Now introduce $\Gamma^k$ instead
$\Gamma^k = {\Gamma^k}_{ij} \gamma^{ij}$
$= \gamma^{kl} \Gamma_{lij} \gamma^{ij}$
$= \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}$
$= 2 {d_i}^{ik} - {d^{ki}}_i$
$= {d_i}^{ik} - V^k$
so
$V^k = {d_i}^{ik} - \Gamma^k$
${\Gamma^k}_{,\mu}$
$= (\gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij})_{,\mu}$
$= \gamma^{kl}_{,\mu} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}
+ \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij})_{,\mu} \gamma^{ij}
+ \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}_{,\mu} $
$= \gamma^{kl} \gamma^{ij} (2 d_{ilj,\mu} - d_{lij,\mu})
- \gamma^{km} \gamma_{mn,\mu} \gamma^{nl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}
- \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{im} \gamma_{mn,\mu} \gamma^{nj}$
$= \gamma^{kl} \gamma^{ij} (2 d_{ijl,\mu} - d_{lij,\mu})
- \gamma^{km} \gamma_{mn,\mu} \Gamma^n
- (2 d^{mnk} - d^{kmn}) \gamma_{mn,\mu}$
${\Gamma^k}_{,r}
= \gamma^{kl} \gamma^{ij} (2 d_{ilj,r} - d_{lij,r})
- \gamma^{km} \gamma_{mn,r} \gamma^{nl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}
- \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{im} \gamma_{mn,r} \gamma^{nj}$
$= (2 \gamma^{kp} \gamma^{mq} - \gamma^{km} \gamma^{pq}) d_{mpq,r}
- 2 {d_r}^{kj} \Gamma_j
- 2 {d_r}^{ij} (2 {d_{ij}}^k - {d^k}_{ij}) $
${\Gamma^k}_{,t}
= \gamma^{kl} \gamma^{ij} (2 d_{ilj,t} - d_{lij,t})
- \gamma^{km} \gamma_{mn,t} \gamma^{nl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}
- \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{im} \gamma_{mn,t} \gamma^{nj} $
$ = \gamma^{kl} \gamma^{ij} (
2 \delta^m_i \delta^p_l \delta^q_j
- \delta^m_l \delta^p_i \delta^q_j
) d_{mpq,t}
- (d_{jli} + d_{ilj} - d_{lij}) (
\gamma^{km} \gamma^{nl} \gamma^{ij}
+ \gamma^{kl} \gamma^{im} \gamma^{nj}
) \gamma_{mn,t} $
$ = (
2 \gamma^{kp} \gamma^{mq}
- \gamma^{km} \gamma^{pq}
) (
d_{mpq,l} \beta^l
- \alpha K_{pq,m}
+ d_{lpq} {\beta^l}_{,m}
+ 2 d_{ml(p} {\beta^l}_{,q)}
+ \gamma_{l(p} {\beta^l}_{,q)m}
- \alpha a_m K_{pq}
)
- (d_{jli} + d_{ilj} - d_{lij}) (
\gamma^{km} \gamma^{nl} \gamma^{ij}
+ \gamma^{kl} \gamma^{im} \gamma^{nj}
) (
\gamma_{mn,p} \beta^p
+ 2 \gamma_{p(m} {\beta^p}_{,n)}
- 2 \alpha K_{mn}
)$
$ =
2 \gamma^{kp} \gamma^{mq} d_{mpq,l} \beta^l
- \gamma^{km} \gamma^{pq} d_{mpq,l} \beta^l
- 2 \gamma^{kp} \gamma^{mq} \alpha K_{pq,m}
+ \gamma^{km} \gamma^{pq} \alpha K_{pq,m}
+ 2 \gamma^{kp} \gamma^{mq} d_{lpq} {\beta^l}_{,m}
- \gamma^{km} \gamma^{pq} d_{lpq} {\beta^l}_{,m}
+ 2 \gamma^{kp} \gamma^{mq} 2 d_{ml(p} {\beta^l}_{,q)}
- \gamma^{km} \gamma^{pq} 2 d_{ml(p} {\beta^l}_{,q)}
+ 2 \gamma^{kp} \gamma^{mq} \gamma_{l(p} {\beta^l}_{,q)m}
- \gamma^{km} \gamma^{pq} \gamma_{l(p} {\beta^l}_{,q)m}
- 2 \gamma^{kp} \gamma^{mq} \alpha a_m K_{pq}
+ \gamma^{km} \gamma^{pq} \alpha a_m K_{pq}
- d_{jli} \gamma^{km} \gamma^{nl} \gamma^{ij} \gamma_{mn,p} \beta^p
- d_{jli} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \gamma_{p(m} {\beta^p}_{,n)}
+ d_{jli} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \alpha K_{mn}
- d_{ilj} \gamma^{km} \gamma^{nl} \gamma^{ij} \gamma_{mn,p} \beta^p
- d_{ilj} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \gamma_{p(m} {\beta^p}_{,n)}
+ d_{ilj} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \alpha K_{mn}
+ d_{lij} \gamma^{km} \gamma^{nl} \gamma^{ij} \gamma_{mn,p} \beta^p
+ d_{lij} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \gamma_{p(m} {\beta^p}_{,n)}
- d_{lij} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \alpha K_{mn}
- d_{jli} \gamma^{kl} \gamma^{im} \gamma^{nj} \gamma_{mn,p} \beta^p
- d_{jli} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \gamma_{p(m} {\beta^p}_{,n)}
+ d_{jli} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \alpha K_{mn}
- d_{ilj} \gamma^{kl} \gamma^{im} \gamma^{nj} \gamma_{mn,p} \beta^p
- d_{ilj} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \gamma_{p(m} {\beta^p}_{,n)}
+ d_{ilj} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \alpha K_{mn}
+ d_{lij} \gamma^{kl} \gamma^{im} \gamma^{nj} \gamma_{mn,p} \beta^p
+ d_{lij} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \gamma_{p(m} {\beta^p}_{,n)}
- d_{lij} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \alpha K_{mn}
$
$ =
(2 \gamma^{kp} \gamma^{mq} - \gamma^{km} \gamma^{pq}) \beta^r d_{mpq,r}
- \alpha (2 \gamma^{kp} \gamma^{qr} - \gamma^{kr} \gamma^{pq}) K_{pq,r}
- 2 ({d_l}^{kr} + {d^{rk}}_l) {\beta^l}_{,r}
- (2 {d_i}^{ir} - {d^{ri}}_i) \delta^k_l {\beta^l}_{,r}
+ {\beta^k}_{,ij} \gamma^{ij}
- 2 {d_l}^{jk} (2 {d^i}_{ij} - {d_{ji}}^i) \beta^l
- 2 d_{lij} (2 d^{ijk} - d^{kij}) \beta^l
- 2 \alpha a_i K^{ki}
+ \alpha a^k K
+ 4 \alpha {d_i}^{il} {K^k}_l
- 2 \alpha {d^{li}}_i {K^k}_l
+ 4 \alpha d^{ijk} K_{ij}
- 2 \alpha d^{kij} K_{ij}
$
...and for the lower form
$\Gamma_k = \Gamma_{kij} \gamma^{ij}$
$= (d_{jki} + d_{ikj} - d_{kij}) \gamma^{ij}$
$= 2 {d^i}_{ik} - {d_{ki}}^i$
$= {d^i}_{ik} - V_k$
${\Gamma_k}_{,\mu}$
$= {\gamma_{ki} \Gamma^i}_{,\mu}$
$= \gamma_{ki,\mu} \Gamma^i + \gamma_{ki} \Gamma^{i,\mu}$
$\Gamma_{k,r} = \gamma_{ki,r} \Gamma^i + \gamma_{ki} {\Gamma^i}_{,r}$
$= 2 d_{rki} \Gamma^i + \gamma_{ki} (
(2 \gamma^{ip} \gamma^{mq} - \gamma^{im} \gamma^{pq}) d_{mpq,r}
- 2 {d_r}^{ij} \Gamma_j
- 2 {d_r}^{kj} (2 {d_{kj}}^i - {d^i}_{kj})
)$
$= 2 d_{rki} \Gamma^i
+ (2 \delta^p_k \gamma^{mq} - \delta^m_k \gamma^{pq}) d_{mpq,r}
- 2 d_{rkj} \Gamma^j
- 2 {d_r}^{ij} (2 d_{ijk} - d_{kij})
$
$\Gamma_{k,t} = (\gamma_{ki} \Gamma^i)_{,t}$
$= \gamma_{ki,t} \Gamma^i + \gamma_{ki} {\Gamma^i}_{,t}$
$= (
\gamma_{ik,j} \beta^j + \gamma_{jk} {\beta^j}_{,i} + \gamma_{ij} {\beta^j}_{,k} - 2 \alpha K_{ik}
) \Gamma^i
+ \gamma_{ki} (
(2 \gamma^{ip} \gamma^{mq} - \gamma^{im} \gamma^{pq}) \beta^r d_{mpq,r}
- \alpha (2 \gamma^{ip} \gamma^{qr} - \gamma^{ir} \gamma^{pq}) K_{pq,r}
- 2 ({d_l}^{ir} + {d^{ri}}_l) {\beta^l}_{,r}
- (2 {d_n}^{nr} - {d^{rn}}_n) \delta^i_l {\beta^l}_{,r}
+ {\beta^i}_{,nj} \gamma^{nj}
- 2 {d_l}^{ji} (2 {d^n}_{nj} - {d_{jn}}^n) \beta^l
- 2 d_{lnj} (2 d^{nji} - d^{inj}) \beta^l
- 2 \alpha a_n K^{in}
+ \alpha a^i K
+ 4 \alpha {d_n}^{nl} {K^i}_l
- 2 \alpha {d^{ln}}_n {K^i}_l
+ 4 \alpha d^{nji} K_{nj}
- 2 \alpha d^{inj} K_{nj}
)$
$=
+ \gamma_{jk} {\beta^j}_{,i} \Gamma^i
+ \gamma_{ij} {\beta^j}_{,k} \Gamma^i
- 2 \alpha K_{ik} \Gamma^i
+ (
2 d_{rki} \Gamma^i
+ (2 \delta^p_k \gamma^{mq} - \delta^m_k \gamma^{pq}) d_{mpq,r}
- 2 d_{rkj} \Gamma^j
- 2 {d_r}^{ij} (2 d_{ijk} - d_{kij})
) \beta^r
- \alpha (2 \delta^p_k \gamma^{qr} - \delta^r_k \gamma^{pq}) K_{pq,r}
- 2 ({d_{lk}}^r + {d^r}_{kl}) {\beta^l}_{,r}
- (2 {d_n}^{nr} - {d^{rn}}_n) \gamma_{kl} {\beta^l}_{,r}
+ \gamma_{kl} {\beta^l}_{,ij} \gamma^{ij}
- 2 \alpha a^i K_{ki}
+ \alpha a_k K
+ 4 \alpha {d_i}^{ij} K_{kj}
- 2 \alpha {d^{ji}}_i K_{kj}
+ 4 \alpha {d^{ij}}_k K_{ij}
- 2 \alpha {d_k}^{ij} K_{ij}
$
$=
- \alpha (2 \delta^p_k \gamma^{qr} - \delta^r_k \gamma^{pq}) K_{pq,r}
- 2 ({d_{lk}}^r + {d^r}_{kl}) {\beta^l}_{,r}
- (2 {d_n}^{nr} - {d^{rn}}_n) \gamma_{kl} {\beta^l}_{,r}
+ \delta^m_k \beta^r \Gamma_{m,r}
+ \gamma_{kl} \Gamma^r {\beta^l}_{,r}
+ \delta^r_k \Gamma_l {\beta^l}_{,r}
+ \gamma_{kl} {\beta^l}_{,ij} \gamma^{ij}
- 2 \alpha K_{ik} \Gamma^i
- 2 \alpha a^i K_{ki}
+ \alpha a_k K
+ 4 \alpha {d_i}^{ij} K_{kj}
- 2 \alpha {d^{ji}}_i K_{kj}
+ 4 \alpha {d^{ij}}_k K_{ij}
- 2 \alpha {d_k}^{ij} K_{ij}
$
neglecting shift...
$ =
- 2 \gamma^{kp} \gamma^{qr} \alpha K_{pq,r}
+ \gamma^{kr} \gamma^{pq} \alpha K_{pq,r}
- 2 \alpha a_i K^{ki}
+ \alpha a^k K
+ 2 \alpha (2 {d_i}^{il} - {d^{li}}_i) {K^k}_l
+ 2 \alpha (2 d^{ijk} - d^{kij}) K_{ij}
$
neglecting source...
$ = \alpha (\gamma^{kr} \gamma^{pq} - 2 \gamma^{kp} \gamma^{qr}) K_{pq,r}$
Now for application to $K_{ij,t}$...
$K_{ij,t} =
- \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r}
+ \alpha (
\gamma^{pq} \delta^m_{(i} \delta^r_{j)}
- \gamma^{mr} \delta^p_{(i} \delta^q_{j)}
) d_{mpq,r}
- 2 \alpha \delta^m_{(i} \delta^r_{j)} V_{m,r}
+ \beta^k K_{(ij),k}
+ 2 K_{k(i} {\beta^k}_{,j)}
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + V_k - {d^l}_{lk})
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli}
+ 4 {d_{(i}}^{kl} d_{kl|j)}
- 3 {d_{(i}}^{kl} d_{j)kl}
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
$=
- \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r}
+ \alpha (
\gamma^{pq} \delta^m_{(i} \delta^r_{j)}
- \gamma^{mr} \delta^p_{(i} \delta^q_{j)}
) d_{mpq,r}
- 2 \alpha \delta^m_{(i} \delta^r_{j)} ({d^k}_{km} - \Gamma_m)_{,r}
+ \beta^k K_{(ij),k}
+ 2 K_{k(i} {\beta^k}_{,j)}
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + {d^l}_{lk} - \Gamma_k - {d^l}_{lk})
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli}
+ 4 {d_{(i}}^{kl} d_{kl|j)}
- 3 {d_{(i}}^{kl} d_{j)kl}
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
$=
- \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r}
+ \alpha (
\gamma^{pq} \delta^m_{(i} \delta^r_{j)}
- \gamma^{mr} \delta^p_{(i} \delta^q_{j)}
) d_{mpq,r}
- 2 \alpha \delta^m_{(i} \delta^r_{j)} (\gamma^{kl} d_{klm})_{,r}
+ 2 \alpha \delta^m_{(i} \delta^r_{j)} \Gamma_{m,r}
+ \beta^k K_{(ij),k}
+ 2 K_{k(i} {\beta^k}_{,j)}
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k - \Gamma_k)
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli}
+ 4 {d_{(i}}^{kl} d_{kl|j)}
- 3 {d_{(i}}^{kl} d_{j)kl}
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
$=
- \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r}
+ \alpha (
\gamma^{pq} \delta^m_{(i} \delta^r_{j)}
- \gamma^{mr} \delta^p_{(i} \delta^q_{j)}
- 2 \gamma^{mp} \delta^q_{(i} \delta^r_{j)}
) d_{mpq,r}
+ 2 \alpha \delta^m_{(i} \delta^r_{j)} \Gamma_{m,r}
+ \beta^k K_{(ij),k}
+ 2 K_{k(i} {\beta^k}_{,j)}
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k - \Gamma_k)
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli}
- 3 {d_{(i}}^{kl} d_{j)kl}
+ 8 {d_{(i}}^{kl} d_{kl|j)}
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
matrix form
$\left[\matrix{
a_k \\ d_{kij} \\ K_{ij} \\ \Gamma_k
}\right]_{,t}
+ \left[\matrix{
-\delta^m_k \beta^r &
0 &
\alpha f \gamma^{pq} \delta^r_k &
0 \\
0 &
-\delta^m_k \delta^p_i \delta^q_j \beta^r &
\alpha \delta^p_i \delta^q_j \delta^r_k &
0 \\
\alpha \delta^m_{(i} \delta^r_{j)} &
\alpha (
\gamma^{(mr)} \delta^{(p}_{(i} \delta^{q)}_{j)}
- \gamma^{(pq)} \delta^{(m}_{(i} \delta^{r)}_{j)}
+ 2 \gamma^{mp} \delta^q_{(i} \delta^r_{j)}
) &
-\delta^p_i \delta^q_j \beta^r &
-2 \alpha \delta^m_{(i} \delta^r_{j)} \\
0 &
0 &
\alpha (2 \delta^p_k \gamma^{qr} - \delta^r_k \gamma^{pq}) &
-\delta^m_k \beta^r
}\right]
\left[\matrix{
a_m \\ d_{mpq} \\ K_{pq} \\ \Gamma_m
}\right]_{,r}
= \left[\matrix{
a_i {\beta^i}_{,k}
+ \alpha (a_k K (f - \alpha f') - 2 f {d_k}^{ij} K_{ij}) \\
d_{lij} {\beta^l}_{,k}
+ 2 d_{kl(i} {\beta^l}_{,j)}
+ \gamma_{l(i} {\beta^l}_{,j)k}
- \alpha a_k K_{ij} \\
2 K_{k(i} {\beta^k}_{,j)}
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k - \Gamma_k)
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli}
- 3 {d_{(i}}^{kl} d_{j)kl}
+ 8 {d_{(i}}^{kl} d_{kl|j)}
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
\\
- 2 ({d_{lk}}^r + {d^r}_{kl}) {\beta^l}_{,r}
- (2 {d_n}^{nr} - {d^{rn}}_n) \gamma_{kl} {\beta^l}_{,r}
+ \gamma_{kl} \Gamma^r {\beta^l}_{,r}
+ \delta^r_k \Gamma_l {\beta^l}_{,r}
+ \gamma_{kl} {\beta^l}_{,ij} \gamma^{ij}
- 2 \alpha K_{ik} \Gamma^i
- 2 \alpha a^i K_{ki}
+ \alpha a_k K
+ 4 \alpha {d_i}^{ij} K_{kj}
- 2 \alpha {d^{ji}}_i K_{kj}
+ 4 \alpha {d^{ij}}_k K_{ij}
- 2 \alpha {d_k}^{ij} K_{ij}
}\right]$