Following Alcubierre "Introduction to 3+1 Numerical Relativity" section 5.4
ADM without any neglected terms
Connection coefficients in terms of first-order variables:
${\Gamma^i}_{jk} = \frac{1}{2} \gamma^{il} (\gamma_{lj,k} + \gamma_{lk,j} - \gamma_{jk,l})$
$ = \gamma^{il} (d_{kjl} + d_{jkl} - d_{ljk})$
$ = {d_{kj}}^i + {d_{jk}}^i - {d^i}_{jk}$
Ricci curvature in terms of first-order variables:
$R_{ij} = {R^k}_{ikj}$
$= {\Gamma^k}_{ij,k} - {\Gamma^k}_{ik,j} + {\Gamma^k}_{lk} {\Gamma^l}_{ij} - {\Gamma^k}_{lj} {\Gamma^l}_{ik}$
$= 2 ({\Gamma^k}_{i[j,k]} + {\Gamma^k}_{l[k} {\Gamma^l}_{j]i})$
$= 2 ((\gamma^{km} \Gamma_{mi[j})_{,k]} + {\Gamma^k}_{l[k} {\Gamma^l}_{j]i})$
$= 2 (\Gamma_{mi[j} {\gamma^{km}}_{,k]} + \gamma^{km} \Gamma_{mi[j,k]} + {\Gamma^k}_{l[k} {\Gamma^l}_{j]i})$
$= 2 (-2 (d_{[j|im} + d_{im[j} - d_{mi[j}) {d_{k]}}^{mk} + \gamma^{km} (d_{[j|im} + d_{im[j} - d_{mi[j})_{,k]}
+ ({{d_{[k|}}^k}_l + {d_{l[k}}^k - {d^k}_{l[k}) ({d_{i|j]}}^l + {d_{j]i}}^l - {d^l}_{j]i}))$
$= 2 (-2 d_{[j|il} {d_{k]}}^{lk} - 2 d_{il[j} {d_{k]}}^{lk} + 2 d_{li[j} {d_{k]}}^{lk}
+ \gamma^{kl} d_{[j|il,|k]} + \gamma^{kl} d_{il[j,k]} - \gamma^{kl} d_{li[j,k]}
+ {{d_{[k|}}^k}_l {d_{i|j]}}^l + {{d_{[k|}}^k}_l {d_{j]i}}^l - {{d_{[k|}}^k}_l {d^l}_{j]i}
+ {d_{l[k}}^k {d_{i|j]}}^l + {d_{l[k}}^k {d_{j]i}}^l - {d_{l[k}}^k {d^l}_{j]i}
- {d^k}_{l[k} {d_{i|j]}}^l - {d^k}_{l[k} {d_{j]i}}^l + {d^k}_{l[k} {d^l}_{j]i} )$
$= 2 (\gamma^{kl} d_{[j|il,|k]} + \gamma^{kl} d_{il[j,k]} - \gamma^{kl} d_{li[j,k]}
- d_{[j|il} {d_{k]}}^{lk} - d_{il[j} {d_{k]}}^{lk} + d_{li[j} {d_{k]}}^{kl}
+ {d_{l[k}}^k {d_{i|j]}}^l + {d_{l[k}}^k {d_{j]i}}^l - {d_{l[k}}^k {d^l}_{j]i}
- {d^k}_{l[k} {d_{i|j]}}^l - {d^k}_{l[k} {d_{j]i}}^l + {d^k}_{l[k} {d^l}_{j]i} )$
$= \gamma^{kl} d_{jil,k} - \gamma^{kl} d_{kil,j}
+ \gamma^{kl} d_{ilj,k} - \gamma^{kl} d_{ilk,j}
- \gamma^{kl} d_{lij,k} + \gamma^{kl} d_{lik,j}
- d_{jil} {d_k}^{lk} + d_{kil} {d_j}^{lk}
- d_{ilj} {d_k}^{lk} + d_{ilk} {d_j}^{lk}
+ d_{lij} {d_k}^{kl} - d_{lik} {d_j}^{kl}
+ {d_{lk}}^k {d_{ij}}^l - {d_{lj}}^k {d_{ik}}^l
+ {d_{lk}}^k {d_{ji}}^l - {d_{lj}}^k {d_{ki}}^l
- {d_{lk}}^k {d^l}_{ji} + {d_{lj}}^k {d^l}_{ki}
- {d^k}_{lk} {d_{ij}}^l + {d^k}_{lj} {d_{ik}}^l
- {d^k}_{lk} {d_{ji}}^l + {d^k}_{lj} {d_{ki}}^l
+ {d^k}_{lk} {d^l}_{ji} - {d^k}_{lj} {d^l}_{ki}$
$= \gamma^{kl} (d_{ilj,k} - d_{ikl,j} - d_{kij,l} + d_{jik,l})
+ d_{ijl} ({d^{lk}}_k - 2 {d_k}^{kl})
+ d_{jil} ({d^{lk}}_k - 2 {d_k}^{kl})
- d_{lij} ({d^{lk}}_k - 2 {d_k}^{kl})
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ d_{ikl} {d_j}^{kl}
$
ADM
substituting Bona-Masso slicing, $Q = f(\alpha) \cdot K$...
$\frac{d}{dt} \alpha = -\alpha^2 Q = -\alpha^2 f \gamma^{ij} K_{ij}$
$\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
$\frac{d}{dt} K_{ij} = -D_i D_j \alpha + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
expanded, in terms of $\partial_t$:
$\alpha_{,t} = \alpha_{,i} \beta^i - \alpha^2 f \gamma^{ij} K_{ij}$
$\gamma_{ij,t} = \gamma_{ij,k} \beta^k + \gamma_{kj} {\beta^k}_{,i} + \gamma_{ik} {\beta^k}_{,j} - 2 \alpha K_{ij}$
$K_{ij,t} = K_{ij,k} \beta^k + K_{kj} {\beta^k}_{,i} + K_{ik} {\beta^k}_{,j}
- \alpha_{,ij} + {\Gamma^k}_{ij} \alpha_{,k} + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
hyperbolic variables:
$a_k = (ln\alpha)_{,k} = \alpha_{,k} / \alpha$
$d_{kij} = \frac{1}{2} \gamma_{ij,k}$
hyperbolic variable time derivatives:
$a_{k,t} = (\alpha_{,k} / \alpha)_{,t}$
$= \alpha_{,kt} / \alpha - \alpha_{,k} \alpha_{,t} / \alpha^2$
$= (\alpha_{,i} \beta^i - \alpha^2 Q)_{,k} / \alpha
- \alpha_{,k} (\alpha_{,i} \beta^i - \alpha^2 Q) / \alpha^2$
$= ( \alpha_{,ik} \beta^i
+ \alpha_{,i} {\beta^i}_{,k}
- 2 \alpha \alpha_{,k} Q
- \alpha^2 Q_{,k}
) / \alpha
- \alpha_{,k} (
\alpha_{,i} \beta^i
- \alpha^2 Q
) / \alpha^2$
$= (\alpha a_{i,k} + \alpha_{,i} \alpha_{,k} / \alpha) \beta^i / \alpha
+ \alpha_{,i} {\beta^i}_{,k} / \alpha
- 2 \alpha_{,k} Q
- \alpha Q_{,k}
- \alpha_{,k} \alpha_{,i} \beta^i / \alpha^2
+ \alpha_{,k} Q $
$= a_{i,k} \beta^i + a_i {\beta^i}_{,k} - \alpha a_k Q - \alpha Q_{,k}$
$= a_{i,k} \beta^i - \alpha Q_{,k} + a_i {\beta^i}_{,k} - \alpha a_k Q$
Here, for the sake of diagonalizing the subsequent linearized flux matrix, I take advantage of $a_{i,k} = a_{k,i}$:
$= a_{k,i} \beta^i - \alpha Q_{,k} + a_i {\beta^i}_{,k} - \alpha a_k Q$
$\frac{d}{dt} a_k = -\alpha (a_k Q + Q_{,k}) + (a_{k,i} - a_{i,k}) \beta^i$
substituting Bona-Masso slicing, $Q = f(\alpha) \cdot K$...
$a_{k,t} = a_{k,i} \beta^i - \alpha (f \gamma^{ij} K_{ij})_{,k} + a_i {\beta^i}_{,k} - \alpha a_k Q$
$= a_{k,i} \beta^i
- \alpha f \gamma^{ij} K_{ij,k}
+ a_i {\beta^i}_{,k}
- \alpha^2 f' a_k K
+ 2 \alpha f {d_k}^{ij} K_{ij}
- \alpha a_k f K
$
Now $d_{kij}$, which is the first derivative variable of $\gamma_{ij}$:
$d_{kij,t} = \frac{1}{2} \gamma_{ij,kt}$
$= \frac{1}{2} \gamma_{ij,tk}$
$= \frac{1}{2} (\gamma_{ij,l} \beta^l
+ \gamma_{lj} {\beta^l}_{,i}
+ \gamma_{il} {\beta^l}_{,j}
- 2 \alpha K_{ij})_{,k}$
$= \frac{1}{2} \gamma_{ij,lk} \beta^l
+ \frac{1}{2} \gamma_{ij,l} {\beta^l}_{,k}
+ \frac{1}{2} \gamma_{lj,k} {\beta^l}_{,i}
+ \frac{1}{2} \gamma_{lj} {\beta^l}_{,ik}
+ \frac{1}{2} \gamma_{il,k} {\beta^l}_{,j}
+ \frac{1}{2} \gamma_{il} {\beta^l}_{,jk}
- \alpha_{,k} K_{ij}
- \alpha K_{ij,k}$
$= d_{kij,l} \beta^l
- \alpha K_{ij,k}
+ d_{lij} {\beta^l}_{,k}
+ d_{klj} {\beta^l}_{,i}
+ d_{kil} {\beta^l}_{,j}
+ \frac{1}{2} \gamma_{jl} {\beta^l}_{,ik}
+ \frac{1}{2} \gamma_{il} {\beta^l}_{,jk}
- \alpha a_k K_{ij}
$
$= d_{kij,l} \beta^l
- \alpha K_{ij,k}
+ d_{lij} {\beta^l}_{,k}
+ 2 d_{kl(i} {\beta^l}_{,j)}
+ \gamma_{l(i} {\beta^l}_{,j)k}
- \alpha a_k K_{ij}
$
$\frac{d}{dt} d_{kij} =
- \alpha K_{ij,k}
+ \gamma_{l(i} {\beta^l}_{,j)k}
- \alpha a_k K_{ij}$
Now for $K_{ij,t}$
using $a_{i,j} = a_{(i,j)}$...
$K_{ij,t} =
- \alpha \delta^m_i \delta^r_j a_{(m,r)}
+ \alpha \gamma^{kl} (
\delta^m_i \delta^p_l \delta^q_j \delta^r_k
- \delta^m_i \delta^p_k \delta^q_l \delta^r_j
- \delta^m_k \delta^p_i \delta^q_j \delta^r_l
+ \delta^m_j \delta^p_i \delta^q_k \delta^r_l
) d_{mpq,r}
+ \beta^k K_{ij,k}
+ K_{kj} {\beta^k}_{,i}
+ K_{ik} {\beta^k}_{,j}
+ \alpha (
- a_i a_j
+ {\Gamma^k}_{ij} a_k
+ d_{ijl} ({d^{lk}}_k - 2 {d_k}^{kl})
+ d_{jil} ({d^{lk}}_k - 2 {d_k}^{kl})
- d_{lij} ({d^{lk}}_k - 2 {d_k}^{kl})
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ d_{ikl} {d_j}^{kl}
+ K K_{ij}
- 2 K_{ik} {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^{pr} \delta^m_i \delta^q_j
- \gamma^{pq} \delta^m_i \delta^r_j
- \gamma^{mr} \delta^p_i \delta^q_j
+ \gamma^{qr} \delta^m_j \delta^p_i
) d_{mpq,r}
+ \beta^k K_{ij,k}
+ K_{kj} {\beta^k}_{,i}
+ K_{ik} {\beta^k}_{,j}
+ \alpha (
- a_i a_j
+ {\Gamma^k}_{ij} a_k
+ d_{ijl} ({d^{lk}}_k - 2 {d_k}^{kl})
+ d_{jil} ({d^{lk}}_k - 2 {d_k}^{kl})
- d_{lij} ({d^{lk}}_k - 2 {d_k}^{kl})
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ d_{ikl} {d_j}^{kl}
+ K K_{ij}
- 2 K_{ik} {K^k}_j
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
$=
- \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}
+ \beta^k K_{ij,k}
+ K_{kj} {\beta^k}_{,i}
+ K_{ik} {\beta^k}_{,j}
+ \alpha (
- a_i a_j
+ {\Gamma^k}_{ij} a_k
+ d_{ijl} ({d^{lk}}_k - 2 {d_k}^{kl})
+ d_{jil} ({d^{lk}}_k - 2 {d_k}^{kl})
- d_{lij} ({d^{lk}}_k - 2 {d_k}^{kl})
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ d_{ikl} {d_j}^{kl}
+ K K_{ij}
- 2 K_{ik} {K^k}_j
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
grouping symmetric terms...
$=
- \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}
+ \beta^k K_{(ij),k}
+ 2 K_{k(i} {\beta^k}_{,j)}
+ \alpha (
- a_{(i} a_{j)}
+ {\Gamma^k}_{(ij)} a_k
+ 2 d_{(ij)l} ({d^{lk}}_k - 2 {d_k}^{kl})
- d_{l(ij)} ({d^{lk}}_k - 2 {d_k}^{kl})
+ 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)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
substitute $V_k = {d_{km}}^m - {d^m}_{mk}$
$=
- \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}
+ \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
+ (2 d_{(ij)l} - d_{l(ij)}) (V^l - {d_k}^{kl})
+ 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)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
$
$=
- \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}
+ \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_{kil}
+ {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})
$
Now if I wanted, I could introduce the flux variable of $K_{ij,t}$ ...
Let ${\Lambda^k}_{ij} = {d^k}_{ij} + \delta^k_{(i} ( a_{j)} + {d_{j)m}}^m - 2 {d^m}_{mj)})$
$ = {d^k}_{ij} + \delta^k_{(i} ( a_{j)} - {d_{j)m}}^m + 2 V_{j)})$
Then ${\Lambda^k}_{ij,k} = {d^k}_{ij,k} + \delta^k_{(i} ( a_{j)} + {d_{j)m}}^m - 2 {d^m}_{mj)})_{,k}$
But I'm lazy and don't have a need to expand it and substitute pieces just yet.
Hyperbolic conservation form:
${U^i}_{,t} + {F^{ir}}_{,r} = S^i$ for the i'th conservation variable in the r'th direction.
${U^i}_{,t} + \frac{\partial F^{ir}}{\partial U^m} {U^m}_{,r} = S^i$
${U^i}_{,t} + {J^{ir}}_m {U^m}_{,r} = S^i$, for ${J^{ir}}_m = \frac{\partial F^{ir}}{\partial U^m}$
matrix form, favoring flux terms:
$\left[\matrix{
\alpha \\ \gamma_{ij} \\ a_k \\ d_{kij} \\ K_{ij} \\ \beta^i
}\right]_{,t}
+ \left[\matrix{
-\beta^r & 0 & 0 & 0 & 0 & 0 \\
0 & -(\delta_i^p \delta_j^q + \delta_i^q \delta_j^p) \beta^r & 0 & 0 & 0 & 0 \\
-\alpha^2 K (f - f' \alpha) \delta^r_k &
\alpha f K^{pq} \delta^r_k &
-\delta^m_k \beta^r &
0 &
\alpha f \gamma^{pq} \delta^r_k &
-a_m \delta^r_k \\
\alpha^2 K_{ij} \delta^r_k &
0 &
0 &
-\delta^m_k \delta^p_i \delta^q_j \beta^r &
\alpha \delta^p_i \delta^q_j \delta^r_k &
-d_{mij} \delta^r_k - 2 d_{km(i} \delta^r_{j)} \\
\alpha^2 a_{(i} \delta^r_{j)} &
\frac{1}{2} \alpha {d_{(i}}^{pq} \delta^r_{j)} &
\alpha \delta^m_{(i} \delta^r_{j)} &
\alpha (
\gamma^{(pq)} \delta^m_{(i} \delta^r_{j)}
+ \gamma^{mr} \delta^{(p}_{(i} \delta^{q)}_{j)}
- 2 \gamma^{r(p} \delta^{q)}_{(i} \delta^m_{j)}
) &
-\delta^p_i \delta^q_j \beta^r &
-2 K_{m(i} \delta^r_{j)} \\
0 & 0 & 0 & 0 & 0 & 0
}\right]
\left[\matrix{
\alpha \\ \gamma_{pq} \\ a_m \\ d_{mpq} \\ K_{pq} \\ \beta^m
}\right]_{,r}
= \left[\matrix{
-\alpha^2 f K \\
-2 \alpha K_{ij} + \gamma_{kj} {\beta^k}_{,i} + \gamma_{ki} {\beta^k}_{,j} \\
0 \\
\gamma_{l(i} {\beta^l}_{,j)k} \\
\alpha (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + V_k - {d^l}_{lk})
+ 2 \alpha ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ \alpha K K_{(ij)}
- 2 \alpha K_{(i|k} {K^k}_{j)}
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij}) \\
0
}\right]$
...but no one seems to do this, probably because the rank of the flux Jacobian isn't full, so the source-only x conservative terms of the Jacobian are absorbed into the source vector.
matrix form, favoring source terms:
$\left[\matrix{
a_k \\ d_{kij} \\ K_{ij}
}\right]_{,t}
+ \left[\matrix{
-\delta^m_k \beta^r &
0 &
\alpha f \gamma^{pq} \delta^r_k \\
0 &
-\delta^m_k \delta^p_i \delta^q_j \beta^r &
\alpha \delta^p_i \delta^q_j \delta^r_k \\
\alpha \delta^m_{(i} \delta^r_{j)} &
\alpha (
\gamma^{(pq)} \delta^m_{(i} \delta^r_{j)}
+ \gamma^{mr} \delta^{(p}_{(i} \delta^{q)}_{j)}
- 2 \gamma^{r(p} \delta^{q)}_{(i} \delta^m_{j)}
) &
-\delta^p_i \delta^q_j \beta^r
}\right]
\left[\matrix{
a_m \\ d_{mpq} \\ K_{pq}
}\right]_{,r}
= \left[\matrix{
a_i {\beta^i}_{,k}
+ \alpha (a_k K (f - \alpha f') - 2 f {d_k}^{ij} K_{ij}) \\
d_{lij} {\beta^l}_{,k}
+ 2 d_{kl(i} {\beta^l}_{,j)}
+ \gamma_{l(i} {\beta^l}_{,j)k}
- \alpha a_k K_{ij} \\
2 K_{k(i} {\beta^k}_{,j)}
+ \alpha (
- a_{(i} a_{j)}
+ (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + V_k - {d^l}_{lk})
+ 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kil}
+ {d_{(i}}^{kl} d_{j)kl}
+ K K_{(ij)}
- 2 K_{(i|k} {K^k}_{j)}
)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
}\right]$
Separating the acoustic tensor from the flux Jacobian tensor:
${J^{ir}}_m = {A^{ir}}_m + \delta^i_m (-\beta^r)$
$\left[\matrix{
-\delta^m_k \beta^r &
0 &
\alpha f \gamma^{pq} \delta^r_k \\
0 &
-\delta^m_k \delta^p_i \delta^q_j \beta^r &
\alpha \delta^p_i \delta^q_j \delta^r_k \\
\alpha \delta^m_{(i} \delta^r_{j)} &
\alpha (
\gamma^{(pq)} \delta^m_{(i} \delta^r_{j)}
+ \gamma^{mr} \delta^{(p}_{(i} \delta^{q)}_{j)}
- 2 \gamma^{r(p} \delta^{q)}_{(i} \delta^m_{j)}
) &
-\delta^p_i \delta^q_j \beta^r
}\right]
= \left[\matrix{
0 &
0 &
\alpha f \gamma^{pq} \delta^r_k \\
0 &
0 &
\alpha \delta^p_i \delta^q_j \delta^r_k \\
\alpha \delta^m_{(i} \delta^r_{j)} &
\alpha (
\gamma^{(pq)} \delta^m_{(i} \delta^r_{j)}
+ \gamma^{mr} \delta^{(p}_{(i} \delta^{q)}_{j)}
- 2 \gamma^{r(p} \delta^{q)}_{(i} \delta^m_{j)}
) &
0
}\right]
+ \left[\matrix{
\delta^m_k & 0 & 0 \\
0 & \delta^m_k \delta^p_i \delta^q_j & 0 \\
0 & 0 & \delta^p_i \delta^q_j
}\right] \cdot (-\beta^r)$
So
${A^{ir}}_m = \left[\matrix{
0 &
0 &
\alpha f \gamma^{pq} \delta^r_k \\
0 &
0 &
\alpha \delta^p_i \delta^q_j \delta^r_k \\
\alpha \delta^m_{(i} \delta^r_{j)} &
\alpha (
\gamma^{(pq)} \delta^m_{(i} \delta^r_{j)}
+ \gamma^{mr} \delta^{(p}_{(i} \delta^{q)}_{j)}
- 2 \gamma^{r(p} \delta^{q)}_{(i} \delta^m_{j)}
) &
0
}\right]$
Flux matrix in x-direction, fully written out:
$\left[\matrix{{\partial_{{t}}({a_x})}\\{\partial_{{t}}({a_y})}\\{\partial_{{t}}({a_z})}\\{\partial_{{t}}({d_{xxx}})}\\{\partial_{{t}}({d_{xxy}})}\\{\partial_{{t}}({d_{xxz}})}\\{\partial_{{t}}({d_{xyy}})}\\{\partial_{{t}}({d_{xyz}})}\\{\partial_{{t}}({d_{xzz}})}\\{\partial_{{t}}({d_{yxx}})}\\{\partial_{{t}}({d_{yxy}})}\\{\partial_{{t}}({d_{yxz}})}\\{\partial_{{t}}({d_{yyy}})}\\{\partial_{{t}}({d_{yyz}})}\\{\partial_{{t}}({d_{yzz}})}\\{\partial_{{t}}({d_{zxx}})}\\{\partial_{{t}}({d_{zxy}})}\\{\partial_{{t}}({d_{zxz}})}\\{\partial_{{t}}({d_{zyy}})}\\{\partial_{{t}}({d_{zyz}})}\\{\partial_{{t}}({d_{zzz}})}\\{\partial_{{t}}({K_{xx}})}\\{\partial_{{t}}({K_{xy}})}\\{\partial_{{t}}({K_{xz}})}\\{\partial_{{t}}({K_{yy}})}\\{\partial_{{t}}({K_{yz}})}\\{\partial_{{t}}({K_{zz}})}}\right]
+ \left[\matrix{ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &{\alpha f {\gamma^{xx}}}&{2 \alpha f {\gamma^{xy}}}&{2 \alpha f {\gamma^{xz}}}&{\alpha f {\gamma^{yy}}}&{2 \alpha f {\gamma^{yz}}}&{\alpha f {\gamma^{zz}}}\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &\alpha& \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &\alpha& \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &\alpha& \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &\alpha& \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &\alpha& \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &\alpha\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\\alpha& \cdot & \cdot & \cdot & \cdot & \cdot &{\alpha {\gamma^{yy}}}&{2 \alpha {\gamma^{yz}}}&{\alpha {\gamma^{zz}}}& \cdot &{-{\alpha {\gamma^{yy}}}}&{-{\alpha {\gamma^{yz}}}}& \cdot & \cdot & \cdot & \cdot &{-{\alpha {\gamma^{yz}}}}&{-{\alpha {\gamma^{zz}}}}& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot &{\frac{\alpha}{2}}& \cdot & \cdot & \cdot & \cdot &{-{\alpha {\gamma^{xy}}}}&{-{\alpha {\gamma^{xz}}}}& \cdot & \cdot &{\alpha {\gamma^{xy}}}&{\frac{\alpha {\gamma^{xz}}}{2}}& \cdot &{\frac{\alpha {\gamma^{yz}}}{2}}&{\frac{\alpha {\gamma^{zz}}}{2}}& \cdot &{\frac{\alpha {\gamma^{xz}}}{2}}& \cdot &{\frac{-{\alpha {\gamma^{yz}}}}{2}}&{\frac{-{\alpha {\gamma^{zz}}}}{2}}& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot &{\frac{\alpha}{2}}& \cdot & \cdot & \cdot & \cdot &{-{\alpha {\gamma^{xy}}}}&{-{\alpha {\gamma^{xz}}}}& \cdot & \cdot &{\frac{\alpha {\gamma^{xy}}}{2}}& \cdot &{\frac{-{\alpha {\gamma^{yy}}}}{2}}&{\frac{-{\alpha {\gamma^{yz}}}}{2}}& \cdot &{\frac{\alpha {\gamma^{xy}}}{2}}&{\alpha {\gamma^{xz}}}&{\frac{\alpha {\gamma^{yy}}}{2}}&{\frac{\alpha {\gamma^{yz}}}{2}}& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &{\alpha {\gamma^{xx}}}& \cdot & \cdot & \cdot &{-{\alpha {\gamma^{xx}}}}& \cdot & \cdot &{-{\alpha {\gamma^{xz}}}}& \cdot & \cdot & \cdot & \cdot &{\alpha {\gamma^{xz}}}& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &{\alpha {\gamma^{xx}}}& \cdot & \cdot & \cdot &{\frac{-{\alpha {\gamma^{xx}}}}{2}}& \cdot &{\frac{\alpha {\gamma^{xy}}}{2}}&{\frac{-{\alpha {\gamma^{xz}}}}{2}}& \cdot &{\frac{-{\alpha {\gamma^{xx}}}}{2}}& \cdot &{\frac{-{\alpha {\gamma^{xy}}}}{2}}&{\frac{\alpha {\gamma^{xz}}}{2}}& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &{\alpha {\gamma^{xx}}}& \cdot & \cdot & \cdot & \cdot & \cdot &{\alpha {\gamma^{xy}}}& \cdot & \cdot &{-{\alpha {\gamma^{xx}}}}& \cdot &{-{\alpha {\gamma^{xy}}}}& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot }\right]
\left[\matrix{{\partial_{{x}}({a_x})}\\{\partial_{{x}}({a_y})}\\{\partial_{{x}}({a_z})}\\{\partial_{{x}}({d_{xxx}})}\\{\partial_{{x}}({d_{xxy}})}\\{\partial_{{x}}({d_{xxz}})}\\{\partial_{{x}}({d_{xyy}})}\\{\partial_{{x}}({d_{xyz}})}\\{\partial_{{x}}({d_{xzz}})}\\{\partial_{{x}}({d_{yxx}})}\\{\partial_{{x}}({d_{yxy}})}\\{\partial_{{x}}({d_{yxz}})}\\{\partial_{{x}}({d_{yyy}})}\\{\partial_{{x}}({d_{yyz}})}\\{\partial_{{x}}({d_{yzz}})}\\{\partial_{{x}}({d_{zxx}})}\\{\partial_{{x}}({d_{zxy}})}\\{\partial_{{x}}({d_{zxz}})}\\{\partial_{{x}}({d_{zyy}})}\\{\partial_{{x}}({d_{zyz}})}\\{\partial_{{x}}({d_{zzz}})}\\{\partial_{{x}}({K_{xx}})}\\{\partial_{{x}}({K_{xy}})}\\{\partial_{{x}}({K_{xz}})}\\{\partial_{{x}}({K_{yy}})}\\{\partial_{{x}}({K_{yz}})}\\{\partial_{{x}}({K_{zz}})}}\right]
= ... $
characteristic polynomial:
$
\lambda^{19}
(\lambda + \alpha \sqrt{\gamma^{xx}})^3
(\lambda - \alpha \sqrt{\gamma^{xx}})^3
(\lambda + \alpha \sqrt{f \gamma^{xx}})
(\lambda - \alpha \sqrt{f \gamma^{xx}})
$
Remove the $U_{,t} = 0$ rows, and try again:
$\left[\matrix{
{\partial_{{t}}({a_x})} \\
{\partial_{{t}}({d_{xxx}})} \\
{\partial_{{t}}({d_{xxy}})} \\
{\partial_{{t}}({d_{xxz}})} \\
{\partial_{{t}}({d_{xyy}})} \\
{\partial_{{t}}({d_{xyz}})} \\
{\partial_{{t}}({d_{xzz}})} \\
{\partial_{{t}}({K_{xx}})} \\
{\partial_{{t}}({K_{xy}})} \\
{\partial_{{t}}({K_{xz}})} \\
{\partial_{{t}}({K_{yy}})} \\
{\partial_{{t}}({K_{yz}})} \\
{\partial_{{t}}({K_{zz}})}
}\right]
+ \left[\matrix{
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & {\alpha f {\gamma^{xx}}} & {2 \alpha f {\gamma^{xy}}} & {2 \alpha f {\gamma^{xz}}} & {\alpha f {\gamma^{yy}}} & {2 \alpha f {\gamma^{yz}}} & {\alpha f {\gamma^{zz}}} \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \alpha & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \alpha & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \alpha & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \alpha & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \alpha & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \alpha \\
\alpha & \cdot & \cdot & \cdot & {\alpha {\gamma^{yy}}} & {2 \alpha {\gamma^{yz}}} & {\alpha {\gamma^{zz}}} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & {-{\alpha {\gamma^{xy}}}} & {-{\alpha {\gamma^{xz}}}} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & {-{\alpha {\gamma^{xy}}}} & {-{\alpha {\gamma^{xz}}}} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & {\alpha {\gamma^{xx}}} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & {\alpha {\gamma^{xx}}} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & {\alpha {\gamma^{xx}}} & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot
}\right]
\left[\matrix{
{\partial_{{x}}({a_x})} \\
{\partial_{{x}}({d_{xxx}})} \\
{\partial_{{x}}({d_{xxy}})} \\
{\partial_{{x}}({d_{xxz}})} \\
{\partial_{{x}}({d_{xyy}})} \\
{\partial_{{x}}({d_{xyz}})} \\
{\partial_{{x}}({d_{xzz}})} \\
{\partial_{{x}}({K_{xx}})} \\
{\partial_{{x}}({K_{xy}})} \\
{\partial_{{x}}({K_{xz}})} \\
{\partial_{{x}}({K_{yy}})} \\
{\partial_{{x}}({K_{yz}})} \\
{\partial_{{x}}({K_{zz}})}
}\right] = ...$