Start with the z4.html worksheet.
Remap $S_{ij} \rightarrow \frac{1}{8 \pi} S_{ij}$ so we match all the other formalisms' conventions on here.
Rename $\tau \rightarrow \rho$ so naming conventions can match Alcubierre and Baumgarte & Shapiro.
Replace the first-order variable of $d_{ijk}$ with the first-order variable ${\Gamma^i}_{jk}$
Replace ${\Gamma^i}_{jk}$ with ${\Delta^i}_{jk} = {\Gamma^i}_{jk} - {\hat{\Gamma}^i}_{jk}$ for grid metric ${\hat{\Gamma}^i}_{jk}$ and use this as an evolution variable.
Reindex ${b_i}^j \rightarrow {b^i}_j$ to be consistent with comma-derivative order (rather than prefix operator order).
Then make sure to include hyperbolic gamma driver shift (which means adding a new state vector, $B^i$) in the final flux matrix,
and separate out the acoustic matrix from it before computing the eigensystem.
...
Ok, after giving it a go, it seems I can't rewrite the ${\Gamma^i}_{jk,t}$ as a Lie derivative so easily, because I can't easily exchange the ${\Gamma^i}_{jl,k} \beta^l$ into a ${\Gamma^i}_{jk,l} \beta^l$...
... maybe there is some identity I can use ... nope, not a pretty one, but a very ugly one.
...
Come to think of it, where is $\Delta_{ijk}$ even used in BSSN? Other than the contraction of ${B^i}_{,t}$, everywhere else it appears seems to be next to the ${\hat{\Gamma}}_{ijk}$ which cancel it and return it to the original $\Gamma_{ijk}$...
...
So this all represents Z4 with (a) hyperbolic gamma driver shift added and (b) $d_{ijk}$ replaced with $\Gamma_{ijk}$.
How about trying Z4 just with hyperbolic gamma driver, but don't screw up $d_{ijk}$? Then, last, move as many source terms into the flux Jacobian.
For Killing vector $Z_u$ we have $\nabla_u Z_v + \nabla_v Z_u = 0$
add this onto the EFE:
${}^4 R_{ab} + \nabla_a Z_b + \nabla_b Z_a = 8 \pi (T_{ab} - \frac{1}{2} T g_{ab})$
Z4-specific variables:
$\Theta = -Z^a n_a = \alpha Z^a$
stress-energy variables:
$\rho = T^{ab} n_a n_b$
$S_i = {T^a}_i n_a$
$S_{ij} = T_{ij}$
for notation's sake:
$R, R_{ij}, {\Gamma^i}_{jk}$ refer to the hypersurface Gaussian and Ricci curvature, and the hypersurface connection.
hyperbolic variables (as usual):
$a_i = (ln (\alpha))_{,i} = \alpha_{,i} / \alpha$
${b^i}_j = {\beta^i}_{,j}$
${\Delta^i}_{jk} = \frac{1}{2} \gamma^{im} (\gamma_{mj,k} + \gamma_{mk,j} - \gamma_{jk,m}) - {\hat{\Gamma}^i}_{jk} = {\Gamma^i}_{jk} - {\hat{\Gamma}^i}_{jk}$
aux vars:
${\Gamma^i}_{jk} = \frac{1}{2} \gamma^{im} (\gamma_{mj,k} + \gamma_{mk,j} - \gamma_{jk,m})$
$\Gamma^i = {\Gamma^i}_{jk} \gamma^{jk}$
$\Delta^i = {\Delta^i}_{jk} \gamma^{jk}$
Now usually in BSSN one of the last two are also evolution variables - for the sake of imposing extra vector of constraints - but with Z4 we now have the Z vector which does that job (right?) so I don't have to use these as state variables (right?).
parameters:
$m = $ damping of $\alpha$
Z4
$\frac{d}{dt} \alpha = -\alpha^2 Q$
$\frac{d}{dt} \beta^i = -\alpha Q^i$
$\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
$\frac{d}{dt} K_{ij} = -\nabla_i \alpha_{,j} + \alpha (R_{ij} + \nabla_i Z_j + \nabla_j Z_i - 2 K_{ik} {K^k}_j + (K - 2 \Theta) K_{ij} - 8 \pi (S_{ij} - \frac{1}{2} (S - \rho) \gamma_{ij} ))$
$\frac{d}{dt} \Theta = \frac{1}{2} \alpha (R + 2 \nabla_k Z^k + (K - 2 \Theta) K - K_{ij} K^{ij} - 16 \pi \rho) - Z^k \alpha_{,k}$
$\frac{d}{dt} Z_i = \alpha (\nabla_j ({K_i}^j - \delta_i^j K) + \Theta_{,i} - 2 {K_i}^j Z_j - 8 \pi S_i) - \Theta \alpha_{,i}$
Using Bona-Masso slicing condition:
$Q = f (K - m \Theta)$ for $f = f(\alpha)$
$\frac{d}{dt} \alpha = -f \alpha^2 (K - m \Theta)$
Using hyperbolic gamma driver shift:
${\beta^i}_{,t} = B^i$
so $Q^i = -\frac{1}{\alpha} B^i$
${B^i}_{,t} = k {\Delta^i}_{,t} - \eta B^i$
...where it looks like $\eta$ is the damping, and $k$ is typically $\frac{3}{4}$ in BSSN, where it is also scaled by the inverse of the conformal factor as well.
TODO solve ${\Gamma^i}_{,t}$ to find out what this is.
moving the Lie derivatives to one side, to isolate the $\partial_t$:
$\alpha_{,t} = \alpha_{,i} \beta^i - f \alpha^2 (K - m \Theta)$
${\beta^i}_{,t} = B^i$
${B^i}_{,t} = k {\Delta^i}_{,t} - \eta B^i$
$\gamma_{ij,t} = \gamma_{ij,k} \beta^k + \gamma_{kj} {\beta^k}_{,i} + \gamma_{ik} {\beta^k}_{,j} - 2 \alpha K_{ij}$
$K_{ij,t} = K_{ij,k} \beta^k + K_{kj} {\beta^k}_{,i} + K_{ik} {\beta^k}_{,j} - \nabla_i \alpha_{,j} + \alpha (R_{ij} + \nabla_i Z_j + \nabla_j Z_i - 2 K_{ik} {K^k}_j + (K - 2 \Theta) K_{ij} - 8 \pi (S_{ij} - \frac{1}{2} (S - \rho) \gamma_{ij} ))$
$\Theta_{,t} = \Theta_{,i} \beta^i + \frac{1}{2} \alpha (R + 2 \nabla_k Z^k + (K - 2 \Theta) K - K_{ij} K^{ij} - 16 \pi \rho) - Z^k \alpha_{,k}$
$Z_{i,t} = Z_{i,j} \beta^j + Z_j {\beta^j}_{,i} + \alpha (\nabla_j ({K_i}^j - \delta_i^j K) + \Theta_{,i} - 2 {K_i}^j Z_j - 8 \pi S_i) - \Theta \alpha_{,i}$
expanding all covariant derivatives in terms of connections:
$\alpha_{,t} = \alpha_{,i} \beta^i - f \alpha^2 (K - m \Theta)$
${\beta^i}_{,t} = B^i$
${B^i}_{,t} = k {\Delta^i}_{,t} - \eta B^i$
$\gamma_{ij,t} = \gamma_{ij,k} \beta^k + \gamma_{kj} {\beta^k}_{,i} + \gamma_{ik} {\beta^k}_{,j} - 2 \alpha K_{ij}$
$K_{ij,t} = K_{ij,k} \beta^k + K_{kj} {\beta^k}_{,i} + K_{ik} {\beta^k}_{,j} - \alpha_{,ij} + {\Gamma^k}_{ij} \alpha_{,k} + \alpha (R_{ij} + Z_{j,i} + Z_{i,j} - 2 {\Gamma^k}_{ij} Z_k - 2 K_{ik} {K^k}_j + (K - 2 \Theta) K_{ij} - 8 \pi (S_{ij} - \frac{1}{2} (S - \rho) \gamma_{ij} ))$
$\Theta_{,t} = \Theta_{,i} \beta^i + \frac{1}{2} \alpha (R + 2 ({Z^k}_{,k} + {\Gamma^k}_{jk} Z^j) + (K - 2 \Theta) K - K_{ij} K^{ij} - 16 \pi \rho) - Z^k \alpha_{,k}$
$Z_{i,t} = Z_{i,j} \beta^j + Z_j {\beta^j}_{,i} + \alpha ((\gamma^{jk} (K_{ik,j} - {\Gamma^m}_{ij} K_{mk} - {\Gamma^m}_{kj} K_{im}) - K_{,i}) + \Theta_{,i} - 2 {K_i}^j Z_j - 8 \pi S_i) - \Theta \alpha_{,i}$
$Z_{i,t} = Z_{i,j} \beta^j + Z_j {\beta^j}_{,i} + \alpha ((\gamma^{jk} K_{ij,k} - {\Gamma^m}_{ij} \gamma^{jk} K_{km} - \Gamma^l K_{il} - K_{,i}) + \Theta_{,i} - 2 K_{ij} Z^j - 8 \pi S_i) - \Theta \alpha_{,i}$
$\alpha_{,ij}$ in terms of hyperbolic state variable $a_i$:
$\alpha_{,i} = \alpha a_i$
$\alpha_{,(ij)} = (\alpha a_{(i})_{,j)}$
$\alpha_{,(ij)} = \alpha_{,(i} a_{j)} + \alpha a_{(i,j)}$
$\alpha_{,(ij)} = \alpha a_{(i} a_{j)} + \alpha a_{(i,j)}$
inserting hyperbolic first-order variables:
expanding all covariant derivatives in terms of connections:
$\alpha_{,t} =
\alpha a_i \beta^i
- f \alpha^2 K
+ f \alpha^2 m \Theta
$
${\beta^i}_{,t} = B^i
$
${B^i}_{,t} =
k {\Delta^i}_{,t}
- \eta B^i
$
$\gamma_{ij,t} =
2 \Gamma_{(ij)k} \beta^k
+ 2 b_{(ij)}
- 2 \alpha K_{ij}
$
$K_{ij,t} =
K_{ij,k} \beta^k
+ K_{kj} {b^k}_i
+ K_{ik} {b^k}_j
- \alpha a_{(i} a_{j)}
- \alpha a_{(i,j)}
+ \alpha {\Gamma^k}_{ij} a_k
+ \alpha (
R_{ij}
+ Z_{j,i}
+ Z_{i,j}
- 2 {\Gamma^k}_{ij} Z_k
- 2 K_{ik} {K^k}_j
+ (K - 2 \Theta) K_{ij}
+ 4 \pi (S - \rho) \gamma_{ij}
- 8 \pi S_{ij}
)
$
$\Theta_{,t} =
\Theta_{,i} \beta^i
+ \frac{1}{2} \alpha (
R
+ 2 (
{Z^k}_{,k}
+ {\Gamma^k}_{jk} Z^j
)
+ (K - 2 \Theta) K
- K_{ij} K^{ij}
- 16 \pi \rho
- 2 Z^k a_k
)
$
$Z_{i,t} =
Z_{i,j} \beta^j
+ Z_j {b^j}_i
+ \alpha (
- {\Gamma^{kj}}_j K_{ik}
+ \Gamma_{jki} K^{jk}
+ \gamma^{jk} K_{ki,j}
- \gamma^{jk} K_{kj,i}
+ \Theta_{,i}
- 2 K_{ij} Z^j
- 8 \pi S_i
)
- \alpha \Theta a_i
$
hyperbolic lapse variable time derivatives in terms of flux:
$a_{k,t} = (ln \alpha)_{,kt}$
$= a_{i,k} \beta^i
- f \alpha \gamma^{ij} K_{ij,k}
+ f \alpha m \Theta_{,k}
+ a_i {b^i}_k
- \alpha a_k (f' \alpha + f) (K - m \Theta)
+ 2 f \alpha \Gamma_{ijk} K^{ij}$
hyperbolic shift variable time derivatives in terms of flux:
${b^i}_{j,t} = {\beta^i}_{,jt} = {\beta^i}_{,tj}$
$ = {B^i}_{,j}$
Ricci curvature in terms of connection:
${R^i}_{jkl} = {\Gamma^i}_{jl,k} - {\Gamma^i}_{jk,l} + {\Gamma^i}_{mk} {\Gamma^m}_{jl} - {\Gamma^i}_{ml} {\Gamma^m}_{jk}$
$R_{jl} = {R^k}_{jkl} = {\Gamma^k}_{jl,k} - {\Gamma^k}_{jk,l} + {\Gamma^k}_{mk} {\Gamma^m}_{jl} - {\Gamma^k}_{ml} {\Gamma^m}_{jk}$
$R = \gamma^{jl} R_{jl} = \gamma^{jl} {\Gamma^k}_{jl,k} - \gamma^{jl} {\Gamma^k}_{jk,l} + {\Gamma^k}_{mk} \Gamma^m - {\Gamma^k}_{ml} \gamma^{jl} {\Gamma^m}_{jk}$
Time evolution of connection, lower:
$\Gamma_{ijk,t} =
\frac{1}{2} (
\gamma_{ij,kt}
+ \gamma_{ik,jt}
- \gamma_{jk,it}
)
$
$\Gamma_{ijk,t} =
\frac{1}{2} (
(
2 \Gamma_{(ij)l} \beta^l
+ 2 b_{(ij)}
- 2 \alpha K_{ij}
)_{,k}
+ (
2 \Gamma_{(ik)l} \beta^l
+ 2 b_{(ik)}
- 2 \alpha K_{ik}
)_{,j}
- (
2 \Gamma_{(jk)l} \beta^l
+ 2 b_{(jk)}
- 2 \alpha K_{jk}
)_{,i}
)
$
$\Gamma_{ijk,t} =
\Gamma_{(ij)l,k} \beta^l
+ \Gamma_{(ij)l} {\beta^l}_{,k}
+ b_{(ij),k}
- \alpha_{,k} K_{ij}
- \alpha K_{ij,k}
+ \Gamma_{(ik)l,j} \beta^l
+ \Gamma_{(ik)l} {\beta^l}_{,j}
+ b_{(ik),j}
- \alpha_{,j} K_{ik}
- \alpha K_{ik,j}
- \Gamma_{(jk)l,i} \beta^l
- \Gamma_{(jk)l} {\beta^l}_{,i}
- b_{(jk),i}
+ \alpha_{,i} K_{jk}
+ \alpha K_{jk,i}
$
$\Gamma_{ijk,t} =
\Gamma_{(ij)l,k} \beta^l
+ \Gamma_{(ik)l,j} \beta^l
- \Gamma_{(jk)l,i} \beta^l
+ \Gamma_{(ij)l} {b^l}_k
+ \Gamma_{(ik)l} {b^l}_j
- \Gamma_{(jk)l} {b^l}_i
+ b_{(ij),k}
+ b_{(ik),j}
- b_{(jk),i}
- \alpha K_{ij} a_k
- \alpha K_{ik} a_j
+ \alpha K_{jk} a_i
- \alpha K_{ij,k}
- \alpha K_{ik,j}
+ \alpha K_{jk,i}
$
${\Gamma^i}_{jk,t} = (\gamma^{im} \Gamma_{mjk})_{,t}$
${\Gamma^i}_{jk,t} =
- \gamma^{in} \gamma_{nm,t} {\Gamma^m}_{jk}
+ \gamma^{im} \Gamma_{mjk,t}
$
${\Gamma^i}_{jk,t} =
- \gamma^{in} {\Gamma^m}_{jk} (
2 \Gamma_{(mn)l} \beta^l
+ 2 b_{(mn)}
- 2 \alpha K_{mn}
)
+ \gamma^{im} (
\Gamma_{(mj)l,k} \beta^l
+ \Gamma_{(mk)l,j} \beta^l
- \Gamma_{(jk)l,m} \beta^l
+ \Gamma_{(mj)l} {b^l}_k
+ \Gamma_{(mk)l} {b^l}_j
- \Gamma_{(jk)l} {b^l}_m
+ b_{(mj),k}
+ b_{(mk),j}
- b_{(jk),m}
- \alpha K_{mj} a_k
- \alpha K_{mk} a_j
+ \alpha K_{jk} a_m
- \alpha K_{mj,k}
- \alpha K_{mk,j}
+ \alpha K_{jk,m}
)
$
${\Gamma^i}_{jk,t} =
- \gamma^{in} {\Gamma^m}_{jk} (
2 \Gamma_{(mn)l} \beta^l
+ 2 b_{(mn)}
- 2 \alpha K_{mn}
)
+ \gamma^{im} (
\Gamma_{(mj)l,k} \beta^l
+ \Gamma_{(mk)l,j} \beta^l
- \Gamma_{(jk)l,m} \beta^l
+ \Gamma_{(mj)l} {b^l}_k
+ \Gamma_{(mk)l} {b^l}_j
- \Gamma_{(jk)l} {b^l}_m
+ b_{(mj),k}
+ b_{(mk),j}
- b_{(jk),m}
- \alpha K_{mj} a_k
- \alpha K_{mk} a_j
+ \alpha K_{jk} a_m
- \alpha K_{mj,k}
- \alpha K_{mk,j}
+ \alpha K_{jk,m}
)
$
${\Gamma^i}_{jk,t} =
- {\Gamma^m}_{jk} (
+ {\Gamma_{ml}}^i \beta^l
+ {\Gamma^i}_{ml} \beta^l
+ {b_m}^i
+ {b^i}_m
- 2 \alpha {K^i}_m
)
+ \frac{1}{2} (
+ \Gamma_{mjl,k} \beta^l \gamma^{im}
+ \Gamma_{jml,k} \beta^l \gamma^{im}
+ \Gamma_{mkl,j} \beta^l \gamma^{im}
+ \Gamma_{kml,j} \beta^l \gamma^{im}
- \Gamma_{jkl,m} \beta^l \gamma^{im}
- \Gamma_{kjl,m} \beta^l \gamma^{im}
+ \Gamma_{mjl} {b^l}_k \gamma^{im}
+ \Gamma_{jml} {b^l}_k \gamma^{im}
+ \Gamma_{mkl} {b^l}_j \gamma^{im}
+ \Gamma_{kml} {b^l}_j \gamma^{im}
- \Gamma_{jkl} {b^l}_m \gamma^{im}
- \Gamma_{kjl} {b^l}_m \gamma^{im}
+ b_{mj,k} \gamma^{im}
+ b_{jm,k} \gamma^{im}
+ b_{mk,j} \gamma^{im}
+ b_{km,j} \gamma^{im}
- b_{jk,m} \gamma^{im}
- b_{kj,m} \gamma^{im}
)
- \alpha K_{mj} a_k \gamma^{im}
- \alpha K_{mk} a_j \gamma^{im}
+ \alpha K_{jk} a_m \gamma^{im}
- \alpha K_{mj,k} \gamma^{im}
- \alpha K_{mk,j} \gamma^{im}
+ \alpha K_{jk,m} \gamma^{im}
$
${\Gamma^i}_{jk,t} =
- {\Gamma_{ml}}^i {\Gamma^m}_{jk} \beta^l
- {\Gamma^i}_{ml} {\Gamma^m}_{jk} \beta^l
- {\Gamma^m}_{jk} {b_m}^i
- {\Gamma^m}_{jk} {b^i}_m
+ 2 \alpha {K^i}_m {\Gamma^m}_{jk}
+ \frac{1}{2} (
+ \Gamma_{mjl,k} \beta^l \gamma^{im}
+ \Gamma_{jml,k} \beta^l \gamma^{im}
+ \Gamma_{mkl,j} \beta^l \gamma^{im}
+ \Gamma_{kml,j} \beta^l \gamma^{im}
- \Gamma_{jkl,m} \beta^l \gamma^{im}
- \Gamma_{kjl,m} \beta^l \gamma^{im}
+ {\Gamma^i}_{jl} {b^l}_k
+ {\Gamma^i}_{kl} {b^l}_j
+ {\Gamma_{jl}}^i {b^l}_k
+ {\Gamma_{kl}}^i {b^l}_j
- \Gamma_{jkl} b^{li}
- \Gamma_{kjl} b^{li}
+ b_{mj,k} \gamma^{im}
+ b_{jm,k} \gamma^{im}
+ b_{mk,j} \gamma^{im}
+ b_{km,j} \gamma^{im}
- b_{jk,m} \gamma^{im}
- b_{kj,m} \gamma^{im}
)
- \alpha K_{mj} a_k \gamma^{im}
- \alpha K_{mk} a_j \gamma^{im}
+ \alpha K_{jk} a_m \gamma^{im}
- \alpha K_{mj,k} \gamma^{im}
- \alpha K_{mk,j} \gamma^{im}
+ \alpha K_{jk,m} \gamma^{im}
$
${\Gamma^i}_{jk,t} =
- {\Gamma_{ml}}^i {\Gamma^m}_{jk} \beta^l
- {\Gamma^i}_{ml} {\Gamma^m}_{jk} \beta^l
- {\Gamma^m}_{jk} {b_m}^i
- {\Gamma^m}_{jk} {b^i}_m
+ \frac{1}{2} (
+ (\gamma_{nm} {\Gamma^n}_{jl})_{,k} \beta^l \gamma^{im}
+ (\gamma_{nj} {\Gamma^n}_{ml})_{,k} \beta^l \gamma^{im}
+ (\gamma_{nm} {\Gamma^n}_{kl})_{,j} \beta^l \gamma^{im}
+ (\gamma_{nk} {\Gamma^n}_{ml})_{,j} \beta^l \gamma^{im}
- (\gamma_{nj} {\Gamma^n}_{kl})_{,m} \beta^l \gamma^{im}
- (\gamma_{nk} {\Gamma^n}_{jl})_{,m} \beta^l \gamma^{im}
+ {\Gamma^i}_{jl} {b^l}_k
+ {\Gamma^i}_{kl} {b^l}_j
+ {\Gamma_{jl}}^i {b^l}_k
+ {\Gamma_{kl}}^i {b^l}_j
- \Gamma_{jkl} b^{li}
- \Gamma_{kjl} b^{li}
+ (\gamma_{nm} {b^n}_j)_{,k} \gamma^{im}
+ (\gamma_{nj} {b^n}_m)_{,k} \gamma^{im}
+ (\gamma_{nm} {b^n}_k)_{,j} \gamma^{im}
+ (\gamma_{nk} {b^n}_m)_{,j} \gamma^{im}
- (\gamma_{nj} {b^n}_k)_{,m} \gamma^{im}
- (\gamma_{nk} {b^n}_j)_{,m} \gamma^{im}
)
+ 2 \alpha {K^i}_m {\Gamma^m}_{jk}
- \alpha K_{mj} a_k \gamma^{im}
- \alpha K_{mk} a_j \gamma^{im}
+ \alpha K_{jk} a_m \gamma^{im}
- \alpha K_{mj,k} \gamma^{im}
- \alpha K_{mk,j} \gamma^{im}
+ \alpha K_{jk,m} \gamma^{im}
$
${\Gamma^i}_{jk,t} =
- {\Gamma_{ml}}^i {\Gamma^m}_{jk} \beta^l
- {\Gamma^i}_{ml} {\Gamma^m}_{jk} \beta^l
- {\Gamma^m}_{jk} {b_m}^i
- {\Gamma^m}_{jk} {b^i}_m
+ \frac{1}{2} (
+ (\Gamma_{mkn} + \Gamma_{nkm}) {\Gamma^n}_{jl} \beta^l \gamma^{im}
+ (\Gamma_{jkn} + \Gamma_{nkj}) {\Gamma^n}_{ml} \beta^l \gamma^{im}
+ (\Gamma_{mjn} + \Gamma_{njm}) {\Gamma^n}_{kl} \beta^l \gamma^{im}
+ (\Gamma_{kjn} + \Gamma_{njk}) {\Gamma^n}_{ml} \beta^l \gamma^{im}
- (\Gamma_{jmn} + \Gamma_{nmj}) {\Gamma^n}_{kl} \beta^l \gamma^{im}
- (\Gamma_{kmn} + \Gamma_{nmk}) {\Gamma^n}_{jl} \beta^l \gamma^{im}
+ \gamma_{nm} {\Gamma^n}_{jl,k} \beta^l \gamma^{im}
+ \gamma_{nj} {\Gamma^n}_{ml,k} \beta^l \gamma^{im}
+ \gamma_{nm} {\Gamma^n}_{kl,j} \beta^l \gamma^{im}
+ \gamma_{nk} {\Gamma^n}_{ml,j} \beta^l \gamma^{im}
- \gamma_{nj} {\Gamma^n}_{kl,m} \beta^l \gamma^{im}
- \gamma_{nk} {\Gamma^n}_{jl,m} \beta^l \gamma^{im}
+ {\Gamma^i}_{jl} {b^l}_k
+ {\Gamma^i}_{kl} {b^l}_j
+ {\Gamma_{jl}}^i {b^l}_k
+ {\Gamma_{kl}}^i {b^l}_j
- \Gamma_{jkl} b^{li}
- \Gamma_{kjl} b^{li}
+ (\Gamma_{mkn} + \Gamma_{nkm}) {b^n}_j \gamma^{im}
+ (\Gamma_{jkn} + \Gamma_{nkj}) {b^n}_m \gamma^{im}
+ (\Gamma_{mjn} + \Gamma_{njm}) {b^n}_k \gamma^{im}
+ (\Gamma_{kjn} + \Gamma_{njk}) {b^n}_m \gamma^{im}
- (\Gamma_{jmn} + \Gamma_{nmj}) {b^n}_k \gamma^{im}
- (\Gamma_{kmn} + \Gamma_{nmk}) {b^n}_j \gamma^{im}
+ \gamma_{nm} {b^n}_{j,k} \gamma^{im}
+ \gamma_{nj} {b^n}_{m,k} \gamma^{im}
+ \gamma_{nm} {b^n}_{k,j} \gamma^{im}
+ \gamma_{nk} {b^n}_{m,j} \gamma^{im}
- \gamma_{nj} {b^n}_{k,m} \gamma^{im}
- \gamma_{nk} {b^n}_{j,m} \gamma^{im}
)
+ 2 \alpha {K^i}_m {\Gamma^m}_{jk}
- \alpha K_{mj} a_k \gamma^{im}
- \alpha K_{mk} a_j \gamma^{im}
+ \alpha K_{jk} a_m \gamma^{im}
- \alpha K_{mj,k} \gamma^{im}
- \alpha K_{mk,j} \gamma^{im}
+ \alpha K_{jk,m} \gamma^{im}
$
${\Gamma^i}_{jk,t} =
- {\Gamma_{ml}}^i {\Gamma^m}_{jk} \beta^l
- {\Gamma^i}_{ml} {\Gamma^m}_{jk} \beta^l
- {\Gamma^m}_{jk} {b_m}^i
- {\Gamma^m}_{jk} {b^i}_m
+ {\Gamma^i}_{l(j,k)} \beta^l
+ {\Gamma^i}_{n(j} {\Gamma^n}_{k)l} \beta^l
+ \Gamma_{(jk)n} {\Gamma^n}_{lm} \beta^l \gamma^{im}
- \Gamma_{(j|nm} {\Gamma^n}_{k)l} \beta^l \gamma^{im}
+ \Gamma_{nm(j} {\Gamma^n}_{k)l} \beta^l \gamma^{im}
+ \Gamma_{n(jk)} {\Gamma^n}_{lm} \beta^l \gamma^{im}
- {\Gamma_{n(j}}^i {\Gamma^n}_{k)l} \beta^l
+ {\Gamma^n}_{lm,(j} \gamma_{k)n} \beta^l \gamma^{im}
- {\Gamma^n}_{l(j|,m} \beta^l \gamma_{|k)n} \gamma^{im}
+ {\Gamma^i}_{jl} {b^l}_k
+ {\Gamma^i}_{kl} {b^l}_j
+ \Gamma_{n(jk)} b^{ni}
+ {b^i}_{(j,k)}
+ {b^n}_{m,(j} \gamma_{k)n} \gamma^{im}
- {b^n}_{(j|,m} \gamma_{k)n} \gamma^{im}
+ 2 \alpha {K^i}_m {\Gamma^m}_{jk}
- \alpha K_{mj} a_k \gamma^{im}
- \alpha K_{mk} a_j \gamma^{im}
+ \alpha K_{jk} a_m \gamma^{im}
- \alpha K_{mj,k} \gamma^{im}
- \alpha K_{mk,j} \gamma^{im}
+ \alpha K_{jk,m} \gamma^{im}
$
Time evolution of connection trace:
${\Gamma^i}_{,t} = ({\Gamma^i}_{jk} \gamma^{jk})_{,t}$
${\Gamma^i}_{,t} = {\Gamma^i}_{jk,t} \gamma^{jk} + {\Gamma^i}_{jk} {\gamma^{jk}}_{,t}$
${\Gamma^i}_{,t} = {\Gamma^i}_{jk,t} \gamma^{jk} - \Gamma^{ijk} \gamma_{jk,t}$
${\Gamma^i}_{,t} = {\Gamma^i}_{jk,t} \gamma^{jk} - \Gamma^{ijk} (
2 \Gamma_{(jk)l} \beta^l
+ 2 b_{(jk)}
- 2 \alpha K_{jk}
)$
${\Gamma^i}_{,t} =
{\Gamma^i}_{jk,t} \gamma^{jk}
- 2 \Gamma^{ijk} (
\Gamma_{jkl} \beta^l
+ b_{jk}
- \alpha K_{jk}
)
$
${\Gamma^i}_{,t} =
(\gamma^{im} \Gamma_{mjk})_{,t} \gamma^{jk}
- 2 \Gamma^{ijk} (
\Gamma_{jkl} \beta^l
+ b_{jk}
- \alpha K_{jk}
)
$
${\Gamma^i}_{,t} =
(
\gamma^{im} \Gamma_{mjk,t}
- \gamma^{in} \gamma_{nm,t} \gamma^{mp} \Gamma_{pjk}
) \gamma^{jk}
- 2 \Gamma^{ijk} (
\Gamma_{jkl} \beta^l
+ b_{jk}
- \alpha K_{jk}
)
$
${\Gamma^i}_{,t} =
(
\gamma^{im} \Gamma_{mjk,t}
- \gamma^{in} {\Gamma^m}_{jk} (
2 \Gamma_{(mn)p} \beta^p
+ 2 b_{(mn)}
- 2 \alpha K_{mn}
)
) \gamma^{jk}
- 2 \Gamma^{ijk} (
\Gamma_{jkl} \beta^l
+ b_{jk}
- \alpha K_{jk}
)
$
${\Gamma^i}_{,t} =
\gamma^{im} \gamma^{jk} \Gamma_{mjk,t}
- 2 \Gamma_j \delta^i_k (
{\Gamma^{(jk)}}_l \beta^l
+ b^{(jk)}
- \alpha K^{jk}
)
- 2 {\Gamma^i}_{jk} (
{\Gamma^{(jk)}}_l \beta^l
+ b^{(jk)}
- \alpha K^{jk}
)
$
${\Gamma^i}_{,t} =
- 2 (\Gamma_j \delta^i_k + {\Gamma^i}_{jk}) (
{\Gamma^{(jk)}}_l \beta^l
+ b^{(jk)}
- \alpha K^{jk}
)
+ \gamma^{im} \gamma^{jk} (
\Gamma_{(mj)l,k} \beta^l
+ \Gamma_{(mk)l,j} \beta^l
- \Gamma_{(jk)l,m} \beta^l
+ \Gamma_{(mj)l} {b^l}_k
+ \Gamma_{(mk)l} {b^l}_j
- \Gamma_{(jk)l} {b^l}_m
+ b_{(mj),k}
+ b_{(mk),j}
- b_{(jk),m}
- \alpha K_{mj} a_k
- \alpha K_{mk} a_j
+ \alpha K_{jk} a_m
- \alpha K_{mj,k}
- \alpha K_{mk,j}
+ \alpha K_{jk,m}
)
$
${\Gamma^i}_{,t} =
- 2 (\Gamma_j \delta^i_k + {\Gamma^i}_{jk}) (
{\Gamma^{(jk)}}_l \beta^l
+ b^{(jk)}
- \alpha K^{jk}
)
+ (\gamma_{mn} {\Gamma^n}_{jl})_{,k} \beta^l \gamma^{im} \gamma^{jk}
+ (\gamma_{jn} {\Gamma^n}_{ml})_{,k} \beta^l \gamma^{im} \gamma^{jk}
- (\gamma_{jn} {\Gamma^n}_{kl})_{,m} \beta^l \gamma^{im} \gamma^{jk}
+ {\Gamma^i}_{jk} {b^j}_l \gamma^{lk}
+ {\Gamma^k}_{lm} {b^l}_k \gamma^{im}
- {\Gamma^j}_{jk} {b^k}_l \gamma^{li}
+ (\gamma_{n(m} {b^n}_{j)})_{,k} \gamma^{im} \gamma^{jk}
+ (\gamma_{n(m} {b^n}_{k)})_{,j} \gamma^{im} \gamma^{jk}
- (\gamma_{n(j} {b^n}_{k)})_{,m} \gamma^{im} \gamma^{jk}
+ \alpha a^i K
- 2 \alpha K^{ij} a_j
- \alpha K_{mj,k} \gamma^{im} \gamma^{jk}
- \alpha K_{mk,j} \gamma^{im} \gamma^{jk}
+ \alpha K_{jk,m} \gamma^{im} \gamma^{jk}
$
${\Gamma^i}_{,t} =
- 2 (\Gamma_j \delta^i_k + {\Gamma^i}_{jk}) (
{\Gamma^{(jk)}}_l \beta^l
+ b^{(jk)}
- \alpha K^{jk}
)
+ \gamma_{mn} {\Gamma^n}_{jl,k} \beta^l \gamma^{im} \gamma^{jk}
+ \gamma_{jn} {\Gamma^n}_{ml,k} \beta^l \gamma^{im} \gamma^{jk}
- \gamma_{jn} {\Gamma^n}_{kl,m} \beta^l \gamma^{im} \gamma^{jk}
+ (\Gamma_{mkn} + \Gamma_{nkm}) {\Gamma^n}_{jl} \beta^l \gamma^{im} \gamma^{jk}
+ (\Gamma_{jkn} + \Gamma_{nkj}) {\Gamma^n}_{ml} \beta^l \gamma^{im} \gamma^{jk}
- (\Gamma_{jmn} + \Gamma_{nmj}) {\Gamma^n}_{kl} \beta^l \gamma^{im} \gamma^{jk}
+ {\Gamma^i}_{jk} {b^j}_l \gamma^{lk}
+ {\Gamma^k}_{lm} {b^l}_k \gamma^{im}
- {\Gamma^j}_{jk} {b^k}_l \gamma^{li}
+ \gamma_{n(m|,k} {b^n}_{|j)} \gamma^{im} \gamma^{jk}
+ \gamma_{n(m|,j} {b^n}_{|k)} \gamma^{im} \gamma^{jk}
- \gamma_{n(j|,m} {b^n}_{|k)} \gamma^{im} \gamma^{jk}
+ \gamma_{n(m} {b^n}_{j),k} \gamma^{im} \gamma^{jk}
+ \gamma_{n(m} {b^n}_{k),j} \gamma^{im} \gamma^{jk}
- \gamma_{n(j} {b^n}_{k),m} \gamma^{im} \gamma^{jk}
+ \alpha a^i K
- 2 \alpha K^{ij} a_j
- \alpha K_{mj,k} \gamma^{im} \gamma^{jk}
- \alpha K_{mk,j} \gamma^{im} \gamma^{jk}
+ \alpha K_{jk,m} \gamma^{im} \gamma^{jk}
$
${\Gamma^i}_{,t} =
- 2 (\Gamma_j \delta^i_k + {\Gamma^i}_{jk}) (
{\Gamma^{(jk)}}_l \beta^l
+ b^{(jk)}
- \alpha K^{jk}
)
+ \gamma_{mn} {\Gamma^n}_{jl,k} \beta^l \gamma^{im} \gamma^{jk}
+ \gamma_{jn} {\Gamma^n}_{ml,k} \beta^l \gamma^{im} \gamma^{jk}
- \gamma_{jn} {\Gamma^n}_{kl,m} \beta^l \gamma^{im} \gamma^{jk}
+ {\Gamma^i}_{kn} {\Gamma^{nk}}_l \beta^l
+ {\Gamma^k}_{kn} {\Gamma^{ni}}_l \beta^l
- {\Gamma^{ki}}_n {\Gamma^n}_{kl} \beta^l
+ {\Gamma_n}^{ik} {\Gamma^n}_{kl} \beta^l
+ {\Gamma_{nk}}^k {\Gamma^{ni}}_l \beta^l
- {\Gamma_{nk}}^i {\Gamma^{nk}}_l \beta^l
+ {\Gamma^i}_{jk} {b^j}_l \gamma^{lk}
+ {\Gamma^k}_{lm} {b^l}_k \gamma^{im}
- {\Gamma^j}_{jk} {b^k}_l \gamma^{li}
+ \frac{1}{2} (
+ \gamma_{nm,k} {b^n}_j \gamma^{im} \gamma^{jk}
+ \gamma_{nj,k} {b^n}_m \gamma^{im} \gamma^{jk}
+ \gamma_{nm,j} {b^n}_k \gamma^{im} \gamma^{jk}
+ \gamma_{nk,j} {b^n}_m \gamma^{im} \gamma^{jk}
- \gamma_{nj,m} {b^n}_k \gamma^{im} \gamma^{jk}
- \gamma_{nk,m} {b^n}_j \gamma^{im} \gamma^{jk}
+ \gamma_{nm} {b^n}_{j,k} \gamma^{im} \gamma^{jk}
+ \gamma_{nj} {b^n}_{m,k} \gamma^{im} \gamma^{jk}
+ \gamma_{nm} {b^n}_{k,j} \gamma^{im} \gamma^{jk}
+ \gamma_{nk} {b^n}_{m,j} \gamma^{im} \gamma^{jk}
- \gamma_{nj} {b^n}_{k,m} \gamma^{im} \gamma^{jk}
- \gamma_{nk} {b^n}_{j,m} \gamma^{im} \gamma^{jk}
)
+ \alpha a^i K
- 2 \alpha K^{ij} a_j
- \alpha K_{mj,k} \gamma^{im} \gamma^{jk}
- \alpha K_{mk,j} \gamma^{im} \gamma^{jk}
+ \alpha K_{jk,m} \gamma^{im} \gamma^{jk}
$
${\Gamma^i}_{,t} =
+ {\Gamma^i}_{lj,k} \gamma^{jk} \beta^l
+ {\Gamma^k}_{ml,k} \gamma^{im} \beta^l
- {\Gamma^k}_{lk,m} \gamma^{im} \beta^l
- {\Gamma^i}_{jl} \Gamma^j \beta^l
- {\Gamma^i}_{jk} {\Gamma^{jk}}_l \beta^l
+ {\Gamma^k}_{kn} {\Gamma^{ni}}_l \beta^l
- {\Gamma^{ki}}_n {\Gamma^n}_{kl} \beta^l
+ {\Gamma_n}^{ik} {\Gamma^n}_{kl} \beta^l
- {\Gamma_{nk}}^i {\Gamma^{nk}}_l \beta^l
+ \frac{1}{2} {\Gamma_{jk}}^i b^{jk}
- \Gamma^j {b^i}_j
+ {b^i}_{j,k} \gamma^{jk}
+ {b^j}_{k,j} \gamma^{ik}
- {b^j}_{j,k} \gamma^{ik}
+ \alpha a^i K
- 2 \alpha K^{ij} a_j
+ 2 \alpha \Gamma_j K^{ij}
+ 2 \alpha {\Gamma^i}_{jk} K^{jk}
- \alpha K_{mj,k} \gamma^{im} \gamma^{jk}
- \alpha K_{mk,j} \gamma^{im} \gamma^{jk}
+ \alpha K_{jk,m} \gamma^{im} \gamma^{jk}
$
...and that's basically ${B^i}_{,t}$.
Any spatial derivative identities I can use with ${\Gamma^i}_{jk}$?
$\Gamma_{ijk,l} = \frac{1}{2} (\gamma_{ij,k} + \gamma_{ik,j} - \gamma_{jk,i})_{,l}$
$\Gamma_{ijk,l} = \frac{1}{2} (\gamma_{ij,kl} + \gamma_{ik,jl} - \gamma_{jk,il})$
$\Gamma_{ijk,l} = \frac{1}{2} (\gamma_{ij,lk} + \gamma_{ik,lj} - \gamma_{jk,li})$
$\Gamma_{ijk,l} = \Gamma_{(ij)l,k} + \Gamma_{(ik)l,j} - \Gamma_{(jk)l,i}$
$\Gamma_{ijk,l} = \frac{1}{2} (
\Gamma_{ijl,k}
+ \Gamma_{jil,k}
+ \Gamma_{ikl,j}
+ \Gamma_{kil,j}
- \Gamma_{jkl,i}
- \Gamma_{kjl,i}
)$
${\Gamma^i}_{jk,l} = (\gamma^{im} \Gamma_{mjk})_{,l}$
${\Gamma^i}_{jk,l} = {\gamma^{im}}_{,l} \Gamma_{mjk} + \gamma^{im} \Gamma_{mjk,l}$
${\Gamma^i}_{jk,l} = -\gamma^{in} \gamma_{nm,l} {\Gamma^m}_{jk} + \gamma^{im} \Gamma_{mjk,l}$
${\Gamma^i}_{jk,l} =
-{\Gamma^i}_{ml} {\Gamma^m}_{jk}
-{\Gamma_{ml}}^i {\Gamma^m}_{jk}
+ \frac{1}{2} \gamma^{im} (
\Gamma_{ijl,k}
+ \Gamma_{jil,k}
+ \Gamma_{ikl,j}
+ \Gamma_{kil,j}
- \Gamma_{jkl,i}
- \Gamma_{kjl,i}
)
$
${\Gamma^i}_{jk,l} =
-{\Gamma^i}_{ml} {\Gamma^m}_{jk}
-{\Gamma_{ml}}^i {\Gamma^m}_{jk}
+ \frac{1}{2} \gamma^{im} (
\Gamma_{mjl,k}
+ \Gamma_{jml,k}
+ \Gamma_{mkl,j}
+ \Gamma_{kml,j}
- \Gamma_{jkl,m}
- \Gamma_{kjl,m}
)
$
so there is ... but it looks ugly. I think it's easier to just write everything in terms of $d_{ijk} = \frac{1}{2} \gamma_{ij,k}$ rather than ${\Gamma^i}_{jk}$...
combine all terms:
$\left[ \begin{matrix}
\alpha \\
a_i \\
\beta^k \\
{b^k}_i \\
B^k \\
\gamma_{ij} \\
{\Gamma^k}_{ij} \\
K_{ij} \\
\Theta \\
Z_i
\end{matrix} \right]_{,t}
= \left[ \begin{matrix}
\alpha a_i \beta^i
- f \alpha^2 K
+ f \alpha^2 m \Theta
\\
a_{(k,i)} \beta^k
- f \alpha \gamma^{pq} K_{pq,s}
+ f \alpha m \Theta_{,i}
+ a_k {b^k}_i
- \alpha a_i (f' \alpha + f) (K - m \Theta)
+ 2 f \alpha \Gamma_{pqi} K^{pq}
\\
B^k
\\
{B^k}_{,i}
\\
- \eta B^k
+ k (
{\Gamma^k}_{lj,r} \gamma^{jr} \beta^l
+ {\Gamma^r}_{ml,r} \gamma^{km} \beta^l
- {\Gamma^r}_{lr,m} \gamma^{km} \beta^l
- {\Gamma^k}_{jl} \Gamma^j \beta^l
- {\Gamma^k}_{jr} {\Gamma^{jr}}_l \beta^l
+ {\Gamma^r}_{rn} {\Gamma^{nk}}_l \beta^l
- {\Gamma^{rk}}_n {\Gamma^n}_{rl} \beta^l
+ {\Gamma_n}^{kr} {\Gamma^n}_{rl} \beta^l
- {\Gamma_{nr}}^k {\Gamma^{nr}}_l \beta^l
+ \frac{1}{2} {\Gamma_{jr}}^k b^{jr}
- \Gamma^j {b^k}_j
+ {b^k}_{j,r} \gamma^{jr}
+ {b^j}_{r,j} \gamma^{kr}
- {b^j}_{j,r} \gamma^{kr}
+ \alpha a^k K
- 2 \alpha K^{kj} a_j
+ 2 \alpha \Gamma_j K^{kj}
+ 2 \alpha {\Gamma^k}_{jr} K^{jr}
- \alpha K_{mj,r} \gamma^{km} \gamma^{jr}
- \alpha K_{mr,j} \gamma^{km} \gamma^{jr}
+ \alpha K_{jr,m} \gamma^{km} \gamma^{jr}
)
\\
2 \Gamma_{(ij)r} \beta^r
+ 2 b_{(ij)}
- 2 \alpha K_{ij}
\\
- {\Gamma_{ml}}^k {\Gamma^m}_{ij} \beta^l
- {\Gamma^k}_{ml} {\Gamma^m}_{ij} \beta^l
- {\Gamma^m}_{ij} {b_m}^k
- {\Gamma^m}_{ij} {b^k}_m
+ {\Gamma^k}_{l(i,j)} \beta^l
+ {\Gamma^k}_{n(i} {\Gamma^n}_{j)l} \beta^l
+ \Gamma_{(ij)n} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- \Gamma_{(i|nm} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{nm(i} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{n(ij)} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- {\Gamma_{n(i}}^k {\Gamma^n}_{j)l} \beta^l
+ {\Gamma^n}_{lm,(i} \gamma_{j)n} \beta^l \gamma^{km}
- {\Gamma^n}_{l(i|,m} \beta^l \gamma_{|j)n} \gamma^{km}
+ {\Gamma^k}_{il} {b^l}_j
+ {\Gamma^k}_{jl} {b^l}_i
+ \Gamma_{n(ij)} b^{nk}
+ {b^k}_{(i,j)}
+ {b^n}_{m,(i} \gamma_{j)n} \gamma^{km}
- {b^n}_{(i|,m} \gamma_{j)n} \gamma^{km}
+ 2 \alpha {K^k}_m {\Gamma^m}_{ij}
- \alpha K_{mi} a_j \gamma^{km}
- \alpha K_{mj} a_i \gamma^{km}
+ \alpha K_{ij} a_m \gamma^{km}
- \alpha K_{mi,j} \gamma^{km}
- \alpha K_{mj,i} \gamma^{km}
+ \alpha K_{ij,m} \gamma^{km}
\\
K_{ij,r} \beta^r
+ K_{rj} {b^r}_i
+ K_{ir} {b^r}_j
- \alpha a_{(i} a_{j)}
- \alpha a_{(i,j)}
+ \alpha {\Gamma^r}_{ij} a_r
+ \alpha (
R_{ij}
+ Z_{j,i}
+ Z_{i,j}
- 2 {\Gamma^r}_{ij} Z_r
- 2 K_{ir} {K^r}_j
+ (K - 2 \Theta) K_{ij}
+ 4 \pi (S - \rho) \gamma_{ij}
- 8 \pi S_{ij}
)
\\
\Theta_{,i} \beta^i
+ \frac{1}{2} \alpha (
R
+ 2 (
{Z^k}_{,k}
+ {\Gamma^k}_{jk} Z^j
)
+ (K - 2 \Theta) K
- K_{ij} K^{ij}
- 16 \pi \rho
- 2 Z^k a_k
)
\\
Z_{i,j} \beta^j
+ Z_j {b^j}_i
+ \alpha (
- {\Gamma^{kj}}_j K_{ik}
+ \Gamma_{jki} K^{jk}
+ \gamma^{jk} K_{ki,j}
- \gamma^{jk} K_{kj,i}
+ \Theta_{,i}
- 2 K_{ij} Z^j
- 8 \pi S_i
)
- \alpha \Theta a_i
\end{matrix} \right]
$
separate out partial derivatives:
$\left[ \begin{matrix}
\alpha \\
\beta^k \\
\gamma_{ij} \\
a_i \\
B^k \\
{b^k}_i \\
{\Gamma^k}_{ij} \\
K_{ij} \\
\Theta \\
Z_i
\end{matrix} \right]_{,t}
+ \left[ \begin{matrix}
0
\\
0
\\
0
\\
- a_{(k,i)} \beta^k
+ f \alpha \gamma^{pq} K_{pq,s}
- f \alpha m \Theta_{,i}
\\
k (
- {\Gamma^k}_{lj,r} \gamma^{jr} \beta^l
- {\Gamma^r}_{ml,r} \gamma^{km} \beta^l
+ {\Gamma^r}_{lr,m} \gamma^{km} \beta^l
- {b^k}_{j,r} \gamma^{jr}
- {b^j}_{r,j} \gamma^{kr}
+ {b^j}_{j,r} \gamma^{kr}
+ \alpha K_{mj,r} \gamma^{km} \gamma^{jr}
+ \alpha K_{mr,j} \gamma^{km} \gamma^{jr}
- \alpha K_{jr,m} \gamma^{km} \gamma^{jr}
)
\\
-{B^k}_{,i}
\\
- {\Gamma^k}_{l(i,j)} \beta^l
- {\Gamma^n}_{lm,(i} \gamma_{j)n} \beta^l \gamma^{km}
+ {\Gamma^n}_{l(i|,m} \beta^l \gamma_{|j)n} \gamma^{km}
- {b^k}_{(i,j)}
- {b^n}_{m,(i} \gamma_{j)n} \gamma^{km}
+ {b^n}_{(i|,m} \gamma_{j)n} \gamma^{km}
+ \alpha K_{mi,j} \gamma^{km}
+ \alpha K_{mj,i} \gamma^{km}
- \alpha K_{ij,m} \gamma^{km}
\\
- K_{ij,r} \beta^r
+ \alpha a_{(i,j)}
- \alpha (
Z_{j,i}
+ Z_{i,j}
+ {\Gamma^k}_{ij,k}
- {\Gamma^k}_{ik,j}
)
\\
- \Theta_{,i} \beta^i
- \alpha \gamma^{kl} Z_{l,k}
- \frac{1}{2} \alpha (
\gamma^{jl} {\Gamma^k}_{jl,k}
- \gamma^{jl} {\Gamma^k}_{jk,l}
)
\\
- Z_{i,j} \beta^j
- \alpha (
\gamma^{jk} K_{ki,j}
- \gamma^{jk} K_{kj,i}
+ \Theta_{,i}
)
\end{matrix} \right]
= \left[ \begin{matrix}
\alpha a_i \beta^i
- f \alpha^2 K
+ f \alpha^2 m \Theta
\\
B^k
\\
2 \Gamma_{(ij)r} \beta^r
+ 2 b_{(ij)}
- 2 \alpha K_{ij}
\\
a_k {b^k}_i
- \alpha a_i (f' \alpha + f) (K - m \Theta)
+ 2 f \alpha \Gamma_{pqi} K^{pq}
\\
- \eta B^k
+ k (
- {\Gamma^k}_{jl} \Gamma^j \beta^l
- {\Gamma^k}_{jr} {\Gamma^{jr}}_l \beta^l
+ {\Gamma^r}_{rn} {\Gamma^{nk}}_l \beta^l
- {\Gamma^{rk}}_n {\Gamma^n}_{rl} \beta^l
+ {\Gamma_n}^{kr} {\Gamma^n}_{rl} \beta^l
- {\Gamma_{nr}}^k {\Gamma^{nr}}_l \beta^l
+ \frac{1}{2} {\Gamma_{jr}}^k b^{jr}
- \Gamma^j {b^k}_j
+ \alpha a^k K
- 2 \alpha K^{kj} a_j
+ 2 \alpha \Gamma_j K^{kj}
+ 2 \alpha {\Gamma^k}_{jr} K^{jr}
)
\\
0
\\
- {\Gamma_{ml}}^k {\Gamma^m}_{ij} \beta^l
- {\Gamma^k}_{ml} {\Gamma^m}_{ij} \beta^l
- {\Gamma^m}_{ij} {b_m}^k
- {\Gamma^m}_{ij} {b^k}_m
+ {\Gamma^k}_{n(i} {\Gamma^n}_{j)l} \beta^l
+ \Gamma_{(ij)n} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- \Gamma_{(i|nm} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{nm(i} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{n(ij)} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- {\Gamma_{n(i}}^k {\Gamma^n}_{j)l} \beta^l
+ {\Gamma^k}_{il} {b^l}_j
+ {\Gamma^k}_{jl} {b^l}_i
+ \Gamma_{n(ij)} b^{nk}
+ 2 \alpha {K^k}_m {\Gamma^m}_{ij}
- \alpha K_{mi} a_j \gamma^{km}
- \alpha K_{mj} a_i \gamma^{km}
+ \alpha K_{ij} a_m \gamma^{km}
\\
K_{rj} {b^r}_i
+ K_{ir} {b^r}_j
- \alpha a_{(i} a_{j)}
+ \alpha {\Gamma^r}_{ij} a_r
+ \alpha (
{\Gamma^k}_{mk} {\Gamma^m}_{ij}
- {\Gamma^k}_{mj} {\Gamma^m}_{ik}
- 2 {\Gamma^r}_{ij} Z_r
- 2 K_{ir} {K^r}_j
+ (K - 2 \Theta) K_{ij}
+ 4 \pi (S - \rho) \gamma_{ij}
- 8 \pi S_{ij}
)
\\
\frac{1}{2} \alpha (
{\Gamma^k}_{km} \Gamma^m
- \Gamma^{kjm} \Gamma_{mjk}
- 2 Z_l \Gamma^l
+ (K - 2 \Theta) K
- K_{ij} K^{ij}
- 16 \pi \rho
- 2 Z^k a_k
)
\\
Z_j {b^j}_i
+ \alpha (
- {\Gamma^{kj}}_j K_{ik}
+ \Gamma_{jki} K^{jk}
- 2 K_{ij} Z^j
- 8 \pi S_i
)
- \alpha \Theta a_i
\end{matrix} \right]
$
factor out linear system:
$\left[ \begin{matrix}
\alpha \\
\beta^k \\
\gamma_{ij} \\
a_i \\
B^k \\
{\Gamma^k}_{ij} \\
K_{ij} \\
\Theta \\
Z_i \\
{b^k}_i
\end{matrix} \right]_{,t}
+ \left[ \begin{matrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
\\
0 &
0 &
0 &
-\frac{1}{2} (
\delta_i^s \beta^p
+ \delta_i^p \beta^s
) &
0 &
0 &
f \alpha \delta_i^s \gamma^{pq} &
- f \alpha m \delta_i^s &
0 &
0
\\
0 & 0 & 0 & 0 & 0 &
- k \delta^k_r \gamma^{qs} \beta^p
- k \delta^s_r \gamma^{pk} \beta^q
+ k \delta^q_r \gamma^{sk} \beta^p
&
2 k \alpha \gamma^{kp} \gamma^{qs}
- k \alpha \gamma^{ks} \gamma^{pq}
&
0 &
0 &
- k \delta^k_r \gamma^{ps}
- k \delta^s_r \gamma^{kp}
+ k \delta^p_r \gamma^{ks}
&
\\
0 & 0 & 0 & 0 & 0 &
- \frac{1}{2} \beta^p \delta^k_r \delta^q_i \delta^s_j
- \frac{1}{2} \beta^p \delta^k_r \delta^q_j \delta^s_i
- \frac{1}{2} \beta^p \gamma_{jr} \gamma^{kq} \delta^s_i
- \frac{1}{2} \beta^p \gamma_{ir} \gamma^{kq} \delta^s_j
+ \frac{1}{2} \beta^p \gamma_{jr} \gamma^{ks} \delta^q_i
+ \frac{1}{2} \beta^p \gamma_{ir} \gamma^{ks} \delta^q_j
&
\alpha \gamma^{kp} \delta^q_i \delta^s_j
+ \alpha \gamma^{kp} \delta^q_j \delta^s_i
- \alpha \gamma^{ks} \delta^p_i \delta^q_j
&
0 & 0 &
- \frac{1}{2} \delta^k_r \delta_i^p \delta_j^s
- \frac{1}{2} \delta^k_r \delta_j^p \delta_i^s
- \frac{1}{2} \gamma_{jr} \gamma^{kp} \delta^s_i
- \frac{1}{2} \gamma_{ir} \gamma^{kp} \delta^s_j
+ \frac{1}{2} \gamma_{jr} \gamma^{ks} \delta^p_i
+ \frac{1}{2} \gamma_{ir} \gamma^{ks} \delta^p_j
\\
0 & 0 & 0 &
\frac{1}{2} \alpha (\delta^s_i \delta^p_j + \delta^s_j \delta^p_i) &
0 &
\alpha \delta^q_r \delta^p_i \delta^s_j
- \alpha \delta^s_r \delta^p_i \delta^q_j
&
- \beta^s \delta^p_i \delta^q_j
&
0 &
- \alpha (\delta^p_i \delta^s_j + \delta^p_j \delta^s_i)
&
0
\\
0 & 0 & 0 & 0 & 0 &
- \frac{1}{2} \alpha (
\gamma^{pq} \delta_r^s
- \gamma^{ps} \delta^q_r
)
&
0 &
- \beta^s
&
- \alpha \gamma^{ps}
&
0 \\
0 & 0 & 0 & 0 & 0 & 0 &
\alpha (\gamma^{pq} \delta^s_i - \gamma^{ps} \delta^q_i)
&
- \alpha \delta^s_i
&
- \delta^p_i \beta^s
&
0 \\
0 & 0 & 0 & 0 & -\delta^s_i \delta^k_r & 0 & 0 & 0 & 0 & 0
\end{matrix} \right]
\left[ \begin{matrix}
\alpha \\
\beta^r \\
\gamma_{pq} \\
a_p \\
B^r \\
{\Gamma^r}_{pq} \\
K_{pq} \\
\Theta \\
Z_p \\
{b^r}_p
\end{matrix} \right]_{,s}
= \left[ \begin{matrix}
\alpha a_i \beta^i
- f \alpha^2 K
+ f \alpha^2 m \Theta
\\
B^k
\\
2 \Gamma_{(ij)r} \beta^r
+ 2 b_{(ij)}
- 2 \alpha K_{ij}
\\
+ a_k {b^k}_i
- \alpha a_i (f' \alpha + f) (K - m \Theta)
+ 2 f \alpha \Gamma_{pqi} K^{pq}
\\
- \eta B^k
+ k (
- {\Gamma^k}_{jl} \Gamma^j \beta^l
- {\Gamma^k}_{jr} {\Gamma^{jr}}_l \beta^l
+ {\Gamma^r}_{rn} {\Gamma^{nk}}_l \beta^l
- {\Gamma^{rk}}_n {\Gamma^n}_{rl} \beta^l
+ {\Gamma_n}^{kr} {\Gamma^n}_{rl} \beta^l
- {\Gamma_{nr}}^k {\Gamma^{nr}}_l \beta^l
+ \frac{1}{2} {\Gamma_{jr}}^k b^{jr}
- \Gamma^j {b^k}_j
+ \alpha a^k K
- 2 \alpha K^{kj} a_j
+ 2 \alpha \Gamma_j K^{kj}
+ 2 \alpha {\Gamma^k}_{jr} K^{jr}
)
\\
- {\Gamma_{ml}}^k {\Gamma^m}_{ij} \beta^l
- {\Gamma^k}_{ml} {\Gamma^m}_{ij} \beta^l
- {\Gamma^m}_{ij} {b_m}^k
- {\Gamma^m}_{ij} {b^k}_m
+ {\Gamma^k}_{n(i} {\Gamma^n}_{j)l} \beta^l
+ \Gamma_{(ij)n} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- \Gamma_{(i|nm} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{nm(i} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{n(ij)} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- {\Gamma_{n(i}}^k {\Gamma^n}_{j)l} \beta^l
+ {\Gamma^k}_{il} {b^l}_j
+ {\Gamma^k}_{jl} {b^l}_i
+ \Gamma_{n(ij)} b^{nk}
+ 2 \alpha {K^k}_m {\Gamma^m}_{ij}
- \alpha K_{mi} a_j \gamma^{km}
- \alpha K_{mj} a_i \gamma^{km}
+ \alpha K_{ij} a_m \gamma^{km}
\\
K_{rj} {b^r}_i
+ K_{ir} {b^r}_j
- \alpha a_{(i} a_{j)}
+ \alpha {\Gamma^r}_{ij} a_r
+ \alpha (
{\Gamma^k}_{mk} {\Gamma^m}_{ij}
- {\Gamma^k}_{mj} {\Gamma^m}_{ik}
- 2 {\Gamma^r}_{ij} Z_r
- 2 K_{ir} {K^r}_j
+ (K - 2 \Theta) K_{ij}
+ 4 \pi (S - \rho) \gamma_{ij}
- 8 \pi S_{ij}
)
\\
\frac{1}{2} \alpha (
{\Gamma^k}_{km} \Gamma^m
- \Gamma^{kjm} \Gamma_{mjk}
- 2 Z_l \Gamma^l
+ (K - 2 \Theta) K
- K_{ij} K^{ij}
- 16 \pi \rho
- 2 Z^k a_k
)
\\
Z_j {b^j}_i
+ \alpha (
- {\Gamma^{kj}}_j K_{ik}
+ \Gamma_{jki} K^{jk}
- 2 K_{ij} Z^j
- 8 \pi S_i
)
- \alpha \Theta a_i
\\
0
\end{matrix} \right]
$
get rid of the purely-source gauge terms: $\alpha, \beta^k, \gamma_{ij}$.
favor the $\beta^s$ diagonal:
$\left[ \begin{matrix}
a_i \\
B^k \\
{\Gamma^k}_{ij} \\
K_{ij} \\
\Theta \\
Z_i \\
{b^k}_i
\end{matrix} \right]_{,t}
+ \left[ \begin{matrix}
-\delta_i^p \beta^s &
0 &
0 &
f \alpha \delta_i^s \gamma^{pq} &
- f \alpha m \delta_i^s &
0 &
0
\\
0 & 0 &
- k \delta^k_r \gamma^{qs} \beta^p
- k \delta^s_r \gamma^{pk} \beta^q
+ k \delta^q_r \gamma^{sk} \beta^p
&
2 k \alpha \gamma^{kp} \gamma^{qs}
- k \alpha \gamma^{ks} \gamma^{pq}
&
0 &
0 &
- k \delta^k_r \gamma^{ps}
- k \delta^s_r \gamma^{kp}
+ k \delta^p_r \gamma^{ks}
&
\\
0 & 0 &
- \frac{1}{2} \beta^p \delta^k_r \delta^q_i \delta^s_j
- \frac{1}{2} \beta^p \delta^k_r \delta^q_j \delta^s_i
- \frac{1}{2} \beta^p \gamma_{jr} \gamma^{kq} \delta^s_i
- \frac{1}{2} \beta^p \gamma_{ir} \gamma^{kq} \delta^s_j
+ \frac{1}{2} \beta^p \gamma_{jr} \gamma^{ks} \delta^q_i
+ \frac{1}{2} \beta^p \gamma_{ir} \gamma^{ks} \delta^q_j
&
\alpha \gamma^{kp} \delta^q_i \delta^s_j
+ \alpha \gamma^{kp} \delta^q_j \delta^s_i
- \alpha \gamma^{ks} \delta^p_i \delta^q_j
&
0 & 0 &
- \frac{1}{2} \delta^k_r \delta_i^p \delta_j^s
- \frac{1}{2} \delta^k_r \delta_j^p \delta_i^s
- \frac{1}{2} \gamma_{jr} \gamma^{kp} \delta^s_i
- \frac{1}{2} \gamma_{ir} \gamma^{kp} \delta^s_j
+ \frac{1}{2} \gamma_{jr} \gamma^{ks} \delta^p_i
+ \frac{1}{2} \gamma_{ir} \gamma^{ks} \delta^p_j
\\
\frac{1}{2} \alpha (\delta^s_i \delta^p_j + \delta^s_j \delta^p_i) &
0 &
\alpha \delta^q_r \delta^p_i \delta^s_j
- \alpha \delta^s_r \delta^p_i \delta^q_j
&
- \beta^s \delta^p_i \delta^q_j
&
0 &
- \alpha (\delta^p_i \delta^s_j + \delta^p_j \delta^s_i)
&
0
\\
0 & 0 &
- \frac{1}{2} \alpha (
\gamma^{pq} \delta_r^s
- \gamma^{ps} \delta^q_r
)
&
0 &
- \beta^s
&
- \alpha \gamma^{ps}
&
0 \\
0 & 0 & 0 &
\alpha (\gamma^{pq} \delta^s_i - \gamma^{ps} \delta^q_i)
&
- \alpha \delta^s_i
&
- \delta^p_i \beta^s
&
0 \\
0 & -\delta^s_i \delta^k_r & 0 & 0 & 0 & 0 & 0
\end{matrix} \right]
\left[ \begin{matrix}
a_p \\
B^r \\
{\Gamma^r}_{pq} \\
K_{pq} \\
\Theta \\
Z_p \\
{b^r}_p
\end{matrix} \right]_{,s}
= \left[ \begin{matrix}
a_k {b^k}_i
- \alpha a_i (f' \alpha + f) (K - m \Theta)
+ 2 f \alpha \Gamma_{pqi} K^{pq}
\\
- \eta B^k
+ k (
- {\Gamma^k}_{jl} \Gamma^j \beta^l
- {\Gamma^k}_{jr} {\Gamma^{jr}}_l \beta^l
+ {\Gamma^r}_{rn} {\Gamma^{nk}}_l \beta^l
- {\Gamma^{rk}}_n {\Gamma^n}_{rl} \beta^l
+ {\Gamma_n}^{kr} {\Gamma^n}_{rl} \beta^l
- {\Gamma_{nr}}^k {\Gamma^{nr}}_l \beta^l
+ \frac{1}{2} {\Gamma_{jr}}^k b^{jr}
- \Gamma^j {b^k}_j
+ \alpha a^k K
- 2 \alpha K^{kj} a_j
+ 2 \alpha \Gamma_j K^{kj}
+ 2 \alpha {\Gamma^k}_{jr} K^{jr}
)
\\
- {\Gamma_{ml}}^k {\Gamma^m}_{ij} \beta^l
- {\Gamma^k}_{ml} {\Gamma^m}_{ij} \beta^l
- {\Gamma^m}_{ij} {b_m}^k
- {\Gamma^m}_{ij} {b^k}_m
+ {\Gamma^k}_{n(i} {\Gamma^n}_{j)l} \beta^l
+ \Gamma_{(ij)n} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- \Gamma_{(i|nm} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{nm(i} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{n(ij)} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- {\Gamma_{n(i}}^k {\Gamma^n}_{j)l} \beta^l
+ {\Gamma^k}_{il} {b^l}_j
+ {\Gamma^k}_{jl} {b^l}_i
+ \Gamma_{n(ij)} b^{nk}
+ 2 \alpha {K^k}_m {\Gamma^m}_{ij}
- \alpha K_{mi} a_j \gamma^{km}
- \alpha K_{mj} a_i \gamma^{km}
+ \alpha K_{ij} a_m \gamma^{km}
\\
K_{rj} {b^r}_i
+ K_{ir} {b^r}_j
- \alpha a_{(i} a_{j)}
+ \alpha {\Gamma^r}_{ij} a_r
+ \alpha (
{\Gamma^k}_{mk} {\Gamma^m}_{ij}
- {\Gamma^k}_{mj} {\Gamma^m}_{ik}
- 2 {\Gamma^r}_{ij} Z_r
- 2 K_{ir} {K^r}_j
+ (K - 2 \Theta) K_{ij}
+ 4 \pi (S - \rho) \gamma_{ij}
- 8 \pi S_{ij}
)
\\
\frac{1}{2} \alpha (
{\Gamma^k}_{km} \Gamma^m
- \Gamma^{kjm} \Gamma_{mjk}
- 2 Z_l \Gamma^l
+ (K - 2 \Theta) K
- K_{ij} K^{ij}
- 16 \pi \rho
- 2 Z^k a_k
)
\\
Z_j {b^j}_i
+ \alpha (
- {\Gamma^{kj}}_j K_{ik}
+ \Gamma_{jki} K^{jk}
- 2 K_{ij} Z^j
- 8 \pi S_i
)
- \alpha \Theta a_i
\\
0
\end{matrix} \right]
$
back to the linear system, cast everything as a partial derivative, so there is no source:
$\left[ \begin{matrix}
\alpha \\
\beta^k \\
\gamma_{ij} \\
a_i \\
B^k \\
{\Gamma^k}_{ij} \\
K_{ij} \\
\Theta \\
Z_i \\
{b^k}_i
\end{matrix} \right]_{,t}
+ \left[ \begin{matrix}
-\beta^s &
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
\\
0 &
-\gamma_{ir} \delta^s_j
-\gamma_{jr} \delta^s_i
&
-(\delta^p_i \delta^q_j + \delta^p_j \delta^q_i) \beta^s &
0 & 0 & 0 & 0 & 0 & 0 & 0 &
\\
\delta^s_i (f' \alpha + f) (K - m \Theta) &
-a_r \delta^s_i &
- f \alpha K^{pq} \delta^s_i &
-\delta_i^p \beta^s &
0 &
0 &
f \alpha \delta_i^s \gamma^{pq} &
- f \alpha m \delta_i^s &
0 &
0
\\
0 & 0 & 0 & 0 & 0 &
- k \delta^k_r \gamma^{qs} \beta^p
- k \delta^s_r \gamma^{pk} \beta^q
+ k \delta^q_r \gamma^{sk} \beta^p
&
2 k \alpha \gamma^{kp} \gamma^{qs}
- k \alpha \gamma^{ks} \gamma^{pq}
&
0 &
0 &
- k \delta^k_r \gamma^{ps}
- k \delta^s_r \gamma^{kp}
+ k \delta^p_r \gamma^{ks}
&
\\
0 & 0 & 0 & 0 & 0 &
- \frac{1}{2} \beta^p \delta^k_r \delta^q_i \delta^s_j
- \frac{1}{2} \beta^p \delta^k_r \delta^q_j \delta^s_i
- \frac{1}{2} \beta^p \gamma_{jr} \gamma^{kq} \delta^s_i
- \frac{1}{2} \beta^p \gamma_{ir} \gamma^{kq} \delta^s_j
+ \frac{1}{2} \beta^p \gamma_{jr} \gamma^{ks} \delta^q_i
+ \frac{1}{2} \beta^p \gamma_{ir} \gamma^{ks} \delta^q_j
&
\alpha \gamma^{kp} \delta^q_i \delta^s_j
+ \alpha \gamma^{kp} \delta^q_j \delta^s_i
- \alpha \gamma^{ks} \delta^p_i \delta^q_j
&
0 & 0 &
- \frac{1}{2} \delta^k_r \delta_i^p \delta_j^s
- \frac{1}{2} \delta^k_r \delta_j^p \delta_i^s
- \frac{1}{2} \gamma_{jr} \gamma^{kp} \delta^s_i
- \frac{1}{2} \gamma_{ir} \gamma^{kp} \delta^s_j
+ \frac{1}{2} \gamma_{jr} \gamma^{ks} \delta^p_i
+ \frac{1}{2} \gamma_{ir} \gamma^{ks} \delta^p_j
\\
0 & 0 & 0 &
\frac{1}{2} \alpha (\delta^s_i \delta^p_j + \delta^s_j \delta^p_i) &
0 &
\alpha \delta^q_r \delta^p_i \delta^s_j
- \alpha \delta^s_r \delta^p_i \delta^q_j
&
- \beta^s \delta^p_i \delta^q_j
&
0 &
- \alpha (\delta^p_i \delta^s_j + \delta^p_j \delta^s_i)
&
0
\\
0 & 0 & 0 & 0 & 0 &
- \frac{1}{2} \alpha (
\gamma^{pq} \delta_r^s
- \gamma^{ps} \delta^q_r
)
&
0 &
- \beta^s
&
- \alpha \gamma^{ps}
&
0 \\
0 & 0 & 0 & 0 & 0 & 0 &
\alpha (\gamma^{pq} \delta^s_i - \gamma^{ps} \delta^q_i)
&
- \alpha \delta^s_i
&
- \delta^p_i \beta^s
&
0 \\
0 & 0 & 0 & 0 & -\delta^s_i \delta^k_r & 0 & 0 & 0 & 0 & 0
\end{matrix} \right]
\left[ \begin{matrix}
\alpha \\
\beta^r \\
\gamma_{pq} \\
a_p \\
B^r \\
{\Gamma^r}_{pq} \\
K_{pq} \\
\Theta \\
Z_p \\
{b^r}_p
\end{matrix} \right]_{,s}
= \left[ \begin{matrix}
- f \alpha^2 (K - m \Theta)
\\
B^k
\\
- 2 \alpha K_{ij}
\\
0 \\
- \eta B^k
+ k (
- {\Gamma^k}_{jl} \Gamma^j \beta^l
- {\Gamma^k}_{jr} {\Gamma^{jr}}_l \beta^l
+ {\Gamma^r}_{rn} {\Gamma^{nk}}_l \beta^l
- {\Gamma^{rk}}_n {\Gamma^n}_{rl} \beta^l
+ {\Gamma_n}^{kr} {\Gamma^n}_{rl} \beta^l
- {\Gamma_{nr}}^k {\Gamma^{nr}}_l \beta^l
+ \frac{1}{2} {\Gamma_{jr}}^k b^{jr}
- \Gamma^j {b^k}_j
+ \alpha a^k K
- 2 \alpha K^{kj} a_j
+ 2 \alpha \Gamma_j K^{kj}
+ 2 \alpha {\Gamma^k}_{jr} K^{jr}
)
\\
- {\Gamma_{ml}}^k {\Gamma^m}_{ij} \beta^l
- {\Gamma^k}_{ml} {\Gamma^m}_{ij} \beta^l
- {\Gamma^m}_{ij} {b_m}^k
- {\Gamma^m}_{ij} {b^k}_m
+ {\Gamma^k}_{n(i} {\Gamma^n}_{j)l} \beta^l
+ \Gamma_{(ij)n} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- \Gamma_{(i|nm} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{nm(i} {\Gamma^n}_{j)l} \beta^l \gamma^{km}
+ \Gamma_{n(ij)} {\Gamma^n}_{lm} \beta^l \gamma^{km}
- {\Gamma_{n(i}}^k {\Gamma^n}_{j)l} \beta^l
+ {\Gamma^k}_{il} {b^l}_j
+ {\Gamma^k}_{jl} {b^l}_i
+ \Gamma_{n(ij)} b^{nk}
+ 2 \alpha {K^k}_m {\Gamma^m}_{ij}
- \alpha K_{mi} a_j \gamma^{km}
- \alpha K_{mj} a_i \gamma^{km}
+ \alpha K_{ij} a_m \gamma^{km}
\\
K_{rj} {b^r}_i
+ K_{ir} {b^r}_j
- \alpha a_{(i} a_{j)}
+ \alpha {\Gamma^r}_{ij} a_r
+ \alpha (
{\Gamma^k}_{mk} {\Gamma^m}_{ij}
- {\Gamma^k}_{mj} {\Gamma^m}_{ik}
- 2 {\Gamma^r}_{ij} Z_r
- 2 K_{ir} {K^r}_j
+ (K - 2 \Theta) K_{ij}
+ 4 \pi (S - \rho) \gamma_{ij}
- 8 \pi S_{ij}
)
\\
\frac{1}{2} \alpha (
{\Gamma^k}_{km} \Gamma^m
- \Gamma^{kjm} \Gamma_{mjk}
- 2 Z_l \Gamma^l
+ (K - 2 \Theta) K
- K_{ij} K^{ij}
- 16 \pi \rho
- 2 Z^k a_k
)
\\
Z_j {b^j}_i
+ \alpha (
- {\Gamma^{kj}}_j K_{ik}
+ \Gamma_{jki} K^{jk}
- 2 K_{ij} Z^j
- 8 \pi S_i
)
- \alpha \Theta a_i
\\
0
\end{matrix} \right]
$