Sources:
2008 Alcubierre. "Introduction to 3+1 Numerical Relativity"
2010 Baumgarte, Shapiro. "Numerical Relativity: Solving Einstein's Equations on the Computer"
2017 Ruchlin, Etienne, Baumgarte. "SENR-NRPy+: Numerical relativity in singular curvilinearcoordinate systems"
Taken from my Differential Geometry ADM formalism worksheet:
ADM lapse evolution:
$\alpha_{,t} = -\alpha^2 f K + \alpha_{,j} \beta^j$ (2017 Ruchlin eqn. 13)
or, specifically, $\alpha_{,t} = -\alpha(1 - \alpha) K + \alpha_{,i} \beta^i$ (2017 Ruchlin eqn. 66)
ADM spatial metric evolution:
$\gamma_{ij,t} = -2 \alpha K_{ij} + 2 D_{(i} \beta_{j)}$
$= -2 \alpha K_{ij} + \gamma_{ij,k} \beta^k + 2 \gamma_{k(i} {\beta^k}_{,j)}$
ADM extrinsic curvature evolution:
$K_{ij,t} = -D_i D_j \alpha + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j) + 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
+ K_{ij,k} \beta^k + 2 K_{k(i} {\beta^k}_{,j)}$
non-advecting Gamma driver function:
2017 Ruchlin eqn. 67:
${\beta^i}_{,t} = B^i$
${B^i}_{,t} = \frac{3}{4} {\bar{\Lambda}^i}_{,t} - \eta B^i$
spatial metric / ADM variable:
$\gamma_{ij}$
spatial metric inverse:
$\gamma^{ij}$ such that $[\gamma_{ij}]^{-1} = [\gamma^{ij}]$
determinant of spatial metric:
$\gamma = det[\gamma_{ij}]$
metric of the grid coordinate system:
$\hat{\gamma}_{ij}$
grid metric inverse:
$\hat{\gamma}^{ij}$ such that $[\hat{\gamma}_{ij}]^{-1} = [\hat{\gamma}^{ij}]$
determinant of the grid coordinate system metric:
$\hat{\gamma} = det[\hat{\gamma}_{ij}]$
Connection of the grid coordinate system:
${\hat{\Gamma}^i}_{ij} = \frac{1}{2} \hat{\gamma}^{il} (\hat{\gamma}_{lj,k} + \hat{\gamma}_{lk,j} - \hat{\gamma}_{jk,l}) =$ Levi-Civita connection of the grid coordinate system.
conformally rescaled metric:
$\bar{\gamma}_{ij} = $ some scalar $ \cdot \gamma_{ij}$
determinant of the conformally rescaled metric:
$\bar{\gamma} = det(\bar{\gamma}_{ij})$
constrain the conformal metric determinant to equal the grid metric determinant:
$\bar{\gamma} = \hat{\gamma}$
Notice that tensors marked with either $\bar{}$ or $\hat{}$ will be raised and lowered by their respective metrices: either $\bar{\gamma}_{ij}$ or $\hat{\gamma}_{ij}$
This means that the original contra- or co-variance of a metric is important to distinguish, especially if it is being applied to a metric other than the one it is denoted to raise/lower with (i.e. $\bar{\gamma}_{jk} \beta^k$).
derivative of conformal and grid metric determinants:
$\bar{\gamma}_{,\mu} = \hat{\gamma}_{,\mu}$
$\bar{\gamma} \bar{\gamma}^{ij} \bar{\gamma}_{ij,\mu}
= \hat{\gamma} \hat{\gamma}^{ij} \hat{\gamma}_{ij,\mu}$
$\bar{\gamma}^{ij} (\bar{\Gamma}_{i \mu j} + \bar{\Gamma}_{j \mu i})
= \hat{\gamma}^{ij} (\hat{\Gamma}_{i \mu j} + \hat{\Gamma}_{j \mu i})$
$\bar{\gamma}^{ij} \bar{\Gamma}_{i \mu j} = \hat{\gamma}^{ij} \hat{\Gamma}_{i \mu j}$
${\bar{\Gamma}^i}_{\mu i} = {\hat{\Gamma}^i}_{\mu i}$
Conformal rescaling factor.
This is a state variable of the 2008 Alcubierre and 2010 Baumgarte, Shapiro.
$\phi = \frac{1}{12} ln \frac{\gamma}{\hat{\gamma}}$
here's a convenient value to have around:
$\chi = exp(-4 \phi)$
$= exp(-4 \cdot \frac{1}{12} ln \frac{\gamma}{\hat{\gamma}})$
$= (\frac{\hat{\gamma}}{\gamma})^{\frac{1}{3}}$
and here's another, which is a state variable of 2017 Ruchlin:
$W = \sqrt{\chi} = exp(-2 \phi)$
$W = (\frac{\hat{\gamma}}{\gamma})^{\frac{1}{6}}$
$W^2 = \chi = exp(-4 \phi)$
$\gamma = \hat{\gamma} W^{-6}$
$W_{,\mu} = (exp(-2 \phi))_{,\mu}$
$= -2 \phi_{,\mu} W$
so $\phi_{,\mu} = -\frac{1}{2 W} W_{,\mu}$
$\phi_{,\mu\nu} = -\frac{1}{2} (\frac{1}{W} W_{,\mu})_{,\nu}$
$= \frac{1}{2} (W^{-2} W_{,\mu} W_{,\nu} - \frac{1}{W} W_{,\mu\nu})$
conformally rescaled metric definition:
$\bar{\gamma}_{ij} = exp(-4 \phi) \gamma_{ij} = W^2 \gamma_{ij}$
$= (\frac{\hat{\gamma}}{\gamma})^{\frac{1}{3}} \gamma_{ij}$
$\gamma_{ij} = exp(4 \phi) \bar{\gamma}_{ij} = W^{-2} \bar{\gamma}_{ij}$
$\gamma^{ij} = exp(-4 \phi) \bar{\gamma}^{ij} = W^2 \bar{\gamma}^{ij}$
$\gamma = det(\gamma_{ij})$
$= det(exp(4\phi) \bar{\gamma}_{ij})$
$= exp(12 \phi) det(\bar{\gamma}_{ij})$
$= exp(12 \phi) \bar{\gamma}$
verify that the determinant of the conformally rescaled metric is equal to its constrained value:
$det(\bar{\gamma}_{ij}) = det(exp(-4 \phi) \gamma_{ij})$
$= det( (\frac{\hat{\gamma}}{\gamma})^{\frac{1}{3}} \gamma_{ij})$
$= \frac{\hat{\gamma}}{\gamma} \gamma$
$= \hat{\gamma}$
partial derivative of conformal metric:
$\bar{\gamma}_{ij,\mu} = (exp(-4 \phi) \gamma_{ij})_{,\mu}$
$= exp(-4 \phi) \gamma_{ij,\mu} - 4 exp(-4 \phi) \gamma_{ij} \phi_{,\mu}$
$= exp(-4 \phi) (\gamma_{ij,\mu} - 4 \gamma_{ij} \phi_{,\mu})$
$= exp(-4 \phi) \gamma_{ij,\mu} - 4 \bar{\gamma}_{ij} \phi_{,\mu}$
so $\gamma_{ij,\mu} = exp(4 \phi) (\bar{\gamma}_{ij,\mu} + 4 \bar{\gamma}_{ij} \phi_{,\mu})$
so $\gamma_{ij,\mu} = W^{-2} (\bar{\gamma}_{ij,\mu} - 2 W^{-1} \bar{\gamma}_{ij} W_{,\mu})$
Connection of the conformal metric:
$\bar{\Gamma}_{ijk} = \frac{1}{2} (\bar{\gamma}_{ij,k} + \bar{\gamma}_{ik,j} - \bar{\gamma}_{jk,k})$
$= \frac{1}{2} exp(-4 \phi) (\gamma_{ij,k} + \gamma_{ik,j} - \gamma_{jk,k} - 4 (\gamma_{ij} \phi_{,k} + \gamma_{ik} \phi_{,j} - \gamma_{jk} \phi_{,i}))$
$= exp(-4 \phi) (\Gamma_{ijk} - 2 (\gamma_{ij} \phi_{,k} + \gamma_{ik} \phi_{,j} - \gamma_{jk} \phi_{,i}))$
$\bar{\Gamma}_{ijk} = exp(-4 \phi) (\Gamma_{ijk} - 2 (\gamma_{ij} \phi_{,k} + \gamma_{ik} \phi_{,j} - \gamma_{jk} \phi_{,i}))$
$\Gamma_{ijk} = exp(4 \phi) \bar{\Gamma}_{ijk} + 2 (\gamma_{ij} \phi_{,k} + \gamma_{ik} \phi_{,j} - \gamma_{jk} \phi_{,i})$
$\Gamma_{ijk} = exp(4 \phi) (\bar{\Gamma}_{ijk} + 2 (\bar{\gamma}_{ij} \phi_{,k} + \bar{\gamma}_{ik} \phi_{,j} - \bar{\gamma}_{jk} \phi_{,i}))$
${\bar{\Gamma}^i}_{jk} = \bar{\gamma}^{il} \bar{\Gamma}_{ljk}$
$= exp(4\phi) \gamma^{il} \cdot exp(-4 \phi) (\Gamma_{ljk} - 2 (\gamma_{lj} \phi_{,k} + \gamma_{lk} \phi_{,j} - \gamma_{jk} \phi_{,l}))$
$= {\Gamma^i}_{jk} - 2 (\delta^i_j \phi_{,k} + \delta^i_k \phi_{,j} - \gamma^{il} \gamma_{jk} \phi_{,l})$
so
${\Gamma^k}_{ij} = {\bar{\Gamma}^k}_{ij}
+ 2 (
\delta^k_i \phi_{,j}
+ \delta^k_j \phi_{,i}
- \gamma_{ij} \gamma^{kl} \phi_{,l}
)$
trace of conformal connection:
$\bar{\Gamma}^i = {\bar{\Gamma}^i}_{jk} \bar{\gamma}^{jk}$
$= (
{\Gamma^i}_{jk} - 2 (
\delta^i_j \phi_{,k} + \delta^i_k \phi_{,j} - \gamma_{jk} \gamma^{il} \phi_{,l}
)
) \bar{\gamma}^{jk}$
$=
{\Gamma^i}_{jk} exp(4 \phi) \gamma^{jk}
- 2 exp(4 \phi) \gamma^{ij} \phi_{,j}
- 2 exp(4 \phi) \gamma^{ij} \phi_{,j}
+ 2 exp(4 \phi) \gamma^{jk} \gamma_{jk} \gamma^{il} \phi_{,l}
$
$= exp(4 \phi) \Gamma^i
- 4 \bar{\gamma}^{ij} \phi_{,j}
+ 6 \bar{\gamma}^{il} \phi_{,l}
$
$= exp(4 \phi) \Gamma^i + 2 \bar{\gamma}^{ij} \phi_{,j}$
so
$\Gamma^i = exp(-4 \phi) (\bar{\Gamma}^i - 2 \bar{\gamma}^{ij} \phi_{,j})$
also
$\bar{\Gamma}^i = {\bar{\Gamma}^i}_{jk} \bar{\gamma}^{jk}$
$= {\bar{\Gamma}^i}_{jk} \bar{\gamma}^{jk}
+ {\bar{\Gamma}^k}_{jk} \bar{\gamma}^{ij}
- {\bar{\Gamma}^k}_{jk} \bar{\gamma}^{ij}
$
$= \bar{\gamma}^{ik} \bar{\Gamma}_{kjl} \bar{\gamma}^{lj}
+ \bar{\gamma}^{ik} \bar{\Gamma}_{ljk} \bar{\gamma}^{lj}
- {\bar{\Gamma}^k}_{jk} \bar{\gamma}^{ij}
$
$= - {\bar{\Gamma}^k}_{jk} \bar{\gamma}^{ij}
+ \bar{\gamma}^{ik} (\bar{\Gamma}_{kjl} + \bar{\Gamma}_{ljk}) \bar{\gamma}^{lj}
$
... using $\Gamma^{ikj} + \Gamma_{jki} = \gamma_{ij,k}$
... using ${\Gamma^j}_{ij} = \frac{1}{\sqrt{\gamma}} (\sqrt{\gamma})_{,i}$ ...
$= - \frac{1}{\sqrt{\bar{\gamma}}} (\sqrt{\bar{\gamma}})_{,j} \bar{\gamma}^{ij}
+ \bar{\gamma}^{ik} \bar{\gamma}_{kl,j} \bar{\gamma}^{lj}
$
$= -\frac{1}{\sqrt{\bar{\gamma}}} (
(\sqrt{\bar{\gamma}})_{,j} \bar{\gamma}^{ij}
+ \sqrt{\bar{\gamma}} {\bar{\gamma}^{ij}}_{,j}
)$
$= -\frac{1}{\sqrt{\bar{\gamma}}} ( \sqrt{\bar{\gamma}} \bar{\gamma}^{ij})_{,j}$
The steps so far work for any Levi-Civita connection.
Now substitute $\bar{\gamma} = \hat{\gamma}$:
$= -\frac{1}{\sqrt{\hat{\gamma}}} ( \sqrt{\hat{\gamma}} \hat{\gamma}^{ij})_{,j}$
Now if our grid metric has a constant determinant then we can use the fact that $\bar{\gamma}$ is constant to simplify the expression to: $\bar{\Gamma}^i = -{\bar{\gamma}^{ij}}_{,j}$. But for other grids (like spherical), we can't use this.
Difference between conformal connection and the grid connection:
While connections themselves are not tensors (they are pseudotensors), the difference between connections is a tensor (citation / evidence?).
${\Delta^i}_{ij} = {\bar{\Gamma}^i}_{ij} - {\hat{\Gamma}^i}_{ij}$
Trace of the difference of the connections:
$\Delta^i = {\Delta^i}_{jk} \bar{\gamma}^{jk}$
$\bar{\Lambda}^i = \Delta^i + \mathcal{C}^i$, for some sort of $\mathcal{C}^i$
TODO find where $\mathcal{C}^i$ is defined.
Trace-free extrinsic curvature:
$A_{ij} = K_{ij} - \frac{1}{3} \gamma_{ij} K$
$K_{ij} = A_{ij} + \frac{1}{3} \gamma_{ij} K$
Difference between conformal metric and grid metric
$\bar{\epsilon}_{ij} = \bar{\gamma}_{ij} - \hat{\gamma}_{ij}$
$\bar{\epsilon}_{ij} = exp(-4 \phi) \gamma_{ij} - \hat{\gamma}_{ij}$
$\bar{\epsilon}_{ij} = W^2 \gamma_{ij} - \hat{\gamma}_{ij}$
$\bar{\gamma}_{ij} = \bar{\epsilon}_{ij} + \hat{\gamma}_{ij}$
$\hat{\gamma}_{ij} = \bar{\gamma}_{ij} - \bar{\epsilon}_{ij}$
Derivative of difference of metrics:
$\bar{\epsilon}_{ij,\mu} = \bar{\gamma}_{ij,\mu} - \hat{\gamma}_{ij,\mu}$
$\bar{\epsilon}_{ij,k} = \bar{\gamma}_{ij,k} - \hat{\gamma}_{ij,k}$
$\bar{\gamma}_{ij,k} = \bar{\epsilon}_{ij,k} + \hat{\gamma}_{ij,k}$
in terms of primitives:
$\bar{\Gamma}_{ijk} = \frac{1}{2} (\bar{\gamma}_{ij,k} + \bar{\gamma}_{ik,j} - \bar{\gamma}_{jk,i})$
$= \frac{1}{2} (
\bar{\epsilon}_{ij,k} + \bar{\epsilon}_{ik,j} - \bar{\epsilon}_{jk,i}
+ \hat{\gamma}_{ij,k} + \hat{\gamma}_{ik,j} - \hat{\gamma}_{jk,i}
)$
Difference of connections:
$\bar{\Gamma}_{ijk} - \hat{\Gamma}_{ijk} = \frac{1}{2} (
\bar{\gamma}_{ij,k} + \bar{\gamma}_{ik,j} - \bar{\gamma}_{jk,i}
- \hat{\gamma}_{ij,k} - \hat{\gamma}_{ik,j} + \hat{\gamma}_{jk,i}
)$
$= \frac{1}{2} (\bar{\epsilon}_{ij,k} + \bar{\epsilon}_{ik,j} - \bar{\epsilon}_{jk,i})$
extrinsic curvature trace:
$K = \gamma^{ij} K_{ij}$
conformally rescaled trace-free extrinsic curvature:
$\bar{A}_{ij} = exp(-4 \phi) A_{ij}$
$\bar{A}_{ij} = (\frac{\hat{\gamma}}{\gamma})^{\frac{1}{3}} A_{ij}$
$A_{ij} = exp(4 \phi) \bar{A}_{ij}$
$K_{ij} = exp(4 \phi) \bar{A}_{ij} + \frac{1}{3} K exp(4 \phi) \bar{\gamma}_{ij}$
$K_{ij} = exp(4 \phi) (\bar{A}_{ij} + \frac{1}{3} K \bar{\gamma}_{ij})$
Gauss' equation:
$R^\perp_{ijkl} + K_{ik} K_{jl} - K_{il} K_{jk} = \perp R_{ijkl}$
Solving the time evolution of these variables:
$\phi_{,\mu} = (\frac{1}{12} ln \frac{\gamma}{\hat{\gamma}})_{,\mu}$
$\phi_{,\mu} = \frac{1}{12} \frac{\hat{\gamma}}{\gamma} (\frac{\gamma}{\hat{\gamma}})_{,\mu}$
$\phi_{,\mu} = \frac{1}{12} \frac{\hat{\gamma}}{\gamma} (
\frac{
\gamma_{,\mu} \hat{\gamma} - \gamma \hat{\gamma}_{,\mu}
}{\hat{\gamma}^2}
)$
$\phi_{,\mu} = \frac{1}{12} \frac{1}{\gamma} (
\gamma_{,\mu}
- \frac{\gamma}{\hat{\gamma}} \hat{\gamma}_{,\mu}
)$
using $\gamma_{,\mu} = \gamma \gamma^{ij} \gamma_{ij,\mu}$
$\phi_{,\mu} = \frac{1}{12} \frac{1}{\gamma} (
\gamma \gamma^{ij} \gamma_{ij,\mu}
- \frac{\gamma}{\hat{\gamma}} \hat{\gamma} \hat{\gamma}^{ij} \hat{\gamma}_{ij,\mu}
)$
$\phi_{,\mu} = \frac{1}{12} (\gamma^{ij} \gamma_{ij,\mu} - \hat{\gamma}^{ij} \hat{\gamma}_{ij,\mu})$
equivalently, since $\bar{\gamma} = \hat{\gamma}$, we can say:
$\phi_{,\mu} = \frac{1}{12} (\gamma^{ij} \gamma_{ij,\mu} - \bar{\gamma}^{ij} \bar{\gamma}_{ij,\mu})$
$\phi_{,k} = \frac{1}{12} (\gamma^{ij} \gamma_{ij,k} - \hat{\gamma}^{ij} \hat{\gamma}_{ij,k})$
or $\phi_{,k} = \frac{1}{12} (\gamma^{ij} \gamma_{ij,k} - \bar{\gamma}^{ij} \bar{\gamma}_{ij,k})$
substituting $\gamma_{ij,k} = \Gamma_{ikj} + \Gamma_{jki}$:
or $\phi_{,k} = \frac{1}{6} ({\Gamma^j}_{k j} - {\hat{\Gamma}^j}_{k j})$
or $\phi_{,k} = \frac{1}{6} ({\Gamma^j}_{k j} - {\bar{\Gamma}^j}_{k j})$
therefore
$\gamma^{ij} \gamma_{ij,k} = 12 \phi_{,k} + \hat{\gamma}^{ij} \hat{\gamma}_{ij,k}$
or $\gamma^{ij} \gamma_{ij,k} = 12 \phi_{,k} + \bar{\gamma}^{ij} \bar{\gamma}_{ij,k}$
or ${\Gamma^j}_{kj} = 6 \phi_{,k} + {\hat{\Gamma}^j}_{kj}$
or ${\Gamma^j}_{kj} = 6 \phi_{,k} + {\bar{\Gamma}^j}_{kj}$
$\phi_{,t} = \frac{1}{12} (\gamma^{ij} \gamma_{ij,t} - \hat{\gamma}^{ij} \hat{\gamma}_{ij,t})$
For static meshes, $\hat{\gamma}_{ij,t} = 0$:
$\phi_{,t} = \frac{1}{12} \gamma^{ij} \gamma_{ij,t}$
substitute $\gamma_{ij,t}$:
$\phi_{,t} = \frac{1}{12} \gamma^{ij} (
-2 \alpha K_{ij}
+ D_i \beta_j
+ D_j \beta_i)$
$\phi_{,t} = \frac{1}{6} \gamma^{ij} (
-\alpha K_{ij}
+ D_i \beta_j
)$
This is the time evolution in 2008 Alcubierre and 2010 Baumgarte, Shapiro.
alternatively...
$\phi_{,t} = \frac{1}{6} (
-\alpha K
+ D_k \beta^k
)$
$\phi_{,t} = \frac{1}{6} (
-\alpha K
+ {\beta^k}_{,k}
+ {\Gamma^j}_{kj} \beta^k
)$
using ${\Gamma^j}_{kj} = 6 \phi_{,k} + {\bar{\Gamma}^j}_{kj}$
$\phi_{,t} =
-\frac{1}{6} \alpha K
+ \beta^k \phi_{,k}
+ \frac{1}{6} {\beta^k}_{,k}
+ \frac{1}{6} {\bar{\Gamma}^j}_{kj} \beta^k
$
$W_{,k} = -2 \phi_{,k} W$
$= -\frac{1}{3} W ({\Gamma^j}_{kj} - {\bar{\Gamma}^j}_{kj})$
$W_{,t} = -2 \phi_{,t} W$
substituting $\phi_{,t}$
$= -2 W (
-\frac{1}{6} \alpha K
+ \beta^k \phi_{,k}
+ \frac{1}{6} \beta^k {\bar{\Gamma}^j}_{kj}
+ \frac{1}{6} {\beta^k}_{,k}
))$
$= \frac{1}{3} W \alpha K
- 2 W \beta^k \phi_{,k}
- \frac{1}{3} W \beta^k {\bar{\Gamma}^j}_{kj}
- \frac{1}{3} W {\beta^k}_{,k}
$
substituting $-2 W \phi_{,k} = W_{,k}$
$= \frac{1}{3} W \alpha K
+ \beta^k W_{,k}
- \frac{1}{3} W \beta^k {\bar{\Gamma}^j}_{kj}
- \frac{1}{3} W {\beta^k}_{,k}
$
$= -\frac{1}{3} W (
\bar{D}_k \beta^k
- \alpha K
)
+ \beta^k W_{,k}
$
This is the time evolution from 2017 Ruchlin et al., with source terms added.
Try with first transforming to upper...
(Which is $\beta$ naturally - covariant or contravariant? Beyond that raising and lowering, after applying $\bar{D}$, is going to include $exp(-4\phi)$.
This is why it helps to denote the indexes with transformations rather than the tensors themselves.)
$D_{(i} \beta_{j)}$
$= \gamma_{k(j} D_{i)} \beta^k$
$= \gamma_{k(j} ({\beta^k}_{,i)} + {\Gamma^k}_{i)l} \beta^l)$
$= \gamma_{k(j} {\beta^k}_{,i)} + \Gamma_{(ij)k} \beta^k$
using $\gamma_{ij,k} = 2 \Gamma_{(ij)k}$
$= \gamma_{k(j} {\beta^k}_{,i)} + \frac{1}{2} \gamma_{ij,k} \beta^k$
$= \frac{1}{2} \gamma_{ki} {\beta^k}_{,j}
+ \frac{1}{2} \gamma_{kj} {\beta^k}_{,i}
+ \frac{1}{2} \gamma_{ij,k} \beta^k$
$= \frac{1}{2} \mathcal{L}_\vec{\beta} \gamma_{ij}$
$\bar{D}_{(i} \beta_{j)}$
IF we are supposed to assume raising/lowering of $\bar{D}_i \beta_j$ should take place using the metric of the covariant derivative, $\bar{\gamma}_{ij}$:
$= \bar{\gamma}_{k(j} \bar{D}_{i)} \beta^k$
$= \bar{\gamma}_{k(j} ({\beta^k}_{,i)} + {\bar{\Gamma}^k}_{i)l} \beta^l)$
$= \bar{\gamma}_{k(j} {\beta^k}_{,i)} + \bar{\Gamma}_{(ij)k} \beta^k$
OR if we are supposed to be assuming $\beta_i = \gamma_{ij} \beta^j$:
$\bar{D}_{(i} \beta_{j)}$
$= \bar{D}_{(i} (\gamma_{j)k} \beta^k)$
$= \bar{D}_{(i} (exp(4 \phi) \bar{\gamma}_{j)k} \beta^k)$
$= \bar{D}_{(i} (exp(4 \phi)) \bar{\gamma}_{j)k} \beta^k
+ exp(4 \phi) \bar{\gamma}_{j)k} \bar{D}_{(i} (\beta^k)$
$= 4 exp(4 \phi) \phi_{,(i} \bar{\gamma}_{j)k} \beta^k
+ exp(4 \phi) \bar{\gamma}_{k(j} (
{\beta^k}_{,i)} + {\bar{\Gamma}^k}_{i)l} \beta^l
)$
$= 4 \phi_{,(i} \gamma_{j)k} \beta^k
+ exp(4 \phi) \bar{\gamma}_{k(j} (
{\beta^k}_{,i)} + {\bar{\Gamma}^k}_{i)l} \beta^l
)$
$= 4 \phi_{,(i} \beta_{j)}
+ \gamma_{k(j} {\beta^k}_{,i)}
+ exp(4 \phi) \bar{\Gamma}_{(ij)k} \beta^k
$
using $\bar{\Gamma}_{ijk} = exp(-4 \phi) (\Gamma_{ijk} - 2 (\gamma_{ij} \phi_{,k} + \gamma_{ik} \phi_{,j} - \gamma_{jk} \phi_{,i}))$
$= 4 \phi_{,(i} \beta_{j)}
+ \gamma_{k(j} {\beta^k}_{,i)}
+ exp(4 \phi) \beta^k
\cdot exp(-4 \phi) (
\Gamma_{(ij)k}
- 2 (
\gamma_{(ij)} \phi_{,k}
+ \gamma_{k(i} \phi_{,j)}
- \gamma_{k(j} \phi_{,i)}
)
)$
$= 4 \phi_{,(i} \beta_{j)}
+ \gamma_{k(j} {\beta^k}_{,i)}
+ \Gamma_{(ij)k} \beta^k
- 2 \gamma_{ij} \phi_{,k} \beta^k
$
using $D_{(i} \beta_{j)} = \gamma_{k(j} {\beta^k}_{,i)} + \Gamma_{(ij)k} \beta^k$
$= 4 \phi_{,(i} \beta_{j)}
- 2 \gamma_{ij} \phi_{,k} \beta^k
+ D_{(i} \beta_{j)}$
$= D_{(i} \beta_{j)}
+ 2 (
\phi_{,i} \beta_j
+ \phi_{,j} \beta_i
- \gamma_{ij} \phi_{,k} \beta^k
)$
$= D_{(i} \beta_{j)} + ({\Gamma^k}_{ij} - {\bar{\Gamma}^k}_{ij}) \beta_k$
so $D_{(i} \beta_{j)} - {\Gamma^k}_{ij} \beta_k = D_{(i} \beta_{j)} - {\bar{\Gamma}^k}_{ij} \beta_k$
This makes sense, if the remaining term in both after subtracting out connections is the $\beta_{(i,j)}$
$= D_{(i} \beta_{j)} + (\Gamma_{kij} - exp(4 \phi) \bar{\Gamma}_{kij}) \beta^k$
or, alternatively...
$\bar{D}_{(i} \beta_{j)} = 4 \phi_{,(i} \beta_{j)}
+ \gamma_{k(j} {\beta^k}_{,i)}
+ exp(4 \phi) \bar{\Gamma}_{(ij)k} \beta^k
$
using $\bar{\Gamma}_{(ij)k} = \frac{1}{2} \bar{\gamma}_{ij,k}$:
$= 4 \phi_{,(i} \beta_{j)}
+ exp(4 \phi) \bar{\gamma}_{k(j} {\beta^k}_{,i)}
+ \frac{1}{2} exp(4 \phi) \bar{\gamma}_{ij,k} \beta^k
$
$= 4 \phi_{,(i} \beta_{j)}
+ exp(4 \phi) \mathcal{L}_\vec{\beta} \bar{\gamma}_{ij}$
$D_{(i} \beta_{j)}$
IF we are supposed to assume raising/lowering of a covariant derivative should use the metric associated with the covariant derivative:
$= exp(4 \phi) \bar{\gamma}_{k(j} {\beta^k}_{,i)} + (
exp(4 \phi) (
\bar{\Gamma}_{(ij)k}
+ 2 (
\bar{\gamma}_{(ij)} \phi_{,k}
+ \bar{\gamma}_{k(i} \phi_{,j)}
- \bar{\gamma}_{k(j} \phi_{,i)}
)
)
) \beta^k$
$= exp(4 \phi) (
\bar{\gamma}_{k(j} {\beta^k}_{,i)}
+ \bar{\Gamma}_{(ij)k} \beta^k
+ 2 (
\bar{\gamma}_{ij} \phi_{,k}
+ \bar{\gamma}_{k(i} \phi_{,j)}
- \bar{\gamma}_{k(j} \phi_{,i)}
) \beta^k
)$
$= exp(4 \phi) (
\bar{D}_{(i} \beta_{j)}
+ 2 (
\bar{\gamma}_{ij} \phi_{,k}
+ \bar{\gamma}_{k(i} \phi_{,j)}
- \bar{\gamma}_{k(j} \phi_{,i)}
) \beta^k
)$
$exp(-4 \phi) D_{(i} \beta_{j)} =
\bar{D}_{(i} \beta_{j)}
+ 2 \bar{\gamma}_{ij} \phi_{,k} \beta^k
$
OR if we are supposed to raise/lower terms that we are covariant-differentiating with their own respective metrics:
$\bar{D}_{(i} \beta_{j)} =
4 \phi_{,(i} \beta_{j)}
- 2 \gamma_{ij} \phi_{,k} \beta^k
+ D_{(i} \beta_{j)}$
$D_{(i} \beta_{j)} =
\bar{D}_{(i} \beta_{j)}
- 4 \phi_{,(i} \beta_{j)}
+ 2 \gamma_{ij} \phi_{,k} \beta^k$
$\hat{D}_{(i} \beta_{j)}$
IF we are supposed to assume raising/lowering of a covariant derivative should use the metric associated with the covariant derivative:
$= \hat{\gamma}_{k(j} \hat{D}_{i)} \beta^k$
$= \hat{\gamma}_{k(j} ({\beta^k}_{,i)} + {\hat{\Gamma}^k}_{i)l} \beta^l)$
$= \hat{\gamma}_{k(j} {\beta^k}_{,i)} + \hat{\Gamma}_{(ij)k} \beta^k$
OR if we are supposed to raise/lower terms that we are covariant-differentiating with their own respective metrics:
$\hat{D}_{(i} \beta_{j)}$
$ = \hat{D}_{(i} (\gamma_{j)k} \beta^k)$
$ = \hat{D}_{(i} (exp(4 \phi) \bar{\gamma}_{j)k} \beta^k)$
$ = \hat{D}_{(i} (exp(4 \phi) (\bar{\epsilon}_{j)k} + \hat{\gamma}_{j)k}) \beta^k)$
$ = \hat{D}_{(i} exp(4 \phi) (\bar{\epsilon}_{j)k} + \hat{\gamma}_{j)k}) \beta^k
+ exp(4 \phi) \hat{D}_{(i} (\bar{\epsilon}_{j)k} + \hat{\gamma}_{j)k}) \beta^k
+ exp(4 \phi) (\bar{\epsilon}_{k(j} + \hat{\gamma}_{k(j}) \hat{D}_{i)} \beta^k
$
$ = 4 exp(4 \phi) \phi_{,(i} \bar{\gamma}_{j)k} \beta^k
+ exp(4 \phi) \hat{D}_{(i} (\bar{\epsilon}_{j)k}) \beta^k
+ exp(4 \phi) \bar{\gamma}_{k(j} \hat{D}_{i)} \beta^k
$
$ = 4 \phi_{,(i} \gamma_{j)k} \beta^k
+ exp(4 \phi) \beta^k \bar{\epsilon}_{k(j,i)}
- exp(4 \phi) \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\epsilon}_{j)l}
- exp(4 \phi) \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
+ \gamma_{k(j} {\beta^k}_{,i)}
+ \gamma_{k(j} {\hat{\Gamma}^k}_{i)l} \beta^l
$
using $\bar{D}_{(i} \beta_{j)} = 4 \phi_{,(i} \beta_{j)}
+ exp(4 \phi) \bar{\gamma}_{k(j} {\beta^k}_{,i)}
+ \frac{1}{2} exp(4 \phi) \bar{\gamma}_{ij,k} \beta^k
$
$ = exp(4 \phi) \beta^k \bar{\epsilon}_{k(j,i)}
- exp(4 \phi) \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\epsilon}_{j)l}
- exp(4 \phi) \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
+ \gamma_{k(j} {\hat{\Gamma}^k}_{i)l} \beta^l
+ \bar{D}_{(i} \beta_{j)}
- \frac{1}{2} exp(4 \phi) \bar{\gamma}_{ij,k} \beta^k
$
$ = exp(4 \phi) \beta^k \bar{\epsilon}_{k(j,i)}
- exp(4 \phi) \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\epsilon}_{j)l}
+ exp(4 \phi) \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\gamma}_{j)l}
- exp(4 \phi) \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
+ \bar{D}_{(i} \beta_{j)}
- exp(4 \phi) \bar{\Gamma}_{(ij)k} \beta^k
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\epsilon}_{k(j,i)}
- \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\epsilon}_{j)l}
+ \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\gamma}_{j)l}
- \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
- \beta^k {\bar{\Gamma}^l}_{k(j} \bar{\gamma}_{i)l}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\epsilon}_{k(j,i)}
+ \beta^k \hat{\gamma}_{j)l} {\hat{\Gamma}^l}_{k(i}
- \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
- \beta^k {\bar{\Gamma}^l}_{k(j} \bar{\gamma}_{i)l}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\epsilon}_{k(j,i)}
+ \beta^k \hat{\Gamma}_{(ij)k}
- \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
- \beta^k \bar{\gamma}_{i)l} {\bar{\Gamma}^l}_{k(j}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\epsilon}_{k(j,i)}
+ \frac{1}{2} \beta^k \hat{\gamma}_{ij,k}
+ \beta^k {\hat{\Gamma}^l}_{(ij)} \hat{\gamma}_{kl}
- \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\gamma}_{kl}
- \beta^k \bar{\Gamma}_{(ij)k}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\epsilon}_{k(j,i)}
+ \frac{1}{2} \beta^k \bar{\gamma}_{ij,k}
- \frac{1}{2} \beta^k \bar{\epsilon}_{ij,k}
+ \beta^k \hat{\Gamma}_{kij}
- \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\gamma}_{kl}
- \frac{1}{2} \beta^k \bar{\gamma}_{ij,k}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\epsilon}_{k(j,i)}
- \frac{1}{2} \beta^k \bar{\epsilon}_{ij,k}
+ \beta^k {\hat{\Gamma}^l}_{(ij)} \hat{\gamma}_{kl}
- \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\gamma}_{kl}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\frac{1}{2} \beta^k \bar{\epsilon}_{ki,j}
+ \frac{1}{2} \beta^k \bar{\epsilon}_{kj,i}
- \frac{1}{2} \beta^k \bar{\epsilon}_{ij,k}
- \beta^k {\hat{\Gamma}^l}_{ij} \bar{\epsilon}_{kl}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\Gamma}_{kij}
- \beta^k \hat{\Gamma}_{kij}
- \beta^k {\hat{\Gamma}^l}_{ij} \bar{\gamma}_{kl}
+ \beta^k {\hat{\Gamma}^l}_{ij} \hat{\gamma}_{kl}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k \bar{\Gamma}_{kij}
- \beta^k {\hat{\Gamma}^l}_{ij} \bar{\gamma}_{kl}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ exp(4 \phi) (
\beta^k {\bar{\Gamma}^l}_{ij} \bar{\gamma}_{kl}
- \beta^k {\hat{\Gamma}^l}_{ij} \bar{\gamma}_{kl}
)
$
$ = \bar{D}_{(i} \beta_{j)}
+ \beta_k ({\bar{\Gamma}^l}_{ij} - {\hat{\Gamma}^l}_{ij})
$
$\hat{D}_{(i} \beta_{j)} - \bar{D}_{(i} \beta_{j)} = \beta_k ({\bar{\Gamma}^l}_{ij} - {\hat{\Gamma}^l}_{ij})$
This, once again, looks correct, as the difference of two covariant derivatives is the difference of their respective connections.
TODO get this result to match the result of the alternative assumption.
I honestly feel better about this assumption than the other (especially since this assumption let's the $D_{(i} \beta_{j)} - \bar{D}_{(i} \beta_{j)}$ connection coefficients match up).
$\bar{D}_{(i} \beta_{j)} - \hat{D}_{(i} \beta_{j)}$
IF we are supposed to assume raising/lowering of a covariant derivative should use the metric associated with the covariant derivative:
$= \bar{\gamma}_{k(j} \bar{D}_{i)} \beta^k - \hat{\gamma}_{k(j} \hat{D}_{i)} \beta^k$
$= \bar{\gamma}_{k(j} ({\beta^k}_{,i)} + {\bar{\Gamma}^k}_{i)l} \beta^l) - \hat{\gamma}_{k(j} ({\beta^k}_{,i)} + {\hat{\Gamma}^k}_{i)l} \beta^l)$
$= \bar{\gamma}_{k(j} {\beta^k}_{,i)} + \bar{\Gamma}_{(ij)k} \beta^k - \hat{\gamma}_{k(j} {\beta^k}_{,i)} - \hat{\Gamma}_{(ij)k} \beta^k$
$= (\bar{\gamma}_{k(j} - \hat{\gamma}_{k(j}) {\beta^k}_{,i)} + (\bar{\Gamma}_{(ij)k} - \hat{\Gamma}_{(ij)k}) \beta^k$
$= \bar{\epsilon}_{k(j} {\beta^k}_{,i)} + \frac{1}{2} (\bar{\epsilon}_{ij,k} + \bar{\epsilon}_{k(i,j)} - \bar{\epsilon}_{k(j,i)}) \beta^k$
$= \bar{\epsilon}_{k(j} {\beta^k}_{,i)} + \frac{1}{2} \bar{\epsilon}_{ij,k} \beta^k$
so $\bar{D}_{(i} \beta_{j)} = \hat{D}_{(i} \beta_{j)} + \bar{\epsilon}_{k(j} {\beta^k}_{,i)} + \frac{1}{2} \bar{\epsilon}_{ij,k} \beta^k$
OR if we are supposed to raise/lower terms that we are covariant-differentiating with their own respective metrics:
$\bar{D}_{(i} \beta_{j)} - \hat{D}_{(i} \beta_{j)}$
$= \beta_k ({\hat{\Gamma}^l}_{ij} - {\bar{\Gamma}^l}_{ij})$
so
IF we are supposed to assume raising/lowering of a covariant derivative should use the metric associated with the covariant derivative
$exp(-4 \phi) D_{(i} \beta_{j)} =
\bar{D}_{(i} \beta_{j)}
+ 2 (
\bar{\gamma}_{ij} \phi_{,k}
+ \bar{\gamma}_{k(i} \phi_{,j)}
- \bar{\gamma}_{k(j} \phi_{,i)}
) \beta^k
$
$exp(-4 \phi) D_{(i} \beta_{j)} =
\hat{D}_{(i} \beta_{j)}
+ \bar{\epsilon}_{k(j} {\beta^k}_{,i)}
+ \frac{1}{2} \bar{\epsilon}_{ij,k} \beta^k
+ 2 \bar{\gamma}_{ij} \phi_{,k} \beta^k
$
OR if we are supposed to raise/lower terms that we are covariant-differentiating with their own respective metrics:
$exp(-4 \phi) D_{(i} \beta_{j)}$
$ = exp(-4 \phi) (
\bar{D}_{(i} \beta_{j)}
- 4 \phi_{,(i} \beta_{j)}
+ 2 \gamma_{ij} \phi_{,k} \beta^k
)$
$ = exp(-4 \phi) (
- 4 \phi_{,(i} \beta_{j)}
+ 2 \gamma_{ij} \phi_{,k} \beta^k
+ \bar{D}_{(i} \beta_{j)}
)$
$ = exp(-4 \phi) (
- 4 \phi_{,(i} \beta_{j)}
+ 2 \gamma_{ij} \phi_{,k} \beta^k
+ \hat{D}_{(i} \beta_{j)}
- exp(4 \phi) (
\beta^k \bar{\epsilon}_{k(j,i)}
- \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\epsilon}_{j)l}
- \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
- \beta^k {\bar{\Gamma}^l}_{k(j} \bar{\gamma}_{i)l}
)
+ \gamma_{k(j} {\hat{\Gamma}^k}_{i)l} \beta^l
)$
$ = exp(-4 \phi) (
- 4 \phi_{,(i} \beta_{j)}
+ 2 \gamma_{ij} \phi_{,k} \beta^k
+ \hat{D}_{(i} \beta_{j)}
+ \gamma_{k(j} {\hat{\Gamma}^k}_{i)l} \beta^l
)
- \beta^k \bar{\epsilon}_{k(j,i)}
+ \beta^k {\hat{\Gamma}^l}_{k(i} \bar{\epsilon}_{j)l}
+ \beta^k {\hat{\Gamma}^l}_{(ij)} \bar{\epsilon}_{kl}
+ \beta^k {\bar{\Gamma}^l}_{k(j} \bar{\gamma}_{i)l}
$
$\bar{\gamma}_{ij,t} = (exp(-4 \phi) \gamma_{ij})_{,t}$
$= (W^2 \gamma_{ij})_{,t}$
$= 2 W W_{,t} \gamma_{ij} + W^2 \gamma_{ij,t}$
substitute $W_{,t}$:
$= 2 W \gamma_{ij} (
-\frac{1}{3} W (
\bar{D}_k \beta^k
- \alpha K
)
+ \beta^k W_{,k}
)
+ W^2 \gamma_{ij,t}
$
substitute $\gamma_{ij,t}$
$= - \frac{2}{3} W^2 \gamma_{ij} \bar{D}_k \beta^k
+ \frac{2}{3} W^2 \gamma_{ij} \alpha K
+ 2 W \gamma_{ij} \beta^k W_{,k}
+ W^2 (
-2 \alpha K_{ij}
+ 2 D_{(i} \beta_{j)}
)
$
$= - \frac{2}{3} W^2 \gamma_{ij} \bar{D}_k \beta^k
+ 2 W^2 D_{(i} \beta_{j)}
- 2 \alpha W^2 K_{ij}
+ \frac{2}{3} W^2 \gamma_{ij} \alpha K
+ 2 W \gamma_{ij} \beta^k W_{,k}
$
add in a scalar of ${\bar{A}^k}_k = 0$:
$= \frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
+ 2 W^2 D_{(i} \beta_{j)}
-2 \alpha W^2 (
K_{ij}
- \frac{1}{3} \gamma_{ij} K
)
+ 2 W \gamma_{ij} \beta^k W_{,k}
$
substitute $W_{,k}$
$= \frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
+ 2 W^2 D_{(i} \beta_{j)}
-2 \alpha W^2 (
K_{ij}
- \frac{1}{3} \gamma_{ij} K
)
- 4 W^2 \gamma_{ij} \phi_{,k} \beta^k
$
substitute the definition of $W^2 D_{(i} \beta_{j)}$
$= \frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
+ 2 (
\hat{D}_{(i} \beta_{j)}
+ \bar{\epsilon}_{k(j} {\beta^k}_{,i)}
+ \frac{1}{2} \bar{\epsilon}_{ij,k} \beta^k
+ 2 \bar{\gamma}_{ij} \phi_{,k} \beta^k
)
- 2 \alpha \bar{A}_{ij}
- 4 W^2 \gamma_{ij} \phi_{,k} \beta^k
$
$= \frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
+ 2 \hat{D}_{(i} \beta_{j)}
- 2 \alpha \bar{A}_{ij}
+ \bar{\epsilon}_{ki} {\beta^k}_{,j}
+ \bar{\epsilon}_{kj} {\beta^k}_{,i}
+ \bar{\epsilon}_{ij,k} \beta^k
$
$\bar{\epsilon}_{ij,t} = \bar{\gamma}_{ij,t}$
$= \frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
+ 2 \hat{D}_{(i} \beta_{j)}
- 2 \alpha \bar{A}_{ij}
+ \bar{\epsilon}_{ki} {\beta^k}_{,j}
+ \bar{\epsilon}_{kj} {\beta^k}_{,i}
+ \bar{\epsilon}_{ij,k} \beta^k
$
This is the time evolution from 2017 Ruchlin et al., with source terms added.
substitute $\bar{D}_{(i} \beta_{j)} = \hat{D}_{(i} \beta_{j)} + \bar{\epsilon}_{k(j} {\beta^k}_{,i)} + \frac{1}{2} \bar{\epsilon}_{ij,k} \beta^k$
$= \frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
- 2 \alpha \bar{A}_{ij}
+ 2 \bar{D}_{(i} \beta_{j)}$
substitute
$= \frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
- 2 \alpha \bar{A}_{ij}
+ \bar{D}_i (\gamma_{jk} \beta^k)
+ \bar{D}_j (\gamma_{ik} \beta^k)$
$D_i D_j \alpha$
$= D_i \alpha_{,j}$
$= \alpha_{,ij} - {\Gamma^k}_{ij} \alpha_{,k}$
$= \alpha_{,ij} - (
{\bar{\Gamma}^k}_{ij}
+ 2 \delta^k_i \phi_{,j}
+ 2 \delta^k_j \phi_{,i}
- 2 \gamma_{ij} \gamma^{kl} \phi_{,l}
) \alpha_{,k}$
$= \alpha_{,ij}
- {\bar{\Gamma}^k}_{ij} \alpha_{,k}
- 2 \phi_{,j} \alpha_{,i}
- 2 \phi_{,i} \alpha_{,j}
+ 2 \gamma_{ij} \gamma^{kl} \phi_{,l} \alpha_{,k}$
$\gamma^{ij} D_i D_j \alpha$
$= \gamma^{ij} (\alpha_{,ij} - {\Gamma^k}_{ij} \alpha_{,k})$
$= \gamma^{ij} \alpha_{,ij} - \Gamma^k \alpha_{,k}$
$= \gamma^{ij} \alpha_{,ij} - exp(-4 \phi) (\bar{\Gamma}^k - 2 \bar{\gamma}^{kl} \phi_{,l}) \alpha_{,k}$
$= \gamma^{ij} \alpha_{,ij} - exp(-4 \phi) \bar{\Gamma}^i \alpha_{,i} + 2 exp(-4 \phi) \bar{\gamma}^{ij} \phi_{,i} \alpha_{,j}$
$\bar{\gamma}^{ij} \bar{D}_i \bar{D}_j \alpha$
$= \bar{\gamma}^{ij} \bar{D}_i \alpha_{,j}$
$= \bar{\gamma}^{ij} \alpha_{,ij} - \bar{\Gamma}^k \alpha_{,k}$
Therefore
$exp(4 \phi) \gamma^{ij} D_i D_j \alpha$
$= \bar{\gamma}^{ij} \alpha_{,ij}
- \bar{\Gamma}^i \alpha_{,i}
+ 2 \bar{\gamma}^{ij} \phi_{,i} \alpha_{,j}$
$= \bar{\gamma}^{ij} \bar{D}_i \bar{D}_j \alpha
+ 2 \bar{\gamma}^{ij} \phi_{,i} \alpha_{,j}$
$\bar{D}_i \bar{D}_j \alpha$
$= \bar{D}_i \alpha_{,j}$
$= \alpha_{,ij} - {\bar{\Gamma}^k}_{ij} \alpha_{,k}$
Therefore
$D_i D_j \alpha$
$= \bar{D}_i \bar{D}_j \alpha
- 2 \phi_{,j} \alpha_{,i}
- 2 \phi_{,i} \alpha_{,j}
+ 2 \gamma_{ij} \gamma^{kl} \phi_{,l} \alpha_{,k}$
time evolution of extrinsic curvature trace:
$K_{,t} = (\gamma^{ij} K_{ij})_{,t}$
$= {\gamma^{ij}}_{,t} K_{ij} + \gamma^{ij} K_{ij,t}$
$= -\gamma^{ik} \gamma_{kl,t} \gamma^{lj} K_{ij} + \gamma^{ij} K_{ij,t}$
substitute $\gamma_{ij,t}$ and $K_{ij,t}$:
$= -\gamma^{ik} (
-2 \alpha K_{kl}
+ \gamma_{kl,m} \beta^m
+ 2 \gamma_{m(k} {\beta^m}_{,l)}
) \gamma^{lj} K_{ij} + \gamma^{ij} (
-D_i D_j \alpha
+ \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
+ K_{ij,k} \beta^k
+ 2 K_{k(i} {\beta^k}_{,j)}
)$
$=
2 \alpha {K^i}_j {K^j}_i
- \gamma^{ik} \gamma_{kl,m} \beta^m \gamma^{lj} K_{ij}
- \gamma^{ik} \gamma_{mk} {\beta^m}_{,l} \gamma^{lj} K_{ij}
- \gamma^{ik} \gamma_{ml} {\beta^m}_{,k} \gamma^{lj} K_{ij}
- \gamma^{ij} D_i D_j \alpha
+ \alpha \gamma^{ij} (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)
+ 4 \pi \alpha \gamma^{ij} (\gamma_{ij} (S - \rho) - 2 S_{ij})
+ \gamma^{ij} K_{ij,k} \beta^k
+ \gamma^{ij} K_{ki} {\beta^k}_{,j}
+ \gamma^{ij} K_{kj} {\beta^k}_{,i}
$
$=
2 \alpha {K^i}_j {K^j}_i
+ {\gamma^{ij}}_{,k} \beta^k K_{ij}
- {\beta^i}_{,j} {K^j}_i
- {\beta^i}_{,j} {K^j}_i
- \gamma^{ij} D_i D_j \alpha
+ \alpha (R + K^2 - 2 {K^i}_j {K^j}_i)
+ 4 \pi \alpha (3 (S - \rho) - 2 S)
+ \gamma^{ij} K_{ij,k} \beta^k
+ {\beta^i}_{,j} {K^j}_i
+ {\beta^i}_{,j} {K^j}_i
$
$=
- \gamma^{ij} D_i D_j \alpha
+ 2 \alpha {K^i}_j {K^j}_i
+ \alpha (R + K^2 - 2 {K^i}_j {K^j}_i)
+ 4 \pi \alpha (3 (S - \rho) - 2 S)
+ \beta^k {\gamma^{ij}}_{,k} K_{ij}
+ \beta^k \gamma^{ij} K_{ij,k}
$
$= - \gamma^{ij} D_i D_j \alpha
+ \alpha R
+ \alpha K^2
+ 4 \pi \alpha (S - 3 \rho)
+ \beta^k K_{,k}
$
using $R = \bar{A}_{ij} \bar{A}^{ij} - \frac{2}{3} K^2 + 16 \pi \rho$
$= - \gamma^{ij} D_i D_j \alpha
+ \alpha (\bar{A}_{ij} \bar{A}^{ij} - \frac{2}{3} K^2 + 16 \pi \rho)
+ \alpha K^2
+ 4 \pi \alpha (S - 3 \rho)
+ \beta^k K_{,k}
$
$= - \gamma^{ij} D_i D_j \alpha
+ \alpha (\bar{A}_{ij} \bar{A}^{ij} + \frac{1}{3} K^2)
+ 4 \pi \alpha (S + \rho)
+ \beta^k K_{,k}
$
This is the time evolution in 2008 Alcubierre and 2010 Baumgarte, Shapiro.
alternatively, further substituting gets us...
using $exp(4 \phi) \gamma^{ij} D_i D_j \alpha = \bar{\gamma}^{ij} \bar{D}_i \bar{D}_j \alpha + 2 \bar{\gamma}^{ij} \phi_{,i} \alpha_{,j}$...
$= - exp(-4 \phi) (
\bar{\gamma}^{ij} \bar{D}_i \bar{D}_j \alpha
+ 2 \bar{\gamma}^{ij} \phi_{,i} \alpha_{,j}
)
+ \alpha (\bar{A}_{ij} \bar{A}^{ij} + \frac{1}{3} K^2)
+ 4 \pi \alpha (S + \rho)
+ \beta^k K_{,k}
$
$= \frac{1}{3} \alpha K^2
+ \alpha \bar{A}_{ij} \bar{A}^{ij}
- exp(-4 \phi) (
\bar{D}^i \bar{D}_i \alpha
+ 2 \bar{D}^i \alpha \bar{D}_i \phi
)
+ 4 \pi \alpha (S + \rho)
+ \beta^k K_{,k}
$
This is the time evolution from 2017 Ruchlin et al., with source terms added.
$\mathcal{L}_{\vec\beta} \bar{\gamma}_{ij}
= \bar{\gamma}_{ij,k} \beta^k
+ \bar{\gamma}_{kj} {\beta^k}_{,i}
+ \bar{\gamma}_{ik} {\beta^k}_{,j}$
$ = (exp(-4 \phi) \gamma_{ij})_{,k} \beta^k
+ exp(-4 \phi) \gamma_{kj} {\beta^k}_{,i}
+ exp(-4 \phi) \gamma_{ik} {\beta^k}_{,j}$
$ = (-4 \phi_{,k} exp(-4 \phi) \gamma_{ij} + exp(-4 \phi) \gamma_{ij,k}) \beta^k
+ exp(-4 \phi) \gamma_{kj} {\beta^k}_{,i}
+ exp(-4 \phi) \gamma_{ik} {\beta^k}_{,j}$
$ = exp(-4 \phi) (
-4 \gamma_{ij} \phi_{,k} \beta^k
+ \gamma_{ij,k} \beta^k
+ \gamma_{kj} {\beta^k}_{,i}
+ \gamma_{ik} {\beta^k}_{,j})$
$(D_i D_j \alpha)^{TF}$
$= D_i D_j \alpha - \frac{1}{3} \gamma_{ij} \gamma^{kl} D_k D_l \alpha $
$= D_i D_j \alpha - \frac{1}{3} \gamma_{ij} (
\gamma^{kl} \alpha_{,kl}
- exp(-4 \phi) \bar{\Gamma}^k \alpha_{,k}
+ 2 exp(-4 \phi) \bar{\gamma}^{kl} \phi_{,k} \alpha_{,l}
)$
$= D_i D_j \alpha
- \frac{1}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \alpha_{,kl}
+ \frac{1}{3} \bar{\gamma}_{ij} \bar{\Gamma}^k \alpha_{,k}
- \frac{2}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \alpha_{,l}$
$\mathcal{L}_{\vec\beta} \bar{A}_{ij}
=\bar{A}_{ij,k} \beta^k + \bar{A}_{kj} {\beta^k}_{,i} + \bar{A}_{ik} {\beta^k}_{,j}$
$K \bar{A}_{ij} - 2 \bar{A}_{il} {\bar{A}^l}_j$
$= exp(-4 \phi) K A_{ij} - 2 \bar{A}_{ik} \bar{\gamma}^{kl} \bar{A}_{lj}$
$= exp(-4 \phi) K A_{ij} - 2 exp(-4 \phi) \gamma^{kl} A_{ik} A_{lj}$
$= exp(-4 \phi) K (K_{ij} - \frac{1}{3} \gamma_{ij} K)
- 2 exp(-4 \phi) \gamma^{kl} (K_{ik} - \frac{1}{3} \gamma_{ik} K) (K_{lj} - \frac{1}{3} \gamma_{lj} K)$
$= exp(-4 \phi) (
K K_{ij}
- \frac{1}{3} \gamma_{ij} K^2
- 2 \gamma^{kl} K_{ik} K_{lj}
+ \frac{2}{3} \gamma^{kl} \gamma_{ik} K_{lj} K
+ \frac{2}{3} \gamma^{kl} \gamma_{lj} K_{ik} K
- \frac{2}{9} \gamma^{kl} \gamma_{ik} \gamma_{lj} K^2
)$
$= exp(-4 \phi) (
- \frac{1}{3} \gamma_{ij} K^2
- \frac{2}{9} \gamma_{ij} K^2
+ K K_{ij}
+ \frac{2}{3} K_{ij} K
+ \frac{2}{3} K_{ij} K
- 2 K_{ik} {K^k}_j
)$
$= exp(-4 \phi) (
- 2 K_{ik} {K^k}_j
- \frac{5}{9} \gamma_{ij} K^2
+ \frac{7}{3} K_{ij} K
)$
Taken from my 'Differential Geometry' notes on worksheet 15 on ADM
$R = K_{ij} K^{ij} - K^2 + 16 \pi \rho$
from there,
using $K_{ij} = A_{ij} + \frac{1}{3} \gamma_{ij} K$
$= (A_{ij} + \frac{1}{3} \gamma_{ij} K) \gamma^{ik} \gamma^{jl} (A_{kl} + \frac{1}{3} \gamma_{kl} K) - K^2 + 16 \pi \rho$
$= A_{ij} A^{ij}
+ \frac{2}{3} A_{ij} \gamma^{ij} K
+ \frac{1}{9} \gamma_{ij} \gamma^{ij} K^2
- K^2 + 16 \pi \rho$
$= A_{ij} A^{ij}
+ \frac{2}{3} (K_{ij} - \frac{1}{3} \gamma_{ij} K) \gamma^{ij} K
- \frac{2}{3} K^2 + 16 \pi \rho$
$= A_{ij} A^{ij} - \frac{2}{3} K^2 + 16 \pi \rho$
$= \bar{A}_{ij} \bar{A}^{ij} - \frac{2}{3} K^2 + 16 \pi \rho$
Riemann curvature tensor of 3-manifold:
${R^i}_{jkl} = 2 {\Gamma^i}_{j[l,k]} + 2 {\Gamma^i}_{m[k} {\Gamma^m}_{l]j}$
$= {\Gamma^i}_{jl,k} - {\Gamma^i}_{jk,l} + {\Gamma^i}_{mk} {\Gamma^m}_{lj} - {\Gamma^i}_{ml} {\Gamma^m}_{kj}$
Riemann curvature tensor of 3-manifold written in terms of the 3-'conformal'-metric connection...
$= 2 ({\bar{\Gamma}^i}_{j[l} + 2 (\delta^i_j \phi_{,[l} + \delta^i_{[l|} \phi_{,j} - \gamma^{im} \phi_{,m} \gamma_{j[l}))_{,k]}
+ 2 (
{\bar{\Gamma}^i}_{m[k} + 2 (\delta^i_m \phi_{,[k} + \delta^i_{[k|} \phi_{,m} - \gamma^{in} \phi_{,n} \gamma_{m[k})
) (
{\bar{\Gamma}^m}_{l]j} + 2 (\delta^m_{|l]} \phi_{,j} + \delta^m_j \phi_{,|l]} - \gamma^{mp} \phi_{,p} \gamma_{l]j})
)
$
$= 2 (
{\bar{\Gamma}^i}_{j[l,k]}
+ 2 (\delta^i_j \phi_{,[lk]} + \delta^i_{[l} \phi_{,k]j}
- {\gamma^{im}}_{,[k|} \phi_{,m} \gamma_{|l]j}
- \gamma^{im} \phi_{,m[k} \gamma_{l]j}
- \gamma^{im} \phi_{,m} \gamma_{j[l,k]}
))
+ 2 (
{\bar{\Gamma}^i}_{m[k} {\bar{\Gamma}^m}_{l]j}
+ 2 {\bar{\Gamma}^i}_{m[k} (\delta^m_{|l]} \phi_{,j} + \delta^m_j \phi_{,|l]} - \gamma^{mp} \phi_{,p} \gamma_{l]j})
+ 2 (\delta^i_m \phi_{,[k} + \delta^i_{[k|} \phi_{,m} - \gamma^{in} \phi_{,n} \gamma_{m[k}) {\bar{\Gamma}^m}_{l]j}
+ 4 (\delta^i_m \phi_{,[k} + \delta^i_{[k|} \phi_{,m} - \gamma^{in} \phi_{,n} \gamma_{m[k})
(\delta^m_{|l]} \phi_{,j} + \delta^m_j \phi_{,|l]} - \gamma^{mp} \phi_{,p} \gamma_{l]j})
)
$
$= 2 {\bar{\Gamma}^i}_{j[l,k]}
+ 2 {\bar{\Gamma}^i}_{m[k} {\bar{\Gamma}^m}_{l]j}
+ 2 {\bar{\Gamma}^i}_{jk} \phi_{,l}
- 2 {\bar{\Gamma}^i}_{jl} \phi_{,k}
- 2 {\bar{\Gamma}^i}_{km} \gamma_{jl} \gamma^{mp} \phi_{,p}
+ 2 {\bar{\Gamma}^i}_{lm} \gamma_{jk} \gamma^{mp} \phi_{,p}
+ 2 \phi_{,k} {\bar{\Gamma}^i}_{jl}
- 2 \phi_{,l} {\bar{\Gamma}^i}_{jk}
+ 2 \delta^i_k {\bar{\Gamma}^m}_{jl} \phi_{,m}
- 2 \delta^i_l {\bar{\Gamma}^m}_{jk} \phi_{,m}
- 2 \gamma^{in} \phi_{,n} \gamma_{km} {\bar{\Gamma}^m}_{jl}
+ 2 \gamma^{in} \phi_{,n} \gamma_{lm} {\bar{\Gamma}^m}_{jk}
+ 2 \delta^i_l \phi_{,jk}
- 2 \delta^i_k \phi_{,jl}
- 2 \gamma^{im} \phi_{,m} \gamma_{jl,k}
+ 2 \gamma^{im} \phi_{,m} \gamma_{jk,l}
- 2 \gamma^{im} \phi_{,km} \gamma_{jl}
+ 2 \gamma^{im} \phi_{,lm} \gamma_{jk}
+ 2 \gamma^{im} \gamma_{mn,k} \gamma_{jl} \gamma^{np} \phi_{,p}
- 2 \gamma^{im} \gamma_{mn,l} \gamma_{jk} \gamma^{np} \phi_{,p}
+ 4 \delta^i_k \phi_{,l} \phi_{,j}
- 4 \delta^i_l \phi_{,k} \phi_{,j}
+ 4 \delta^i_j \phi_{,k} \phi_{,l}
- 4 \delta^i_j \phi_{,l} \phi_{,k}
- 4 \delta^i_k \gamma_{lj} \gamma^{mn} \phi_{,m} \phi_{,n}
+ 4 \delta^i_l \gamma_{kj} \gamma^{mn} \phi_{,m} \phi_{,n}
+ 4 \gamma^{im} \phi_{,m} \gamma_{jl} \phi_{,k}
- 4 \gamma^{im} \phi_{,m} \gamma_{jk} \phi_{,l}
$
Ricci curvature tensor of 3-manifold:
$R_{ij} = {R^k}_{ikj} = {\Gamma^k}_{ij,k} - {\Gamma^k}_{ik,j} + {\Gamma^k}_{lk} {\Gamma^l}_{ji} - {\Gamma^k}_{lj} {\Gamma^l}_{ki}$
Ricci tensor in terms of $\bar{\gamma}_{ij}$:
$R_{ij} =
{\bar{\Gamma}^k}_{ij,k}
- {\bar{\Gamma}^k}_{ik,j}
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^k}_{lj} {\bar{\Gamma}^l}_{ki}
+ 2 {\bar{\Gamma}^k}_{ik} \phi_{,j}
- 2 {\bar{\Gamma}^k}_{ij} \phi_{,k}
- 2 \gamma_{ij} {\bar{\Gamma}^k}_{kl} \gamma^{lm} \phi_{,m}
+ 2 \gamma_{ik} {\bar{\Gamma}^k}_{jl} \gamma^{lm} \phi_{,m}
+ 2 \phi_{,k} {\bar{\Gamma}^k}_{ij}
- 2 \phi_{,j} {\bar{\Gamma}^k}_{ik}
+ 4 {\bar{\Gamma}^k}_{ij} \phi_{,k}
- 2 \phi_{,k} {\bar{\Gamma}^k}_{ij}
+ 2 \gamma^{km} \phi_{,m} \gamma_{jl} {\bar{\Gamma}^l}_{ik}
- 2 \phi_{,ij}
+ 4 \phi_{,j} \phi_{,i}
- 2 \gamma^{kl} \phi_{,l} \gamma_{ij,k}
- 2 \gamma_{ij} \gamma^{lk} \phi_{,lk}
- 4 \gamma_{ij} \gamma^{lk} \phi_{,k} \phi_{,l}
+ 2 \gamma_{ij} \gamma^{kl} \gamma_{lm,k} \gamma^{mn} \phi_{,n}
$
$ =
\frac{1}{2} (\bar{\gamma}^{kl} (
\bar{\gamma}_{li,j}
+ \bar{\gamma}_{lj,i}
- \bar{\gamma}_{ij,l}
))_{,k}
- \frac{1}{2} (\bar{\gamma}^{kl} (
\bar{\gamma}_{li,k}
+ \bar{\gamma}_{lk,i}
- \bar{\gamma}_{ik,l}
))_{,j}
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^k}_{lj} {\bar{\Gamma}^l}_{ki}
+ 2 {\bar{\Gamma}^k}_{ik} \phi_{,j}
+ 2 \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{jl} \bar{\gamma}^{lm} \phi_{,m}
+ 2 \phi_{,k} {\bar{\Gamma}^k}_{ij}
- 2 \phi_{,j} {\bar{\Gamma}^k}_{ik}
- 2 \phi_{,k} {\bar{\Gamma}^k}_{ij}
+ 2 \bar{\gamma}^{km} \phi_{,m} \bar{\gamma}_{jl} {\bar{\Gamma}^l}_{ik}
- 2 \bar{\gamma}^{kl} \phi_{,l} \bar{\gamma}_{ij,k}
- 2 \bar{\gamma}_{ij} {\bar{\Gamma}^k}_{kl} \bar{\gamma}^{lm} \phi_{,m}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{\gamma}_{lm} \phi_{,k} \bar{\gamma}^{mn} \phi_{,n}
+ 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{\Gamma}_{lkm} \bar{\gamma}^{mn} \phi_{,n}
- 2 \phi_{,ij}
+ 2 {\bar{\Gamma}^k}_{ij} \phi_{,k}
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{lk} \phi_{,kl}
+ 2 \bar{\gamma}_{ij} \bar{\Gamma}^k \phi_{,k}
+ 4 \phi_{,j} \phi_{,i}
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{lk} \phi_{,k} \phi_{,l}
$
$ = \bar{R}_{ij} + R^\phi_{ij}$
...where $R^{\phi}_{ij} =
- 2 \phi_{,ij}
+ 2 {\bar{\Gamma}^k}_{ij} \phi_{,k}
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{lk} \phi_{,kl}
+ 2 \bar{\gamma}_{ij} \bar{\Gamma}^k \phi_{,k}
+ 4 \phi_{,i} \phi_{,j}
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{lk} \phi_{,k} \phi_{,l}
$
$R^\phi_{ij} =
-2 (\phi_{,ij} - {\bar{\Gamma}^k}_{ij} \phi_{,k})
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} (\phi_{,kl} - {\bar{\Gamma}^m}_{kl} \phi_{,m})
+ 4 \phi_{,i} \phi_{,j}
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
$R^\phi_{ij} =
-2 \bar{D}_i \bar{D}_j \phi
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{D}_k \bar{D}_l \phi
+ 4 \bar{D}_i \phi \bar{D}_j \phi
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{D}_k \phi \bar{D}_l \phi
$
This is from 2008 Alcubierre eqn 2.8.18; 2010 Baumgarte, Shapiro eqn 3.10.
...and
$\bar{R}_{ij} =
\frac{1}{2} (\bar{\gamma}^{kl} (
\bar{\gamma}_{li,j}
+ \bar{\gamma}_{lj,i}
- \bar{\gamma}_{ij,l}
))_{,k}
- \frac{1}{2} (\bar{\gamma}^{kl} (
\bar{\gamma}_{li,k}
+ \bar{\gamma}_{lk,i}
- \bar{\gamma}_{ik,l}
))_{,j}
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^k}_{lj} {\bar{\Gamma}^l}_{ki}
+ 2 {\bar{\Gamma}^k}_{ik} \phi_{,j}
+ 2 \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{jl} \bar{\gamma}^{lm} \phi_{,m}
+ 2 \phi_{,k} {\bar{\Gamma}^k}_{ij}
- 2 \phi_{,j} {\bar{\Gamma}^k}_{ik}
- 2 \phi_{,k} {\bar{\Gamma}^k}_{ij}
+ 2 \bar{\gamma}^{km} \phi_{,m} \bar{\gamma}_{jl} {\bar{\Gamma}^l}_{ik}
- 2 \bar{\gamma}^{kl} \phi_{,l} \bar{\gamma}_{ij,k}
- 2 \bar{\gamma}_{ij} {\bar{\Gamma}^k}_{kl} \bar{\gamma}^{lm} \phi_{,m}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{\gamma}_{lm} \phi_{,k} \bar{\gamma}^{mn} \phi_{,n}
+ 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{\Gamma}_{lkm} \bar{\gamma}^{mn} \phi_{,n}
$
$\bar{R}_{ij} =
(
{\bar{\gamma}^{kl}}_{,k} \bar{\Gamma}_{lij}
+ \frac{1}{2} \bar{\gamma}^{kl} (
\bar{\gamma}_{li,jk}
+ \bar{\gamma}_{lj,ik}
- \bar{\gamma}_{ij,lk}
)
)
- (
{\bar{\gamma}^{kl}}_{,j} \bar{\Gamma}_{lik}
+ \frac{1}{2} \bar{\gamma}^{kl} (
\bar{\gamma}_{li,kj}
+ \bar{\gamma}_{lk,ij}
- \bar{\gamma}_{ik,lj}
)
)
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^l}_{ki} {\bar{\Gamma}^k}_{lj}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
$\bar{R}_{ij} =
{\bar{\gamma}^{kl}}_{,k} \bar{\Gamma}_{lij}
+ \frac{1}{2} \bar{\gamma}^{kl} (
\bar{\gamma}_{li,jk}
+ \bar{\gamma}_{lj,ik}
- \bar{\gamma}_{ij,lk}
)
- {\bar{\gamma}^{kl}}_{,j} \bar{\Gamma}_{lik}
- \frac{1}{2} \bar{\gamma}^{kl} (
\bar{\gamma}_{li,kj}
+ \bar{\gamma}_{lk,ij}
- \bar{\gamma}_{ik,lj}
)
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^l}_{ki} {\bar{\Gamma}^k}_{lj}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
$\bar{R}_{ij} =
- \bar{\gamma}^{km} \bar{\gamma}_{mn,k} \bar{\gamma}^{nl} \bar{\Gamma}_{lij}
+ \bar{\gamma}^{km} \bar{\gamma}_{mn,j} \bar{\gamma}^{nl} \bar{\Gamma}_{lik}
- \frac{1}{2} \bar{\gamma}^{kl} (
\bar{\gamma}_{ik,jl}
+ \bar{\gamma}_{jk,il}
- \bar{\gamma}_{ij,kl}
- \bar{\gamma}_{kl,ij}
)
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^l}_{ki} {\bar{\Gamma}^k}_{lj}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
$\bar{R}_{ij} =
- \bar{\gamma}^{km} (\bar{\Gamma}_{mkn} + \bar{\Gamma}_{nkm}) \bar{\gamma}^{nl} \bar{\Gamma}_{lij}
+ \bar{\gamma}^{km} (\bar{\Gamma}_{mjn} + \bar{\Gamma}_{njm}) \bar{\gamma}^{nl} \bar{\Gamma}_{lik}
- \frac{1}{2} \bar{\gamma}^{kl} (
\bar{\gamma}_{ik,jl}
+ \bar{\gamma}_{jk,il}
- \bar{\gamma}_{ij,kl}
- \bar{\gamma}_{kl,ij}
)
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^l}_{ki} {\bar{\Gamma}^k}_{lj}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
$\bar{R}_{ij} =
- {\bar{\Gamma}^k}_{lk} {\bar{\Gamma}^l}_{ij}
- {\bar{\Gamma}^{lk}}_k \bar{\Gamma}_{lij}
+ {\bar{\Gamma}^k}_{jn} \bar{\gamma}^{nl} \bar{\Gamma}_{lik}
+ {\bar{\Gamma}^{lk}}_j \bar{\Gamma}_{lik}
- \frac{1}{2} \bar{\gamma}^{kl} (
\bar{\gamma}_{ik,jl}
+ \bar{\gamma}_{jk,il}
- \bar{\gamma}_{ij,kl}
- \bar{\gamma}_{kl,ij}
)
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^l}_{ki} {\bar{\Gamma}^k}_{lj}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
$\bar{R}_{ij} =
+ {\bar{\Gamma}^l}_{ij} {\bar{\Gamma}^k}_{lk}
- {\bar{\Gamma}^k}_{lj} {\bar{\Gamma}^l}_{ik}
- {\bar{\Gamma}^k}_{lk} {\bar{\Gamma}^l}_{ij}
- {\bar{\Gamma}^{lk}}_k \bar{\Gamma}_{lij}
+ {\bar{\Gamma}^k}_{jl} {\bar{\Gamma}^l}_{ik}
+ {\bar{\Gamma}^{lk}}_j \bar{\Gamma}_{lik}
- \bar{\gamma}^{kl} \bar{\Gamma}_{kij,l}
+ \frac{1}{2} \bar{\gamma}^{kl} \bar{\gamma}_{kl,ij}
+ 8 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \phi_{,k} \phi_{,l}
$
(TODO derive this)
$\bar{R}_{ij} =
-\frac{1}{2} \bar{\gamma}^{kl} \bar{\gamma}_{ij,kl}
+ \frac{1}{2} \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{lm,j} \bar{\gamma}^{lm}
+ \frac{1}{2} \bar{\gamma}_{jk} {\bar{\Gamma}^k}_{lm,i} \bar{\gamma}^{lm}
+ \frac{1}{2} \bar{\Gamma}_{ijk} {\bar{\Gamma}^{kl}}_l
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{jki}
+ {\bar{\Gamma}^{kl}}_i \bar{\Gamma}_{klj}
$
$\bar{R}_{ij} =
-\frac{1}{2} \bar{\gamma}^{kl} \bar{\gamma}_{ij,kl}
+ \frac{1}{2} \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{lm,j} \bar{\gamma}^{lm}
- \frac{1}{2} \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{lm} \bar{\gamma}^{lp} \bar{\gamma}_{pq,j} \bar{\gamma}^{qm}
+ \frac{1}{2} \bar{\gamma}_{jk} {\bar{\Gamma}^k}_{lm,i} \bar{\gamma}^{lm}
- \frac{1}{2} \bar{\gamma}_{jk} {\bar{\Gamma}^k}_{lm} \bar{\gamma}^{lp} \bar{\gamma}_{pq,i} \bar{\gamma}^{qm}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{ijk}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{jik}
+ {\bar{\Gamma}^{kl}}_i \bar{\Gamma}_{jkl}
+ {\bar{\Gamma}^{kl}}_j \bar{\Gamma}_{ikl}
+ {\bar{\Gamma}^{kl}}_i \bar{\Gamma}_{klj}
$
$\bar{R}_{ij} =
-\frac{1}{2} \bar{\gamma}^{kl} \bar{\gamma}_{ij,kl}
+ \frac{1}{2} \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{lm,j} \bar{\gamma}^{lm}
+ \frac{1}{2} \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{lm} {\bar{\gamma}^{lm}}_{,j}
+ \frac{1}{2} \bar{\gamma}_{jk} {\bar{\Gamma}^k}_{lm,i} \bar{\gamma}^{lm}
+ \frac{1}{2} \bar{\gamma}_{jk} {\bar{\Gamma}^k}_{lm} {\bar{\gamma}^{lm}}_{,i}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{ijk}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{jik}
+ {\bar{\Gamma}^{kl}}_i \bar{\Gamma}_{jkl}
+ {\bar{\Gamma}^{kl}}_j \bar{\Gamma}_{ikl}
+ {\bar{\Gamma}^{kl}}_i \bar{\Gamma}_{klj}
$
$\bar{R}_{ij} =
-\frac{1}{2} \bar{\gamma}^{kl} \bar{\gamma}_{ij,kl}
+ \frac{1}{2} \bar{\gamma}_{ik} ({\bar{\Gamma}^k}_{lm} \bar{\gamma}^{lm})_{,j}
+ \frac{1}{2} \bar{\gamma}_{jk} ({\bar{\Gamma}^k}_{lm} \bar{\gamma}^{lm})_{,i}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{ijk}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{jik}
+ {\bar{\Gamma}^{kl}}_i \bar{\Gamma}_{jkl}
+ {\bar{\Gamma}^{kl}}_j \bar{\Gamma}_{ikl}
+ {\bar{\Gamma}^{kl}}_i \bar{\Gamma}_{klj}
$
$\bar{R}_{ij} =
-\frac{1}{2} \bar{\gamma}^{kl} \bar{\gamma}_{ij,kl}
+ \frac{1}{2} \bar{\gamma}_{ik} {\bar{\Gamma}^k}_{,j}
+ \frac{1}{2} \bar{\gamma}_{jk} {\bar{\Gamma}^k}_{,i}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{ijk}
+ \frac{1}{2} \bar{\Gamma}^k \bar{\Gamma}_{jik}
+ \bar{\gamma}^{lm} (
{\bar{\Gamma}^k}_{li} \bar{\Gamma}_{jkm}
+ {\bar{\Gamma}^k}_{lj} \bar{\Gamma}_{ikm}
+ {\bar{\Gamma}^k}_{im} \bar{\Gamma}_{klj}
)
$
This is 2008 Alcubierre eqn 2.8.17; 2010 Baumgarte, Shapiro eqn 11.41.
... TODO more steps ...
$\bar{R}_{ij} =
-\frac{1}{2} \bar{\gamma}^{kl} \hat{D}_k \hat{D}_l \bar{\gamma}_{ij}
+ \bar{\gamma}_{k(i} \hat{D}_{j)} \bar{\Lambda}^k
+ \Delta^k \Delta_{(ij)k}
+ \bar{\gamma}^{kl} (
2 {\Delta^m}_{k(i} \Delta_{j)ml}
+ {\Delta^m}_{ik} \Delta_{mjl}
)
$
This is in 2017 Ruchlin eqn 12.
Gaussian curvature scalar:
$R = \gamma^{ij} R_{ij} = {R^{ij}}_{ij} = \gamma^{ij} ({\Gamma^k}_{ij,k} - {\Gamma^k}_{ik,j}) + {\Gamma^k}_{kl} {\Gamma^{lj}}_{j} - {\Gamma^k}_{lj} {\Gamma^{lj}}_k$
Relations between the 4-metric affine connection and the 3-metric affine connection?
That's just going to be the extrinsic curvature, which is the lapse times the connection in the tangent dimension
$exp(-4 \phi) \alpha (R_{ij})^{TF}$
$= exp(-4 \phi) \alpha (R_{ij}^\phi + \bar{R}_{ij})^{TF}$
$= exp(-4 \phi) \alpha (
- 2 \bar{D}_i \bar{D}_j \phi
- 2 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{D}_k \bar{D}_l \phi
+ 4 \bar{D}_i \phi \bar{D}_j \phi
- 4 \bar{\gamma}_{ij} \bar{\gamma}^{kl} \bar{D}_k \phi \bar{D}_l \phi
+ \bar{R}_{ij}
)^{TF}$
using $(\gamma_{ij})^{TF} = 0$
$= exp(-4 \phi) \alpha (
- 2 \bar{D}_i \bar{D}_j \phi
+ 4 \bar{D}_i \phi \bar{D}_j \phi
+ \bar{R}_{ij}
)^{TF}$
$= exp(-4 \phi) (
- 2 \alpha \bar{D}_i \bar{D}_j \phi
+ 4 \alpha \bar{D}_i \phi \bar{D}_j \phi
+ \alpha \bar{R}_{ij}
)^{TF}$
time evolution of conformally rescaled trace free extrinsic curvature:
$\bar{A}_{ij,t} = (exp(-4 \phi) A_{ij})_{,t}$
$= -4 \phi_{,t} exp(-4 \phi) A_{ij} + exp(-4 \phi) A_{ij,t}$
$= -4 \phi_{,t} \bar{A}_{ij} + exp(-4 \phi) (K_{ij} - \frac{1}{3} \gamma_{ij} K)_{,t}$
$= -4 \bar{A}_{ij} \phi_{,t}
+ exp(-4 \phi) K_{ij,t}
- \frac{1}{3} exp(-4 \phi) K \gamma_{ij,t}
- \frac{1}{3} exp(-4 \phi) \gamma_{ij} K_{,t}$
substitute the definition of $\phi_{,t}, \gamma_{ij,t}, K_{,t}$:
$= -4 \bar{A}_{ij} (
-\frac{1}{6} \alpha K
+ \phi_{,k} \beta^k
+ \frac{1}{6} {\beta^k}_{,k}
+ \frac{1}{6} \beta^k {\bar{\Gamma}^l}_{kl}
)
+ exp(-4 \phi) (
-D_i D_j \alpha
+ \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)
+ 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
+ K_{ij,k} \beta^k
+ K_{ki} {\beta^k}_{,j}
+ K_{kj} {\beta^k}_{,i}
)
- \frac{1}{3} exp(-4 \phi) K (
-2 \alpha K_{ij}
+ \gamma_{ij,k} \beta^k
+ \gamma_{ki} {\beta^k}_{,j}
+ \gamma_{kj} {\beta^k}_{,i}
)
- \frac{1}{3} exp(-4 \phi) \gamma_{ij} (
- \gamma^{kl} D_k D_l \alpha
+ \alpha (\bar{A}_{kl} \bar{A}^{kl} + \frac{1}{3} K^2)
+ 4 \pi \alpha (S + \rho)
+ \beta^k K_{,k}
)$
$=
- exp(-4 \phi) D_i D_j \alpha
+ \frac{1}{3} exp(-4 \phi) \gamma_{ij} \gamma^{kl} D_k D_l \alpha
+ 4 \bar{A}_{ij} \frac{1}{6} \alpha K
+ exp(-4 \phi) \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)
+ \frac{1}{3} exp(-4 \phi) K 2 \alpha K_{ij}
- \frac{1}{3} exp(-4 \phi) \gamma_{ij} \alpha (\bar{A}_{kl} \bar{A}^{kl} + \frac{1}{3} K^2)
+ exp(-4 \phi) 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})
- \frac{1}{3} exp(-4 \phi) \gamma_{ij} 4 \pi \alpha (S + \rho)
- 4 \bar{A}_{ij} \phi_{,k} \beta^k
- 4 \bar{A}_{ij} \frac{1}{6} {\beta^k}_{,k}
+ exp(-4 \phi) K_{ij,k} \beta^k
+ exp(-4 \phi) K_{ki} {\beta^k}_{,j}
+ exp(-4 \phi) K_{kj} {\beta^k}_{,i}
- \frac{1}{3} exp(-4 \phi) K \gamma_{ij,k} \beta^k
- \frac{1}{3} exp(-4 \phi) K \gamma_{ki} {\beta^k}_{,j}
- \frac{1}{3} exp(-4 \phi) K \gamma_{kj} {\beta^k}_{,i}
- \frac{1}{3} exp(-4 \phi) \gamma_{ij} \beta^k K_{,k}
-\frac{2}{3} \bar{A}_{ij} \beta^k {\bar{\Gamma}^l}_{kl}
$
$=
exp(-4 \phi) (-(D_i D_j \alpha)^{TF} - 8 \pi \alpha (S_{ij})^{TF})
+ \alpha \frac{2}{3} \bar{A}_{ij} K
+ \alpha \frac{2}{3} exp(-4 \phi) K_{ij} K
+ exp(-4 \phi) \alpha R_{ij}
- exp(-4 \phi) \alpha \frac{1}{3} \gamma_{ij} R
+ exp(-4 \phi) \alpha \frac{1}{3} \gamma_{ij} R
+ exp(-4 \phi) \alpha K K_{ij}
- exp(-4 \phi) \alpha 2 K_{ik} {K^k}_j
- \frac{1}{3} \alpha \bar{\gamma}_{ij} {K^l}_k {K^k}_l
- \frac{4}{3} \pi \alpha exp(-4 \phi) \gamma_{ij} \rho
- 4 \pi \alpha exp(-4 \phi) \gamma_{ij} \rho
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
- \frac{2}{3} \bar{A}_{ij} {\beta^k}_{,k}
-\frac{2}{3} \bar{A}_{ij} \beta^k {\bar{\Gamma}^l}_{kl}
$
$=
exp(-4 \phi) (-(D_i D_j \alpha)^{TF} + \alpha ((R_{ij})^{TF} - 8 \pi (S_{ij})^{TF}))
+ \alpha exp(-4 \phi) (
\frac{2}{3} K A_{ij}
+ \frac{2}{3} K K_{ij}
+ \frac{1}{3} \gamma_{ij} A_{kl} A^{kl}
- \frac{2}{9} \gamma_{ij} K^2
+ K K_{ij}
- 2 K_{ik} {K^k}_j
- \frac{1}{3} \gamma_{ij} {K^l}_k {K^k}_l
+ \frac{16}{3} \pi \rho \gamma_{ij}
- \frac{4}{3} \pi \gamma_{ij} \rho
- 4 \pi \gamma_{ij} \rho
)
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
- \frac{2}{3} \bar{A}_{ij} {\beta^k}_{,k}
-\frac{2}{3} \bar{A}_{ij} \beta^k {\bar{\Gamma}^l}_{kl}
$
$=
exp(-4 \phi) (-(D_i D_j \alpha)^{TF} + \alpha ((R_{ij})^{TF} - 8 \pi (S_{ij})^{TF}))
+ \alpha exp(-4 \phi) (
- 2 K_{ik} {K^k}_j
+ \frac{7}{3} K K_{ij}
- \frac{5}{9} K^2 \gamma_{ij}
)
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
- \frac{2}{3} \bar{A}_{ij} {\beta^k}_{,k}
-\frac{2}{3} \bar{A}_{ij} \beta^k {\bar{\Gamma}^l}_{kl}
$
$= exp(-4 \phi)(-D_i D_j \alpha + \alpha (R_{ij} - 8 \pi S_{ij}))^{TF}
+ \alpha (K \bar{A}_{ij} - 2 \bar{A}_{ik} {\bar{A}^k}_j)
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
- \frac{2}{3} \bar{A}_{ij} {\beta^k}_{,k}
- \frac{2}{3} \bar{A}_{ij} \beta^k {\bar{\Gamma}^l}_{kl}
$
This is the time evolution in 2008 Alcubierre and 2010 Baumgarte, Shapiro.
alternatively, further substituting gets us...
$=
- \frac{2}{3} \bar{A}_{ij} {\beta^k}_{,k}
- \frac{2}{3} \bar{A}_{ij} \beta^k {\bar{\Gamma}^l}_{kl}
- 2 \alpha \bar{A}_{ik} {\bar{A}^k}_j
+ \alpha K \bar{A}_{ij}
+ exp(-4 \phi)(
- 2 \gamma_{ij} \gamma^{kl} \phi_{,l} \alpha_{,k}
+ 4 \phi_{,(i} \alpha_{,j)}
- \bar{D}_i \bar{D}_j \alpha
+ \alpha (R_{ij} - 8 \pi S_{ij})
)^{TF}
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
$
using $(\gamma_{ij} \phi)^{TF} = 0$
$= - \frac{2}{3} \bar{A}_{ij} \bar{D}_k \beta^k
- 2 \alpha \bar{A}_{ik} {\bar{A}^k}_j
+ \alpha K \bar{A}_{ij}
+ exp(-4 \phi)(
4 \phi_{,(i} \alpha_{,j)}
- \bar{D}_i \bar{D}_j \alpha
+ \alpha (R_{ij} - 8 \pi S_{ij})
)^{TF}
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
$
$= - \frac{2}{3} \bar{A}_{ij} \bar{D}_k \beta^k
- 2 \alpha \bar{A}_{ik} {\bar{A}^k}_j
+ \alpha K \bar{A}_{ij}
+ exp(-4 \phi)(
4 \bar{D}_{(i} \phi \bar{D}_{j)} \alpha
- \bar{D}_i \bar{D}_j \alpha
+ \alpha (R_{ij} - 8 \pi S_{ij})
)^{TF}
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
$
...substitute $exp(-4 \phi) \alpha (R_{ij})^{TF} = exp(-4 \phi) (
- 2 \alpha \bar{D}_i \bar{D}_j \phi
+ 4 \alpha \bar{D}_i \bar{D}_j
+ \alpha \bar{R}_{ij}
)^{TF}$...
$=
- \frac{2}{3} \bar{A}_{ij} \bar{D}_k \beta^k
- 2 \alpha \bar{A}_{ik} {\bar{A}^k}_j
+ \alpha K \bar{A}_{ij}
+ exp(-4 \phi)(
- 2 \alpha \bar{D}_i \bar{D}_j \phi
+ 4 \alpha \bar{D}_i \phi \bar{D}_j \phi
+ 4 \bar{D}_{(i} \phi \bar{D}_{j)} \alpha
- \bar{D}_i \bar{D}_j \alpha
+ \alpha (\bar{R}_{ij} - 8 \pi S_{ij})
)^{TF}
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}
$
This is the time evolution from 2017 Ruchlin et al., with source terms added.
$\bar{\gamma}^{jk} \hat{D}_j \hat{D}_k \beta^i$
$= \bar{\gamma}^{jk} \hat{D}_j ({\beta^i}_{,k} + {\hat{\Gamma}^i}_{lk} \beta^l)$
$= \bar{\gamma}^{jk} (
({\beta^i}_{,k} + {\hat{\Gamma}^i}_{lk} \beta^l)_{,j}
+ {\hat{\Gamma}^i}_{mj} ({\beta^m}_{,k} + {\hat{\Gamma}^m}_{lk} \beta^l)
- {\hat{\Gamma}^m}_{kj} ({\beta^i}_{,m} + {\hat{\Gamma}^i}_{lm} \beta^l)
)$
$= \bar{\gamma}^{jk} (
{\beta^i}_{,kj}
+ {\hat{\Gamma}^i}_{lk,j} \beta^l
+ {\hat{\Gamma}^i}_{lk} {\beta^l}_{,j}
+ {\hat{\Gamma}^i}_{mj} {\beta^m}_{,k}
+ {\hat{\Gamma}^i}_{mj} {\hat{\Gamma}^m}_{lk} \beta^l
- {\hat{\Gamma}^m}_{kj} {\beta^i}_{,m}
- {\hat{\Gamma}^m}_{kj} {\hat{\Gamma}^i}_{lm} \beta^l
)$
$= \bar{\gamma}^{jk} (
{\beta^i}_{,jk}
+ {\hat{\Gamma}^i}_{lj,k} \beta^l
+ 2 {\hat{\Gamma}^i}_{lj} {\beta^l}_{,k}
- {\hat{\Gamma}^l}_{jk} {\beta^i}_{,l}
+ {\hat{\Gamma}^i}_{jm} {\hat{\Gamma}^m}_{lk} \beta^l
- {\hat{\Gamma}^i}_{lm} {\hat{\Gamma}^m}_{jk} \beta^l
)$
${\bar{\Lambda}^i}_{,t} = (\Delta^i - \mathcal{C}^i)_{,t}$
$= {\Delta^i}_{,t}$
$= ({\Delta^i}_{jk} \bar{\gamma}^{jk})_{,t}$
$= (
{\bar{\Gamma}^i}_{jk} \bar{\gamma}^{jk}
- {\hat{\Gamma}^i}_{jk} \bar{\gamma}^{jk}
)_{,t}$
$= (
\bar{\gamma}^{il} \bar{\Gamma}_{ljk} \bar{\gamma}^{jk}
- \hat{\gamma}^{il} \hat{\Gamma}_{ljk} \bar{\gamma}^{jk}
)_{,t}$
$= \frac{1}{2} (
\bar{\gamma}^{il} (
\bar{\gamma}_{lj,k}
+ \bar{\gamma}_{lk,j}
- \bar{\gamma}_{jk,l}
) \bar{\gamma}^{jk}
- \hat{\gamma}^{il} (
\hat{\gamma}_{lj,k}
+ \hat{\gamma}_{lk,j}
- \hat{\gamma}_{jk,l}
) \bar{\gamma}^{jk}
)_{,t}$
...TODO a few more steps...
...TODO TODO ... the 2017 paper has the term $-2 \bar{A}^{jk} {\Delta^i}_{jk}$
but Etienne's SENR Mathematica notebook has the term $-2 \alpha \bar{A}^{jk} {\Delta^i}_{jk}$... so verify which is correct.
Maybe I'm reading an earlier version of the paper?
${\bar{\Lambda}^i}_{,t} =
\bar{\gamma}^{jk} {\beta^i}_{,jk}
+ \frac{1}{3} \bar{\gamma}^{ik} {\beta^j}_{,jk}
+ 2 \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{lj} {\beta^l}_{,k}
+ \frac{1}{3} \bar{\gamma}^{ik} {\bar{\Gamma}^j}_{lj} {\beta^l}_{,k}
- \bar{\gamma}^{jk} {\hat{\Gamma}^l}_{jk} {\beta^i}_{,l}
+ \frac{2}{3} {\bar{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\beta^j}_{,j}
- \frac{2}{3} {\hat{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\beta^j}_{,j}
+ \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{lj,k} \beta^l
+ \frac{1}{3} \bar{\gamma}^{ik} {\bar{\Gamma}^j}_{lj,k} \beta^l
- 2 \bar{A}^{ij} \alpha_{,j}
+ 12 \bar{A}^{ij} \phi_{,j}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{jm} {\hat{\Gamma}^m}_{lk} \beta^l
- \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{lm} {\hat{\Gamma}^m}_{jk} \beta^l
+ \frac{2}{3} {\bar{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\bar{\Gamma}^j}_{kj} \beta^k
- \frac{2}{3} {\hat{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\bar{\Gamma}^j}_{kj} \beta^k
+ 2 \bar{A}^{jk} {\bar{\Gamma}^i}_{jk}
- 2 \bar{A}^{jk} {\hat{\Gamma}^i}_{jk}
+ {\bar{\Lambda}^i}_{,j} \beta^j
- \bar{\Lambda}^j {\beta^i}_{,j}
$
${\bar{\Lambda}^i}_{,t} =
\bar{\gamma}^{jk} {\beta^i}_{,jk}
+ \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{lj,k} \beta^l
+ 2 \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{lj} {\beta^l}_{,k}
- \bar{\gamma}^{jk} {\hat{\Gamma}^l}_{jk} {\beta^i}_{,l}
+ \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{jm} {\hat{\Gamma}^m}_{lk} \beta^l
- \bar{\gamma}^{jk} {\hat{\Gamma}^i}_{lm} {\hat{\Gamma}^m}_{jk} \beta^l
+ \frac{2}{3} {\bar{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\beta^j}_{,j}
+ \frac{2}{3} {\bar{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\bar{\Gamma}^j}_{kj} \beta^k
- \frac{2}{3} {\hat{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\beta^j}_{,j}
- \frac{2}{3} {\hat{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\bar{\Gamma}^j}_{kj} \beta^k
+ \frac{1}{3} \bar{\gamma}^{ik} {\beta^j}_{,jk}
+ \frac{1}{3} \bar{\gamma}^{ik} {\bar{\Gamma}^j}_{lj,k} \beta^l
+ \frac{1}{3} \bar{\gamma}^{ik} {\bar{\Gamma}^j}_{lj} {\beta^l}_{,k}
- 2 \bar{A}^{ij} \alpha_{,j}
+ 12 \bar{A}^{ij} \phi_{,j}
+ 2 \bar{A}^{jk} {\bar{\Gamma}^i}_{jk}
- 2 \bar{A}^{jk} {\hat{\Gamma}^i}_{jk}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ {\bar{\Lambda}^i}_{,j} \beta^j
- \bar{\Lambda}^j {\beta^i}_{,j}
$
${\bar{\Lambda}^i}_{,t} =
\bar{\gamma}^{jk} (
{\beta^i}_{,jk}
+ {\hat{\Gamma}^i}_{lj,k} \beta^l
+ 2 {\hat{\Gamma}^i}_{lj} {\beta^l}_{,k}
- {\hat{\Gamma}^l}_{jk} {\beta^i}_{,l}
+ {\hat{\Gamma}^i}_{jm} {\hat{\Gamma}^m}_{lk} \beta^l
- {\hat{\Gamma}^i}_{lm} {\hat{\Gamma}^m}_{jk} \beta^l
)
+ \frac{2}{3} {\bar{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\beta^j}_{,j}
+ \frac{2}{3} {\bar{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\bar{\Gamma}^j}_{kj} \beta^k
- \frac{2}{3} {\hat{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\beta^j}_{,j}
- \frac{2}{3} {\hat{\Gamma}^i}_{lm} \bar{\gamma}^{lm} {\bar{\Gamma}^j}_{kj} \beta^k
+ \frac{1}{3} \bar{\gamma}^{ik} {\beta^j}_{,jk}
+ \frac{1}{3} \bar{\gamma}^{ik} {\bar{\Gamma}^j}_{lj,k} \beta^l
+ \frac{1}{3} \bar{\gamma}^{ik} {\bar{\Gamma}^j}_{lj} {\beta^l}_{,k}
- 2 \bar{A}^{ij} \alpha_{,j}
+ 12 \bar{A}^{ij} \phi_{,j}
+ 2 \bar{A}^{jk} {\bar{\Gamma}^i}_{jk}
- 2 \bar{A}^{jk} {\hat{\Gamma}^i}_{jk}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ {\bar{\Lambda}^i}_{,j} \beta^j
- \bar{\Lambda}^j {\beta^i}_{,j}
$
${\bar{\Lambda}^i}_{,t} =
\bar{\gamma}^{jk} \hat{D}_j ({\beta^i}_{,k} + {\hat{\Gamma}^i}_{lk} \beta^l)
+ \frac{2}{3} {\bar{\Gamma}^i}_{lm} \bar{\gamma}^{lm} ({\beta^j}_{,j} + {\bar{\Gamma}^j}_{kj} \beta^k)
- \frac{2}{3} {\hat{\Gamma}^i}_{lm} \bar{\gamma}^{lm} ({\beta^j}_{,j} + {\bar{\Gamma}^j}_{kj} \beta^k)
+ \frac{1}{3} \bar{\gamma}^{ik} ({\beta^j}_{,j} + {\bar{\Gamma}^j}_{lj} \beta^l)_{,k}
- 2 \bar{A}^{ij} (\alpha_{,j} - 6 \phi_{,j})
+ 2 \bar{A}^{jk} {\bar{\Gamma}^i}_{jk}
- 2 \bar{A}^{jk} {\hat{\Gamma}^i}_{jk}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ {\bar{\Lambda}^i}_{,j} \beta^j
- \bar{\Lambda}^j {\beta^i}_{,j}
$
This is the time evolution from 2017 Ruchlin et al., with source terms added.
${\bar{\Lambda}^i}_{,t} =
\bar{\gamma}^{jk} \hat{D}_j \hat{D}_k \beta^i
+ \frac{2}{3} \Delta^i \bar{D}_j \beta^j
+ \frac{1}{3} \bar{D}^i \bar{D}_j \beta^j
- 2 \bar{A}^{ij} (\alpha_{,j} - 6 \phi_{,j})
+ 2 \bar{A}^{jk} {\Delta^i}_{jk}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ {\bar{\Lambda}^i}_{,j} \beta^j
- \bar{\Lambda}^j {\beta^i}_{,j}
$
So overall the coordinate-free BSSN evolution equations involve evolution of (matching the paper):
$(\alpha, \beta^i, B^i, \bar{\epsilon}_{ij}, W, K, \bar{A}_{ij}, \bar{\Lambda}^i)$
$\alpha_{,t} = -\alpha^2 f(\alpha) K + \alpha_{,i} \beta^i$
${\beta^i}_{,t} = B^i$
${B^i}_{,t} = \frac{3}{4} {\bar{\Lambda}^i}_{,t} - \eta B^i$
$\bar{\epsilon}_{ij,t} =
\frac{2}{3} \bar{\gamma}_{ij} (\alpha {\bar{A}^k}_k - \bar{D}_k \beta^k)
+ 2 \hat{D}_{(i} \beta_{j)}
- 2 \alpha \bar{A}_{ij}
+ \bar{\epsilon}_{ij,k} \beta^k
+ \bar{\epsilon}_{kj} {\beta^k}_{,i}
+ \bar{\epsilon}_{ki} {\beta^k}_{,j}$
$W_{,t} =
-\frac{1}{3} W (
\bar{D}_k \beta^k
- \alpha K
)
+ W_{,i} \beta^i$
$K_{,t} = \frac{1}{3} \alpha K^2
+ \alpha \bar{A}_{ij} \bar{A}^{ij}
- exp(-4 \phi) (
\bar{D}^i \bar{D}_i \alpha
+ 2 \bar{D}^i \alpha \bar{D}_i \phi
)
+ 4 \pi \alpha (S + \rho)
+ K_{,i} \beta^i$
$\bar{A}_{ij,t} =
- \frac{2}{3} \bar{A}_{ij} \bar{D}_k \beta^k
- 2 \alpha \bar{A}_{ik} {\bar{A}^k}_j
+ \alpha K \bar{A}_{ij}
+ exp(-4 \phi)(
- 2 \alpha \bar{D}_i \bar{D}_j \phi
+ 4 \alpha \bar{D}_i \phi \bar{D}_j \phi
+ 4 \bar{D}_{(i} \phi \bar{D}_{j)} \alpha
- \bar{D}_i \bar{D}_j \alpha
+ \alpha (\bar{R}_{ij} - 8 \pi S_{ij})
)^{TF}
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}$
${\bar{\Lambda}^i}_{,t} =
\bar{\gamma}^{jk} \hat{D}_j \hat{D}_k \beta^i
+ \frac{2}{3} \Delta^i \bar{D}_j \beta^j
+ \frac{1}{3} \bar{D}^i \bar{D}_j \beta^j
- 2 \bar{A}^{ij} (\alpha_{,j} - 6 \phi_{,j})
+ 2 \bar{A}^{jk} {\Delta^i}_{jk}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ {\bar{\Lambda}^i}_{,j} \beta^j
- \bar{\Lambda}^j {\beta^i}_{,j}$
Simplifying all covariant derivatives:
$\alpha_{,t} = -\alpha^2 f(\alpha) K + \alpha_{,i} \beta^i$
${\beta^i}_{,t} = B^i$
${B^i}_{,t} = \frac{3}{4} {\bar{\Lambda}^i}_{,t} - \eta B^i$
(Notice here for $\bar{\epsilon}_{ij,t}$ I'm assuming that $\hat{D}_i \beta_j = \hat{D}_i (\hat{\gamma}_{jk} \beta^k) = \hat{\gamma}_{jk} \hat{D}_i \beta^k$)
(And a few more steps. TODO move these few more steps up to the first derivation of $\bar{\epsilon}_{ij,t}$).
$\bar{\epsilon}_{ij,t} =
\frac{2}{3} \bar{\gamma}_{ij} (
\alpha {\bar{A}^k}_k
- {\beta^k}_{,k}
- {\bar{\Gamma}^k}_{lk} \beta^l
)
- 2 \alpha \bar{A}_{ij}
+ 2 \bar{\gamma}_{k(i} {\beta^k}_{,j)}
+ \bar{\gamma}_{ij,k} \beta^k
$
$W_{,t} =
-\frac{1}{3} W (
{\beta^k}_{,k}
+ {\bar{\Gamma}^k}_{lk} \beta^l
- \alpha K
)
+ W_{,i} \beta^i$
$K_{,t} = \frac{1}{3} \alpha K^2
+ \alpha \bar{A}_{ij} \bar{A}^{ij}
- exp(-4 \phi) (
\bar{\gamma}^{ij} \alpha_{,ij}
- \bar{\Gamma}^k \alpha_{,k}
+ 2 \bar{\gamma}^{ij} \alpha_{,i} \phi_{,j}
)
+ 4 \pi \alpha (S + \rho)
+ K_{,i} \beta^i$
$\bar{A}_{ij,t} =
- \frac{2}{3} \bar{A}_{ij} (
{\beta^k}_{,k}
+ {\bar{\Gamma}^k}_{lk} \beta^l
)
- 2 \alpha \bar{A}_{ik} {\bar{A}^k}_j
+ \alpha K \bar{A}_{ij}
+ exp(-4 \phi)(
- 2 \alpha (\phi_{,ij} - {\bar{\Gamma}^k}_{ij} \phi_{,k})
+ 4 \alpha \phi_{,i} \phi_{,j}
+ 4 \phi_{,(i} \alpha_{,j)}
- \alpha_{,ij}
+ {\bar{\Gamma}^k}_{ij} \alpha_{,k}
+ \alpha (\bar{R}_{ij} - 8 \pi S_{ij})
)^{TF}
+ \bar{A}_{ij,k} \beta^k
+ \bar{A}_{kj} {\beta^k}_{,i}
+ \bar{A}_{ki} {\beta^k}_{,j}$
${\bar{\Lambda}^i}_{,t} =
\bar{\gamma}^{jk} (
{\beta^i}_{,kj}
+ {\hat{\Gamma}^i}_{lk,j} \beta^l
+ {\hat{\Gamma}^i}_{lk} {\beta^l}_{,j}
+ {\hat{\Gamma}^i}_{mj} (
{\beta^m}_{,k}
+ {\hat{\Gamma}^m}_{lk} \beta^l
)
- {\hat{\Gamma}^m}_{kj} (
{\beta^i}_{,m}
+ {\hat{\Gamma}^i}_{lm} \beta^l
)
)
+ \frac{2}{3} \Delta^i (
{\beta^j}_{,j}
+ {\bar{\Gamma}^j}_{kj} \beta^k
)
+ \frac{1}{3} \bar{\gamma}^{il} (
{\beta^j}_{,jl}
+ {\bar{\Gamma}^j}_{kj} {\beta^k}_{,l}
+ {\bar{\Gamma}^j}_{kj,l} \beta^k
)
- 2 \bar{A}^{ij} (\alpha_{,j} - 6 \phi_{,j})
+ 2 \bar{A}^{jk} {\Delta^i}_{jk}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ {\bar{\Lambda}^i}_{,j} \beta^j
- \bar{\Lambda}^j {\beta^i}_{,j}$
Rescaling:
Let ${e_i}^I$ be the basis that transforms from spatial metric ($a-z$) to the kronecker delta diagonal metric ($A-Z$).
Let ${e^i}_I$ be its dual such that ${e_i}^I {e^j}_I = \delta_i^j$ and ${e_i}^I {e^i}_J = \delta^I_J$.
Such that $\gamma_{ij} = {e_i}^I {e_j}^J \delta_{IJ}$
I take that back. The tetrad relativity folks use this definition of ${e_i}^I$, but the numerical relativity folks use $e_i := \sqrt{\gamma_{ii}} \partial_i$.
This at least accomplishes the same goal of diagonalizing the spherical metric $\gamma_{ij} = diag(1, r^2, r^2 sin(\theta)^2) = \{ \gamma_{rr} = 1, \gamma_{\theta\theta} = r^2, \gamma_{\phi\phi} = r^2 sin(\theta)^2 \}$.
Notice, for spherical coordinates that gives us
${e_i}^I = \downarrow i \overset{\rightarrow I}{\left[ \begin{matrix}
1 & 0 & 0 \\
0 & r & 0 \\
0 & 0 & r sin(\theta)
\end{matrix} \right] }$
and
${e^i}_I = \downarrow i \overset{\rightarrow I}{\left[ \begin{matrix}
1 & 0 & 0 \\
0 & \frac{1}{r} & 0 \\
0 & 0 & \frac{1}{r sin(\theta)}
\end{matrix} \right] }$
In other words, divide out r in our initial conditions when preparing $\bar{\epsilon}_{IJ}$, and then subsequently multiply in $r$ as we convert it back into coordinate form $\bar{\epsilon}_{ij}$.
So now we talk about storing our variables in their diagonalized coordinate state. Do we advect them in this diagonalized coordinate state as well?
If all our values are tensoral then we should be fine.
How is that working for the partial derivatives? I suppose they all work out somehow?
It almost seems like this might be the same amount of work as just using a fixed tetrad basis in the first place ... and using a fixed basis might even let us cut some corners in other places (like our Levi-Civita connection definition).
As far as those partial derivatives are concerned, if you compute the partial derivatives in terms of coordinates (i.e. do a finite difference between the coordinate forms of the metrics)
then that will fix that. But won't that re-introduce the singularities that we just went through so much effort to remove?
$\epsilon_{IJ} = {e^i}_I {e^j}_J \epsilon_{ij}$
$\epsilon_{IJ} = {e^i}_I {e^j}_J (\bar{\gamma}_{ij} - \hat{\gamma}_{ij})$
${e_i}^I {e_j}^J \epsilon_{IJ} = \bar{\gamma}_{ij} - \hat{\gamma}_{ij}$
$exp(-4\phi) \gamma_{ij} = {e_i}^I {e_j}^J \epsilon_{IJ} + \hat{\gamma}_{ij}$
$\gamma_{ij} = exp(4\phi) ({e_i}^I {e_j}^J \epsilon_{IJ} + \hat{\gamma}_{ij})$
(notice I call $\epsilon_{IJ}$ what the 2017 Ruchlin paper, and Etienne's SENR code, calls $h_{ij}$)
$\bar{\Lambda}^I = {e_i}^I \bar{\Lambda}^i$ (notice 2017 Ruchlin calls this $\bar{\lambda}^i)$
$\bar{A}_{IJ} = {e^i}_I {e^j}_J \bar{A}_{ij}$
$\beta^I = {e_i}^I \beta^i$ (notice Etienne's SENR calls this $ב^i$)
$B^I = {e_i}^I B^i$ (notice Etienne's SENR calls this $בּ^i$)
Notice $\beta^i$ is the only tensor here not 'conformally rescaled', so should I instead be transforming $\bar{\beta}^i = exp(4\phi) \beta^i$?
Then again, since $\beta^i$ is the vector which we are using the Lie derivative wrt to, maybe I shouldn't be transforming it at all?
(Fwiw, Etienne's SENR Mathematica notebook does rescale $B^i$ and $\beta^i$ and does not conformally rescale it.)
Now, substituting in the rescaled index form of tensors, the evolution equations look like:
$\alpha_{,t} = -\alpha^2 f(\alpha) K + \alpha_{,i} {e^i}_I \beta^I$
$= -\alpha^2 f(\alpha) K + \alpha_{,J} {e_i}^J {e^i}_I \beta^I$
$= -\alpha^2 f(\alpha) K + \alpha_{,I} \beta^I$
Looks the same as it did before, just with coordinate indexes replaced with rescaled indexes.
Notice that $\alpha_{,I} = {e^i}_I \alpha_{,i}$ is the $\alpha$ variable, partial derivative, and then rescaled.
This means $e_I$ is an anholonomic basis, which means it may have commutation coefficients (i.e. there may exist $[e_I, e_J] = {c_{IJ}}^K e_K \ne 0$).
$({e^i}_I \beta^I)_{,t} = B^i$
Let ${e^i}_{I,t} = 0$
${e^i}_I {\beta^I}_{,t} = B^i$
${\beta^I}_{,t} = {e_i}^I B^i$
${\beta^I}_{,t} = B^I$
$({e^i}_I B^I)_{,t} = \frac{3}{4} ({e^i}_I \bar{\Lambda}^I)_{,t} - \eta {e^i}_I B^I$
${B^I}_{,t} = \frac{3}{4} {\bar{\Lambda}^I}_{,t} - \eta B^I$
$({e_i}^I {e_j}^J \bar{\epsilon}_{IJ})_{,t} =
\frac{2}{3} \bar{\gamma}_{ij} (
\alpha \bar{\gamma}^{kl} ({e_k}^K {e_l}^L \bar{A}_{KL})
- ({e^k}_K \beta^K)_{,k}
- {\bar{\Gamma}^k}_{lk} ({e^l}_L \beta^L)
)
+ 2 (\gamma_{k(i} {e^k}_K \beta^K)_{,j)}
- 2 {\hat{\Gamma}^k}_{ij} {e_k}^K \beta_K
- 2 \alpha {e_i}^I {e_j}^J \bar{A}_{IJ}
+ ({e_i}^I {e_j}^J \bar{\epsilon}_{IJ})_{,k} {e^k}_K \beta^K
+ \bar{\epsilon}_{kj} ({e^k}_K \beta^K)_{,i}
+ \bar{\epsilon}_{ki} ({e^k}_K \beta^K)_{,j}$
Notice that whenever I use $\gamma_{IJ}, \gamma^{IJ}, \bar{\gamma}_{IJ}, {\hat{\Gamma}^K}_{IJ}$, etc, I am referring to the coordinate-rescaled metrics / connections,
and not any other metric or connection (i.e. not the geodesic connection using the rescaled basis).
$\bar{\epsilon}_{IJ,t} =
{e^i}_I {e^j}_J \cdot (
\frac{2}{3} \bar{\gamma}_{ij} (
\alpha \bar{\gamma}^{kl} ({e_k}^K {e_l}^L \bar{A}_{KL})
- {e^k}_{K,k} \beta^K
- {e^k}_K {\beta^K}_{,k}
- {\bar{\Gamma}^k}_{lk} ({e^l}_L \beta^L)
)
+ 2 \gamma_{k(i,j)} {e^k}_K \beta^K
+ 2 \gamma_{k(i} {e^k}_{K|,j)} \beta^K
+ 2 \gamma_{k(i} {\beta^K}_{,j)} {e^k}_K
- 2 {\hat{\Gamma}^k}_{ij} {e_k}^K \beta_K
)
- 2 \alpha \bar{A}_{IJ}
+ {e^i}_I {e^j}_J \cdot (
+ (
{{e_i}^L}_{,k} {e_j}^M \bar{\epsilon}_{LM}
+ {e_i}^L {{e_j}^M}_{,k} \bar{\epsilon}_{LM}
+ {e_i}^L {e_j}^M \bar{\epsilon}_{LM,k}
) {e^k}_K \beta^K
+ \bar{\epsilon}_{kj} (
{e^k}_{K,i} \beta^K
+ {e^k}_K {\beta^K}_{,i}
)
+ \bar{\epsilon}_{ki} (
{e^k}_{K,j} \beta^K
+ {e^k}_K {\beta^K}_{,j}
)
)$
$\bar{\epsilon}_{IJ,t} =
\frac{2}{3} \bar{\gamma}_{IJ} (
\alpha \bar{\gamma}^{KL} \bar{A}_{KL}
- {e^k}_{K,k} \beta^K
- {\beta^K}_{,K}
- {\bar{\Gamma}^K}_{LK} \beta^L
)
+ {e^i}_I {e^k}_K \gamma_{ki,J} \beta^K
+ {e^j}_J {e^k}_K \gamma_{kj,I} \beta^K
+ \gamma_{LI} {e_k}^L {e^k}_{K,J} \beta^K
+ \gamma_{LJ} {e_k}^L {e^k}_{K,I} \beta^K
+ 2 \gamma_{K(I} {\beta^K}_{,J)}
- 2 {\hat{\Gamma}^K}_{IJ} \beta_K
- 2 \alpha \bar{A}_{IJ}
+ {e^i}_I {{e_i}^L}_{,K} \bar{\epsilon}_{LJ} \beta^K
+ {e^j}_J {{e_j}^M}_{,K} \bar{\epsilon}_{IM} \beta^K
+ \bar{\epsilon}_{IJ,K} \beta^K
+ \bar{\epsilon}_{LJ} {e_k}^L {e^k}_{K,I} \beta^K
+ \bar{\epsilon}_{KJ} {\beta^K}_{,I}
+ \bar{\epsilon}_{LI} {e_k}^L {e^k}_{K,J} \beta^K
+ \bar{\epsilon}_{KI} {\beta^K}_{,J}
$
$\bar{\epsilon}_{IJ,t} =
\frac{2}{3} \bar{\gamma}_{IJ} (
\alpha \bar{\gamma}^{KL} \bar{A}_{KL}
- {\beta^K}_{,K}
- {\bar{\Gamma}^K}_{LK} \beta^L
)
+ 4 \phi_{,J} exp(4 \phi) (\bar{\epsilon}_{IK} + \hat{\gamma}_{IK}) \beta^K
+ exp(4 \phi) \bar{\epsilon}_{KI,J} \beta^K
+ 4 \phi_{,I} exp(4 \phi) (\bar{\epsilon}_{JK} + \hat{\gamma}_{JK}) \beta^K
+ exp(4 \phi) \bar{\epsilon}_{KJ,I} \beta^K
+ 2 \gamma_{K(I} {\beta^K}_{,J)}
- 2 {\hat{\Gamma}^K}_{IJ} \beta_K
- 2 \alpha \bar{A}_{IJ}
+ \bar{\epsilon}_{IJ,K} \beta^K
+ \bar{\epsilon}_{KJ} {\beta^K}_{,I}
+ \bar{\epsilon}_{KI} {\beta^K}_{,J}
$
$ - \frac{2}{3} \bar{\gamma}_{IJ} {e^k}_{K,k} \beta^K
+ exp(4 \phi) {e^k}_K {{e_k}^L}_{,J} \bar{\epsilon}_{LI} \beta^K
+ exp(4 \phi) {e^i}_I {{e_i}^M}_{,J} \bar{\epsilon}_{KM} \beta^K
+ exp(4 \phi) {e^i}_I {e^k}_K \hat{\gamma}_{kl,J} \beta^K
+ exp(4 \phi) {e^k}_K {{e_k}^L}_{,I} \bar{\epsilon}_{LJ} \beta^K
+ exp(4 \phi) {e^j}_J {{e_j}^M}_{,I} \bar{\epsilon}_{KM} \beta^K
+ exp(4 \phi) {e^j}_J {e^k}_K \hat{\gamma}_{kl,I} \beta^K
+ \gamma_{LI} {e_k}^L {e^k}_{K,J} \beta^K
+ \gamma_{LJ} {e_k}^L {e^k}_{K,I} \beta^K
+ \bar{\epsilon}_{LJ} {e^i}_I {{e_i}^L}_{,K} \beta^K
+ \bar{\epsilon}_{IM} {e^i}_J {{e_i}^M}_{,K} \beta^K
+ \bar{\epsilon}_{LJ} {e_k}^L {e^k}_{K,I} \beta^K
+ \bar{\epsilon}_{LI} {e_k}^L {e^k}_{K,J} \beta^K
$
$W_{,t} =
-\frac{1}{3} W (
({e^k}_K \beta^K)_{,k}
+ {e^k}_K {e_l}^L {e_k}^M {e^l}_N {\bar{\Gamma}^K}_{LM} \beta^N
- \alpha K
)
+ W_{,i} {e^i}_I \beta^I$
$W_{,t} =
-\frac{1}{3} W (
{e^k}_{K,k} \beta^K
+ {e^k}_K {\beta^K}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,I} \beta^I$
$W_{,t} =
-\frac{1}{3} W (
{e^k}_{K,k} \beta^K
+ {\beta^K}_{,K}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,I} \beta^I$
$K_{,t} = \frac{1}{3} \alpha K^2
+ {e_i}^I {e_j}^J {e^i}_K {e^j}_L \alpha \bar{A}_{IJ} \bar{A}^{KL}
- exp(-4 \phi) (
\bar{\gamma}^{ij} \alpha_{,ij}
- \bar{\Gamma}^k \alpha_{,k}
+ 2 \bar{\gamma}^{ij} \alpha_{,i} \phi_{,j}
)
+ 4 \pi \alpha (S + \rho)
+ K_{,i} {e^i}_I \beta^I$
$K_{,t} = \frac{1}{3} \alpha K^2
+ \alpha \bar{A}_{IJ} \bar{A}^{IJ}
- exp(-4 \phi) (
\bar{\gamma}^{ij} \alpha_{,ij}
- \bar{\Gamma}^K \alpha_{,K}
+ 2 \bar{\gamma}^{IJ} \alpha_{,I} \phi_{,J}
)
+ 4 \pi \alpha (S + \rho)
+ K_{,I} \beta^I$
$({e_i}^I {e_j}^J \bar{A}_{IJ})_{,t} =
- \frac{2}{3} {e_i}^I {e_j}^J \bar{A}_{IJ} (
({e^k}_K \beta^K)_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
)
- 2 \alpha {e_i}^I {e_j}^J \bar{A}_{IK} {\bar{A}^K}_J
+ \alpha K {e_i}^I {e_j}^J \bar{A}_{ij}
+ {e_i}^I {e_j}^J exp(-4 \phi)(
- 2 \alpha (\phi_{,ij} - {\bar{\Gamma}^k}_{ij} \phi_{,k})
+ 4 \alpha \phi_{,i} \phi_{,j}
+ 4 \phi_{,(i} \alpha_{,j)}
- \alpha_{,ij}
+ {\bar{\Gamma}^K}_{ij} \alpha_{,K}
+ \alpha (\bar{R}_{ij} - 8 \pi S_{ij})
)^{TF}
+ {e_i}^I {e_j}^J ({e_i}^L {e_j}^M \bar{A}_{LM})_{,K} \beta^K
+ {e_i}^I {e_j}^J \bar{A}_{kj} ({e^k}_K \beta^K)_{,i}
+ {e_i}^I {e_j}^J \bar{A}_{ki} ({e^k}_K \beta^K)_{,j}$
$\bar{A}_{IJ,t} =
- \frac{2}{3} \bar{A}_{IJ} (
({e^k}_K \beta^K)_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
)
- 2 \alpha \bar{A}_{IK} {\bar{A}^K}_J
+ \alpha K \bar{A}_{IJ}
+ exp(-4 \phi)(
- 2 \alpha ({e^i}_I {e^j}_J \phi_{,ij} - {\bar{\Gamma}^K}_{IJ} \phi_{,K})
+ 4 \alpha \phi_{,I} \phi_{,J}
+ 4 \phi_{,(I} \alpha_{,J)}
- {e^i}_I {e^j}_J \alpha_{,IJ}
+ {\bar{\Gamma}^K}_{IJ} \alpha_{,K}
+ \alpha (\bar{R}_{IJ} - 8 \pi S_{IJ})
)^{TF}
+ {e_i}^I {{e_i}^L}_{,K} \bar{A}_{LJ} \beta^K
+ {e_j}^J {{e_j}^M}_{,K} \bar{A}_{IM} \beta^K
+ \bar{A}_{IJ,K} \beta^K
+ \bar{A}_{kJ} {e^k}_{K,I} \beta^K
+ \bar{A}_{KJ} {\beta^K}_{,I}
+ \bar{A}_{kI} {e^k}_{K,J} \beta^K
+ \bar{A}_{KI} {\beta^K}_{,J}$
$({e^i}_I \bar{\Lambda}^I)_{,t} =
\bar{\gamma}^{jk} (
({e^i}_I \beta^I)_{,kj}
+ {\hat{\Gamma}^i}_{lk,j} {e^l}_L \beta^L
+ {\hat{\Gamma}^i}_{lk} ({e^l}_L \beta^L)_{,j}
+ {\hat{\Gamma}^i}_{mj} (
({e^m}_M \beta^M)_{,k}
+ {\hat{\Gamma}^m}_{lk} {e^l}_L \beta^L
)
- {\hat{\Gamma}^m}_{kj} (
({e^i}_I \beta^I)_{,m}
+ {\hat{\Gamma}^i}_{lm} {e^l}_L \beta^L
)
)
+ \frac{2}{3} \Delta^i (
({e^j}_J \beta^J)_{,j}
+ {\bar{\Gamma}^j}_{kj} {e^k}_K \beta^K
)
+ \frac{1}{3} \bar{\gamma}^{il} (
({e^j}_J \beta^J)_{,jl}
+ {\bar{\Gamma}^j}_{kj} ({e^k}_K \beta^K)_{,l}
+ {\bar{\Gamma}^j}_{kj,l} {e^k}_K \beta^K
)
- 2 {e^i}_I {e^j}_J \bar{A}^{IJ} (\alpha_{,j} - 6 \phi_{,j})
+ 2 {e^j}_J {e^k}_K \bar{A}^{JK} {\Delta^i}_{jk}
- \frac{4}{3} \alpha \bar{\gamma}^{ij} K_{,j}
+ ({e^i}_I \bar{\Lambda}^I)_{,j} {e^j}_J \beta^J
- {e^j}_J \bar{\Lambda}^J ({e^i}_I \beta^I)_{,j}$
${\bar{\Lambda}^I}_{,t} =
{e_n}^I \bar{\gamma}^{jk} (
{e^n}_{N,kj} \beta^N
+ 2 {e^n}_{N,k} {\beta^N}_{,j}
+ {e^n}_N {\beta^N}_{,kj}
+ {\hat{\Gamma}^n}_{lk,j} {e^l}_L \beta^L
+ {\hat{\Gamma}^n}_{lk} {e^l}_{L,j} \beta^L
+ {\hat{\Gamma}^n}_{lk} {e^l}_L {\beta^L}_{,j}
+ {\hat{\Gamma}^n}_{mj} (
{e^m}_{M,k} \beta^M
+ {e^m}_M {\beta^M}_{,k}
+ {\hat{\Gamma}^m}_{Lk} \beta^L
)
- {\hat{\Gamma}^m}_{kj} (
{e^n}_{N,m} \beta^N
+ {e^n}_N {\beta^N}_{,m}
+ {\hat{\Gamma}^n}_{lm} {e^l}_L \beta^L
)
)
+ \frac{2}{3} {e_n}^I \Delta^n (
{e^j}_{J,j} \beta^J
+ {e^j}_J {\beta^J}_{,j}
+ {\bar{\Gamma}^j}_{kj} {e^k}_K \beta^K
)
+ \frac{1}{3} {e_n}^I \bar{\gamma}^{nl} (
{e^j}_{J,jl} \beta^J
+ {e^j}_{J,j} {\beta^J}_{,l}
+ {e^j}_{J,l} {\beta^J}_{,j}
+ {e^j}_J {\beta^J}_{,jl}
+ {\bar{\Gamma}^j}_{kj} {e^k}_{K,l} \beta^K
+ {\bar{\Gamma}^j}_{kj} {e^k}_K {\beta^K}_{,l}
+ {\bar{\Gamma}^j}_{kj,l} {e^k}_K \beta^K
)
- 2 {e_n}^I {e^n}_N {e^j}_J \bar{A}^{NJ} (\alpha_{,j} - 6 \phi_{,j})
+ 2 {e_n}^I {e^j}_J {e^k}_K \bar{A}^{JK} {\Delta^n}_{jk}
- \frac{4}{3} {e_n}^I \alpha \bar{\gamma}^{nj} K_{,j}
+ {e_n}^I {e^n}_{N,j} \bar{\Lambda}^N {e^j}_J \beta^J
+ {e_n}^I {e^n}_N {\bar{\Lambda}^N}_{,j} {e^j}_J \beta^J
- {e_n}^I {e^j}_J \bar{\Lambda}^J {e^n}_{N,j} \beta^N
- {e_n}^I {e^j}_J \bar{\Lambda}^J {e^n}_N {\beta^N}_{,j}
$
${\bar{\Lambda}^I}_{,t} =
\bar{\gamma}^{JK} (
{e^j}_J {e^k}_K {e_i}^I {e^i}_{L,jk} \beta^L
+ 2 {e_i}^I {e^i}_{L,K} {\beta^L}_{,J}
+ {e^j}_J {e^k}_K {\beta^I}_{,jk}
+ {e_n}^I {e^k}_K {e^l}_L {\hat{\Gamma}^n}_{lk,J} \beta^L
+ {\hat{\Gamma}^I}_{MK} {e^M}_{L,J} \beta^L
+ {\hat{\Gamma}^I}_{LK} {\beta^L}_{,J}
+ {\hat{\Gamma}^I}_{NJ} (
{e^N}_{M,K} \beta^M
+ {e^N}_M {\beta^M}_{,K}
+ {\hat{\Gamma}^N}_{LK} \beta^L
)
- {\hat{\Gamma}^M}_{KJ} (
{e_n}^I {e^n}_{N,M} \beta^N
+ {e^I}_N {\beta^N}_{,M}
+ {\hat{\Gamma}^I}_{LM} \beta^L
)
)
+ \frac{2}{3} \Delta^I (
{e^j}_{J,j} \beta^J
+ {\beta^J}_{,J}
+ {\bar{\Gamma}^J}_{KJ} \beta^K
)
+ \frac{1}{3} \bar{\gamma}^{IL} (
{e^l}_L {e^j}_{J,jl} \beta^J
+ {e^j}_{J,j} {\beta^J}_{,L}
+ {e_j}^M {e^j}_{J,L} {\beta^J}_{,M}
+ {e^l}_L {e^j}_J {\beta^J}_{,jl}
+ {\bar{\Gamma}^J}_{MJ} {e_k}^M {e^k}_{K,L} \beta^K
+ {\bar{\Gamma}^J}_{KJ} {\beta^K}_{,L}
+ {\bar{\Gamma}^j}_{kj,L} {e^k}_K \beta^K
)
- 2 \bar{A}^{IJ} (\alpha_{,J} - 6 \phi_{,J})
+ 2 \bar{A}^{JK} {\Delta^I}_{JK}
- \frac{4}{3} \alpha \bar{\gamma}^{IJ} K_{,J}
+ {e_n}^I {e^n}_{N,J} \bar{\Lambda}^N \beta^J
+ {\bar{\Lambda}^I}_{,J} \beta^J
- \bar{\Lambda}^J {e_n}^I {e^n}_{N,J} \beta^N
- \bar{\Lambda}^J {\beta^I}_{,J}
$
Nevermind that, let's assume we are converting to coordinates before partial differentating.
This way we don't have to worry about introducing partial derivatives of the rescaling terms.
Nevermind that nevermind. Looks like differentiating the rescaling separately helps cancel out the scale factors.
$\alpha_{,t} = -\alpha^2 f(\alpha) K + \alpha_{,i} {e^i}_I \beta^I$
${\beta^I}_{,t} = B^I$
${B^I}_{,t} = \frac{3}{4} {\bar{\Lambda}^I}_{,t} - \eta B^I$
$\bar{\epsilon}_{IJ,t} =
\frac{2}{3} \bar{\gamma}_{IJ} (
\alpha {\bar{A}^K}_K
- {\beta^k}_{,k}
- {\bar{\Gamma}^K}_{LK} \beta^L
)
- 2 \alpha \bar{A}_{IJ}
+ {e^i}_I {e^j}_J (
\bar{\gamma}_{ij,k} \beta^k
+ \bar{\gamma}_{ki} {\beta^k}_{,j}
+ \bar{\gamma}_{kj} {\beta^k}_{,i}
)
$
$W_{,t} =
-\frac{1}{3} W (
{\beta^k}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,i} {e^i}_I \beta^I$
$K_{,t} = \frac{1}{3} \alpha K^2
+ \alpha \bar{A}_{IJ} \bar{A}^{IJ}
- exp(-4 \phi) (
\bar{\gamma}^{ij} \alpha_{,ij}
- \bar{\Gamma}^k \alpha_{,k}
+ 2 \bar{\gamma}^{ij} \alpha_{,i} \phi_{,j}
)
+ 4 \pi \alpha (S + \rho)
+ K_{,i} {e^i}_I \beta^I$
$\bar{A}_{IJ,t} =
- \frac{2}{3} \bar{A}_{IJ} (
{\beta^k}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
)
- 2 \alpha \bar{A}_{IK} {\bar{A}^K}_J
+ \alpha K \bar{A}_{IJ}
+ exp(-4 \phi)(
- 2 \alpha ({e^i}_I {e^j}_J \phi_{,ij} - {\bar{\Gamma}^K}_{IJ} {e^k}_K \phi_{,k})
+ 4 \alpha {e^i}_I {e^j}_J \phi_{,i} \phi_{,j}
+ 4 {e^i}_I {e^j}_J \phi_{,(i} \alpha_{,j)}
- {e^i}_I {e^j}_J \alpha_{,ij}
+ {\bar{\Gamma}^K}_{IJ} {e^k}_K \alpha_{,k}
+ \alpha (\bar{R}_{IJ} - 8 \pi S_{IJ})
)^{TF}
+ \bar{A}_{ij,k} {e^k}_K \beta^K
+ \bar{A}_{KJ} {e_k}^K {\beta^k}_{,I}
+ \bar{A}_{KI} {e_k}^K {\beta^k}_{,J}$
${\bar{\Lambda}^I}_{,t} =
\bar{\gamma}^{jk} {e_i}^I (
{\beta^i}_{,kj}
+ {\hat{\Gamma}^i}_{lk,j} {e^l}_L \beta^L
+ {\hat{\Gamma}^i}_{lk} {\beta^l}_{,j}
+ {\hat{\Gamma}^i}_{mj} (
{\beta^m}_{,k}
+ {\hat{\Gamma}^m}_{lk} {e^l}_L \beta^L
)
- {\hat{\Gamma}^m}_{kj} (
{\beta^i}_{,m}
+ {\hat{\Gamma}^i}_{lm} {e^l}_L \beta^L
)
)
+ \frac{2}{3} \Delta^I (
{\beta^j}_{,j}
+ {\bar{\Gamma}^J}_{KJ} \beta^K
)
+ \frac{1}{3} \bar{\gamma}^{IL} (
{\beta^j}_{,jl} {e^l}_L
+ {\bar{\Gamma}^j}_{kj} {\beta^k}_{,l} {e^l}_L
+ {\bar{\Gamma}^j}_{kj,l} {e^l}_L {e^k}_K \beta^K
)
- 2 \bar{A}^{IJ} {e^j}_J (\alpha_{,j} - 6 \phi_{,j})
+ 2 \bar{A}^{JK} {\Delta^I}_{JK}
- \frac{4}{3} \alpha {e_i}^I \bar{\gamma}^{ij} K_{,j}
+ {e_i}^I {\bar{\Lambda}^i}_{,j} {e^j}_J \beta^J
- {e_i}^I \bar{\Lambda}^J {e^j}_J {\beta^i}_{,j}$
But what if we assume $\mathcal{L}_\vec\beta$ is a tensor? Then can we just combine those terms into the $\frac{d}{dt}$?
Then the $\beta^i$ of the Lie derivative still needs to be rescaled.
Fwiw, the Etienne SENR Mathematica worksheets do factor out the the chain rule, like so:
TODO things in this form: $\bar{\epsilon}_{ij,k} = \bar{\epsilon}_{IJ,k} ({e_i}^I {e_j}^J) + \bar{\epsilon}_{IJ} ({e_i}^I {e_j}^J)_{,k}$
Notice that, for spherical, we have the following:
Transform covariant from coordinate to non-coordinate / transform contravariant from non-coordinate to coordinate: ${e^i}_I = diag(1, \frac{1}{r}, \frac{1}{r sin(\theta)})$
Derivative thereof: ${{e^i}_I}_{,j}
= \left[ \begin{matrix}
\partial_r {e^i}_I \\
\partial_\theta {e^i}_I \\
\partial_\phi {e^i}_I \\
\end{matrix} \right]
= \left[ \begin{matrix}
diag(0, -\frac{1}{r^2}, -\frac{1}{r^2 sin(\theta)}) \\
diag(0, 0, -\frac{cos(\theta)}{r sin(\theta)^2}) \\
diag(0, 0, 0),
\end{matrix} \right]$
Transform contravariant from coordinate to non-coordinate / transform covariant from non-coordinate to coordinate: ${e_i}^I = diag(1, r, r sin(\theta))$
Derivative thereof: ${{e_i}^I}_{,j}
= \left[ \begin{matrix}
\partial_r {e_i}^I \\
\partial_\theta {e_i}^I \\
\partial_\phi {e_i}^I \\
\end{matrix} \right]
= \left[ \begin{matrix}
diag(0, 1, sin(\theta)) \\
diag(0, 0, r cos(\theta)) \\
diag(0, 0, 0),
\end{matrix} \right]$
Notice that this one, as sparse as it is, and with no inverses (thankfully), is what we use with our chain rule to transform $\bar{\epsilon}_{IJ,k}$ into $\bar{\epsilon}_{ij,k}$.
However it is the ugly one, above, which we use to transform our ${\beta^I}_{,j}$, ${B^I}_{,j}$, and ${\bar{\Lambda}^I}_{,j}$ into their respective coordinate representation.
It is also the ugly one which is used to transform any derivative back into non-coordinate form.
Ok more thinking on this. One pain point was $\bar{A}_{IJ,t}$, caused by $\bar{R}_{ij}$ caused by $\bar{\gamma}^{kl} \bar{\gamma}_{ij,kl}$
It was giving bad values until I factored it out.
I'm going to have to do this with every term.
Time to fire up the index notation CAS?
${e_i}^I = \delta_i^I \cdot f_i$, where $f_i = \sqrt{\gamma_{ii}}$ ... so for spherical we get $f_i = \{1, r, r sin(\theta)\}$ and $f^i = \frac{1}{f_i}$ (f's are a sequence, not a tensor).
${e_i}^I {\beta^i}_{,j} {e^j}_J$
$= {e_i}^I (\beta^M {e^i}_M)_{,j} {e^j}_J$
$= {e_i}^I ({\beta^M}_{,j} {e^i}_M + \beta^M {e^i}_M,j) {e^j}_J$
$= {\beta^I}_{,j} {e^j}_J + \beta^M {e_i}^I {e^i}_M,j {e^j}_J$
$= {\beta^I}_{,j} \delta^j_J f^j + \beta^M \delta_i^I f_i (\delta^i_M f^i)_{,j} \delta^j_J f_j$
$= ({\beta^I}_{,j} + \beta^I f_I (f^I)_{,j}) \delta^j_J f_j$
$= ({\beta^I}_{,j} - \beta^I \frac{f_{I,j}}{f_I}) f_j$
${e_i}^I {\beta^i}_{,jk} {e^j}_J {e^k}_K$
$= {e_i}^I (\beta^M {e^i}_M)_{,jk} {e^j}_J {e^k}_K$
$= {e_i}^I (
{\beta^M}_{,j} {e^i}_M
+ \beta^M {e^i}_{M,j}
)_{,k} {e^j}_J {e^k}_K$
$= (
{\beta^M}_{,jk} {e^i}_M
+ {\beta^M}_{,j} {e^i}_{M,k}
+ {\beta^M}_{,k} {e^i}_{M,j}
+ \beta^M {e^i}_{M,jk}
) {e_i}^I {e^j}_J {e^k}_K$
$= (
{\beta^M}_{,jk} \delta^i_M f^i
+ {\beta^M}_{,j} (\delta^i_M f^i)_{,k}
+ {\beta^M}_{,k} (\delta^i_M f^i)_{,j}
+ \beta^M (\delta^i_M f^i)_{,jk}
) \delta_i^I \delta^j_J \delta^k_K f_i f^j f^k$
$= (
{\beta^I}_{,jk}
- {\beta^I}_{,j} \frac{f_{i,k}}{f_i}
- {\beta^I}_{,k} \frac{f_{i,j}}{f_i}
+ \beta^I (
- \frac{f_{i,jk}}{(f_i)}
+ \frac{2 f_{i,j} f_{i,k}}{(f_i)^2}
)
) \delta^j_J \delta^k_K f^j f^k$
$W_{,t} =
-\frac{1}{3} W (
{\beta^k}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,i} {e^i}_I \beta^I$
$W_{,t} =
-\frac{1}{3} W (
{\beta^K}_{,k} {e^k}_K
+ {\beta^K} {e^k}_{K,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,i} {e^i}_I \beta^I$
$W_{,t} =
-\frac{1}{3} W (
{\beta^K}_{,k} {\delta^k}_K f^k
+ {\beta^K} {\delta^k}_K {f^k}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,i} {e^i}_I \beta^I$
$W_{,t} =
-\frac{1}{3} W (
{\beta^K}_{,k} {\delta^k}_K f^k
+ {\beta^K} {f^k}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,i} {e^i}_I \beta^I$
$({e_i}^I {e_j}^J)_{,k}$
$= {{e_i}^I}_{,k} {e_j}^J + {e_i}^I {{e_j}^J}_{,k}$
$= \delta_i^I f_{i,k} \delta_j^J + \delta_i^I \delta_j^J f_{j,k}$
$({e_i}^I {e_j}^J)_{,kl}$
$= ({{e_i}^I}_{,k} {e_j}^J + {e_i}^I {{e_j}^J}_{,k})_{,l}$
$= {{e_i}^I}_{,kl} {e_j}^J + {e_i}^I {{e_j}^J}_{,kl} + {{e_i}^I}_{,k} {{e_j}^J}_{,l} + {{e_i}^I}_{,l} {{e_j}^J}_{,k}$
${e^i}_I {e^j}_J \bar{A}_{ij,k}$
$= {e^i}_I {e^j}_J ( {e_i}^M {e_j}^N \bar{A}_{MN} )_{,k}$
$= {e^i}_I {e^j}_J (
{e_i}^M {e_j}^N \bar{A}_{MN,k}
+ ({e_i}^M {e_j}^N)_{,k} \bar{A}_{MN}
)$
$=
({e_i}^M {e_j}^N)_{,k} {e^i}_I {e^j}_J \bar{A}_{MN}
+ \bar{A}_{IJ,k}
$
$=
(\delta_i^M \delta_j^N f_i f_j)_{,k} \delta^i_I \delta^j_J f^i f^j \bar{A}_{MN}
+ \bar{A}_{IJ,k}
$
$=
(f_{I,k} / f_I + f_{J,k} / f_J) \bar{A}_{IJ}
+ \bar{A}_{IJ,k}
$
${e^i}_I {e^j}_J \bar{\gamma}_{ij,k}$
$= {e^i}_I {e^j}_J ( {e_i}^M {e_j}^N (\hat{\gamma}_{MN} + \bar{\epsilon}_{MN} ))_{,k}$
$= {e^i}_I {e^j}_J (
{e_i}^M {e_j}^N (\delta_{MN} + \bar{\epsilon}_{MN} )_{,k}
+ ({e_i}^M {e_j}^N)_{,k} (\delta_{MN} + \bar{\epsilon}_{MN} )
)$
$=
({e_i}^M {e_j}^N)_{,k} {e^i}_I {e^j}_J (\delta_{MN} + \bar{\epsilon}_{MN})
+ \bar{\epsilon}_{IJ,k}
$
$=
(\delta_i^M \delta_j^N f_i f_j)_{,k} \delta^i_I \delta^j_J f^i f^j (\delta_{MN} + \bar{\epsilon}_{MN})
+ \bar{\epsilon}_{IJ,k}
$
$=
(f_{I,k} / f_I + f_{J,k} / f_J) (\delta_{IJ} + \bar{\epsilon}_{IJ})
+ \bar{\epsilon}_{IJ,k}
$
${e^i}_I {e^j}_J \hat{\gamma}_{ij,k}$
$= (f_{I,k} / f_I + f_{J,k} / f_J) \delta_{IJ}$
${e_i}^I {e_j}^J {\hat{\gamma}^{ij}}_{,k}$
$= -(f_{I,k} / f_I + f_{J,k} / f_J) \delta^{IJ}$
${e^i}_I {e^j}_J \bar{\gamma}_{ij,kl} \bar{\gamma}^{kl}$
$= {e^i}_I {e^j}_J (\hat{\gamma}_{ij,kl} + \epsilon_{ij,kl}) \bar{\gamma}^{kl}$
$= {e^i}_I {e^j}_J (({e_i}^M {e_j}^N \delta_{MN})_{,kl} + ({e_i}^M {e_j}^N \epsilon_{MN})_{,kl}) {e^k}_K {e^l}_L \bar{\gamma}^{KL}$
$= {e^i}_I {e^j}_J (({e_i}^M {e_j}^N)_{,kl} \delta_{MN} + (({e_i}^M {e_j}^N)_{,k} \epsilon_{MN} + {e_i}^M {e_j}^N \epsilon_{MN,k})_{,l}) {e^k}_K {e^l}_L \bar{\gamma}^{KL}$
$= (
({e_i}^M {e_j}^N)_{,kl} (\delta_{MN} + \epsilon_{MN})
+ ({e_i}^M {e_j}^N)_{,k} \epsilon_{MN,l}
+ ({e_i}^M {e_j}^N)_{,l} \epsilon_{MN,k}
+ {e_i}^M {e_j}^N \epsilon_{MN,kl}
) {e^i}_I {e^j}_J {e^k}_K {e^l}_L \bar{\gamma}^{KL}$
$= (
(\delta_{MN} + \epsilon_{MN}) (
{{e_i}^M}_{,kl} {e_j}^N
+ 2 {{e_i}^M}_{,k} {{e_j}^N}_{,l}
+ {e_i}^M {{e_j}^N}_{,kl}
)
+ 2 ({{e_i}^M}_{,k} {e_j}^N + {e_i}^M {{e_j}^N}_{,k}) \epsilon_{MN},l
+ {e_i}^M {e_j}^N \epsilon_{MN,kl}
) {e^i}_I {e^j}_J {e^k}_K {e^l}_L \bar{\gamma}^{KL}$
$= (
(\delta_{MN} + \epsilon_{MN}) (
\delta_i^M \delta_j^N f_{i,kl} f_j
+ 2 \delta_i^M \delta_j^N f_{i,k} f_{j,l}
+ \delta_i^M \delta_j^N f_i f_{j,kl}
)
+ 2 \delta_i^M \delta_j^N f_{i,k} f_j \epsilon_{MN,l}
+ 2 \delta_i^M \delta_j^N f_i f_{j,k} \epsilon_{MN,l}
+ \delta_i^M \delta_j^N f_i f_j \epsilon_{MN,kl}
) \delta^i_I \delta^j_J \delta^k_K \delta^l_L f^i f^j f^k f^l \bar{\gamma}^{KL}$
$= (
(\delta_{IJ} + \epsilon_{IJ}) (
f_{I,kl} f^I
+ 2 f_{I,k} f^I f_{J,l} f^J
+ f_{J,kl} f^J
)
+ 2 \epsilon_{IJ},l (
f_{I,k} f^I
+ f_{J,k} f^J
)
+ \epsilon_{IJ,kl}
) f^K f^L \delta^k_K \delta^l_L \bar{\gamma}^{KL}$
$\bar{\epsilon}_{IJ,t} =
\frac{2}{3} \bar{\gamma}_{IJ} (
\alpha {\bar{A}^K}_K
- {\beta^k}_{,k}
- {\bar{\Gamma}^K}_{LK} \beta^L
)
- 2 \alpha \bar{A}_{IJ}
+ {e^i}_I {e^j}_J (
\bar{\gamma}_{ij,k} \beta^k
+ \bar{\gamma}_{ki} {\beta^k}_{,j}
+ \bar{\gamma}_{kj} {\beta^k}_{,i}
)
$
And in its final form:
$\alpha_{,t} = -\alpha^2 f(\alpha) K + \alpha_{,i} {e^i}_I \beta^I$
$W_{,t} =
-\frac{1}{3} W (
{\beta^K}_{,k} {\delta^k}_K f^k
+ {\beta^K} {f^k}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
- \alpha K
)
+ W_{,i} {e^i}_I \beta^I$
$K_{,t} = \frac{1}{3} \alpha K^2
+ \alpha \bar{A}_{IJ} \bar{A}^{IJ}
- exp(-4 \phi) (
\bar{\gamma}^{IJ} {e^i}_I {e^j}_J \alpha_{,ij}
- \bar{\Gamma}^K {e^k}_K \alpha_{,k}
+ 2 \bar{\gamma}^{IJ} {e^i}_I {e^j}_J \alpha_{,i} \phi_{,j}
)
+ 4 \pi \alpha (S + \rho)
+ K_{,i} {e^i}_I \beta^I$
${\beta^I}_{,t} = B^I$
${B^I}_{,t} = \frac{3}{4} {\bar{\Lambda}^I}_{,t} - \eta B^I$
$\bar{\epsilon}_{IJ,t} =
- 2 \alpha \bar{A}_{IJ}
\frac{2}{3} \bar{\gamma}_{IJ} (
\alpha {\bar{A}^K}_K
- {\beta^k}_{,k}
- {\bar{\Gamma}^K}_{LK} \beta^L
)
{e^i}_I {e^j}_J (
- 2 {\hat{\Gamma}^k}_{ij} \beta_k
+ {e^k}_K \bar{\epsilon}_{ij,k} \beta^K
+ 2 \gamma_{k(i} {\beta^k}_{(,j)}
+ 2 \bar{\epsilon}_{k(i} {\beta^k}_{,j)}
)
$
$\bar{A}_{IJ,t} =
- \frac{2}{3} \bar{A}_{IJ} (
{\beta^k}_{,k}
+ {\bar{\Gamma}^K}_{LK} \beta^L
)
- 2 \alpha \bar{A}_{IK} {\bar{A}^K}_J
+ \alpha K \bar{A}_{IJ}
+ exp(-4 \phi)(
- 2 \alpha ({e^i}_I {e^j}_J \phi_{,ij} - {\bar{\Gamma}^K}_{IJ} {e^k}_K \phi_{,k})
+ 4 \alpha {e^i}_I {e^j}_J \phi_{,i} \phi_{,j}
+ 4 {e^i}_I {e^j}_J \phi_{,(i} \alpha_{,j)}
- {e^i}_I {e^j}_J \alpha_{,ij}
+ {\bar{\Gamma}^K}_{IJ} {e^k}_K \alpha_{,k}
+ \alpha (\bar{R}_{IJ} - 8 \pi S_{IJ})
)^{TF}
+ \bar{A}_{ij,k} {e^k}_K \beta^K
+ \bar{A}_{KJ} {e_k}^K {\beta^k}_{,I}
+ \bar{A}_{KI} {e_k}^K {\beta^k}_{,J}$
${\bar{\Lambda}^I}_{,t} =
\bar{\gamma}^{jk} {e_i}^I (
{\beta^i}_{,kj}
+ {\hat{\Gamma}^i}_{lk,j} {e^l}_L \beta^L
+ {\hat{\Gamma}^i}_{lk} {\beta^l}_{,j}
+ {\hat{\Gamma}^i}_{mj} (
{\beta^m}_{,k}
+ {\hat{\Gamma}^m}_{lk} {e^l}_L \beta^L
)
- {\hat{\Gamma}^m}_{kj} (
{\beta^i}_{,m}
+ {\hat{\Gamma}^i}_{lm} {e^l}_L \beta^L
)
)
+ \frac{2}{3} \Delta^I (
{\beta^j}_{,j}
+ {\bar{\Gamma}^J}_{KJ} \beta^K
)
+ \frac{1}{3} \bar{\gamma}^{IL} (
{\beta^j}_{,jl} {e^l}_L
+ {\bar{\Gamma}^j}_{kj} {\beta^k}_{,l} {e^l}_L
+ {\bar{\Gamma}^j}_{kj,l} {e^l}_L {e^k}_K \beta^K
)
- 2 \bar{A}^{IJ} {e^j}_J (\alpha_{,j} - 6 \phi_{,j})
+ 2 \bar{A}^{JK} {\Delta^I}_{JK}
- \frac{4}{3} \alpha {e_i}^I \bar{\gamma}^{ij} K_{,j}
+ {e_i}^I {\bar{\Lambda}^i}_{,j} {e^j}_J \beta^J
- {e_i}^I \bar{\Lambda}^J {e^j}_J {\beta^i}_{,j}$
Example:
Minkowski:
$ds^2 = -dt^2 + dr^2 + r^2 (d\theta^2 + sin(\theta)^2 d\phi^2)$
$g_{tt} = -1$, $g_{rr} = 1$, $g_{\theta\theta} = r^2$, $g_{\phi\phi} = r^2 sin(\theta0^2$
$\alpha = 1$, $\beta^i = 0$, $\gamma_{ij} = diag(1, r^2, r^2 sin(\theta)^2)$
$\gamma = det(\gamma_{ij}) = r^4 sin(\theta)^2$
grid metric: $\hat{\gamma}_{ij} = diag(1, r^2, r^2 sin(\theta)^2)$
$\hat{\gamma} = det(\hat{\gamma}_{ij}) = r^4 sin(\theta)^2$
$\frac{\gamma}{\hat{\gamma}} = 1$
$\phi = \frac{1}{12} ln(\frac{\gamma}{\hat{\gamma}}) = 0$
$W = exp(-4 \phi) = 1$
so $\bar{\gamma}_{ij} = exp(-4\phi) \gamma_{ij} = \gamma_{ij} = \hat{\gamma}_{ij}$
$\bar{\epsilon}_{IJ} = {e^i}_I {e^j}_J \bar{\epsilon}_{ij} = {e^i}_I {e^j}_J (\bar{\gamma}_{ij} - \hat{\gamma}_{ij}) = 0$
Schwarzschild:
$ds^2 + -(1 - \frac{R}{r}) dt^2 + (1 - \frac{R}{r})^{-1} dr^2 + r^2 d\theta^2 + r^2 sin(\theta)^2 d\phi^2$
Let $r = r' (1 + \frac{R}{4 r'})^2$
Schwarzschild isotropic coordinates:
$ds^2 = -\frac{(1 - \frac{R}{4 r'})^2}{(1 + \frac{R}{4 r'})^2} dt^2 + (1 + \frac{R}{4 r'})^4 ( dr'^2 + r'^2 (d\theta^2 + sin(\theta)^2 d\phi^2))$
metric components:
$g_{tt} = -\frac{(1 - \frac{R}{4 r'})^2}{(1 + \frac{R}{4 r'})^2}$
$g_{r'r'} = (1 + \frac{R}{4 r'})^4$
$g_{\theta\theta} = (1 + \frac{R}{4 r'})^4 \cdot r'^2$
$g_{\phi\phi} = (1 + \frac{R}{4 r'})^4 \cdot r'^2 sin(\theta)^2$
deriving ADM Metric components:
$g_{ti} = \beta_i = 0$
$g_{tt} = -\alpha^2 + \beta^2 = -\alpha^2 = -\frac{(1 - \frac{R}{4 r'})^2}{(1 + \frac{R}{4 r'})^2}$
$g_{ij} = \gamma_{ij} = (1 + \frac{R}{4 r'})^4 \cdot diag(1, r'^2, r'^2 sin(\theta)^2)$
so
$\alpha = \frac{(1 - \frac{R}{4 r'})}{(1 + \frac{R}{4 r'})}$
$\beta^i = 0$
$\gamma_{ij} = (1 + \frac{R}{4 r'})^4 \cdot diag(1, r'^2, r'^2 sin(\theta)^2)$
$\gamma = det(\gamma_{ij}) = (1 + \frac{R}{4 r'})^{12} r'^4 sin(\theta)^2$
let $\hat{\gamma}_{ij} = diag(1, r'^2, r'^2 sin(\theta)^2)$
so $\hat{\gamma} = det(\hat{\gamma}_{ij}) = r'^4 sin(\theta)^2$
so $\gamma = (1 + \frac{R}{4 r'})^{12} \hat{\gamma}$
so $\frac{\gamma}{\hat{\gamma}} = (1 + \frac{R}{4 r'})^{12}$
$\phi = \frac{1}{12} ln(\frac{\gamma}{\hat{\gamma}})$
$ = \frac{1}{12} ln( (1 + \frac{R}{4 r'})^{12} )$
$ = ln(1 + \frac{R}{4 r'})$
$exp(-4 \phi) = exp(-4 ln(1 - \frac{R}{4 r'}))$
$= (1 - \frac{R}{4 r'})^{-4}$
$\bar{\gamma}_{ij} = exp(-4 \phi) \gamma_{ij}$
$= (1 - \frac{R}{4 r'})^{-4} \cdot (1 - \frac{R}{4 r'})^4 \cdot diag(1, r'^2, r'^2 sin(\theta)^2)$
$= diag(1, r'^2, r'^2 sin(\theta)^2)$
$\bar{\epsilon}_{ij} = \bar{\gamma}_{ij} - \hat{\gamma}_{ij}$
$= diag(1, r'^2, r'^2 sin(\theta)^2) - diag(1, r'^2, r'^2 sin(\theta)^2)$
$= 0$
$\bar{\epsilon}_{IJ} = {e^i}_I {e^j}_J \bar{\epsilon}_{ij} = 0$