \frac{1}{4} {d^{rp}}_i \delta_j^q
- \frac{1}{4} {d^{rq}}_j \delta_i^p
) &
0 &
\frac{1}{2} \alpha (
+ \gamma^{pq} \delta_i^m \delta_j^r
- \gamma^{pr} \delta_j^q \delta_i^m
- \gamma^{qr} \delta_i^p \delta_j^m
+ \gamma^{mr} \delta_i^p \delta_j^q
) &
0 &
- \beta^r &
- \alpha \delta^m_r &
0 &
0 &
0 \\
\delta^r_k \Theta &
- \frac{1}{2} \alpha K^{pq} \delta^r_k
- \frac{1}{2} \alpha {K_k}^r \gamma^{pq}
+ \alpha {K_k}^q \gamma^{pr} &
0 &
0 &
- \alpha (\gamma^{qr} \delta^p_k - \gamma^{qp} \delta^r_k) &
- \alpha \delta^r_k &
- \beta^r \delta^m_k &
-Z_n \delta^r_m &
0 &
0 \\
0 & 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} &
0 &
\alpha \frac{\partial Q^l}{\partial {b^n}_m} &
0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\delta^l_n \delta^r_k \\
... & ... & ... & ... & ... & ... & ... & ... & ... & ...
}\right]
\left[\matrix{
\alpha \\
\gamma_{pq} \\
a_m \\
d_{mpq} \\
K_{pq} \\
\Theta \\
Z_m \\
\beta^n \\
{b^n}_m \\
B^n
}\right]_{,r}
= \left[\matrix{
- f \alpha^2 (K - m \Theta) \\
-2 \alpha K_{ij} \\
0 \\
0 \\
- 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} \\
+ \frac{1}{2} \alpha (K - 2 \Theta) K
- \frac{1}{2} \alpha K_{ij} K^{ij}
- 8 \pi \alpha \rho
\\
- 2 \alpha K_{kj} Z^j
- 8 \pi \alpha S_k \\
-\alpha Q^l \\
0 \\
...
}\right]$